ESTIMATES FOR THE SZEGÓ KERNEL ON UNBOUNDED CONVEX DOMAINS
By
María Soledad Benguria Depassier
A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUJREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
(J'v1ATHEMATICS)
at the
UNIVERSITY OF WISCONSIN - MADISON
2014
Date of final oral examination: April 21, 2014
The dissertation is approved by the following members of the Final Oral Committee:
Professor Alexander J. Nagel, Professor Emeritus, Mathematics
Professor Lindsay E. Stovall, Assistant Professor, l'vlathematics
Professor Andreas Seeger, Professor, Mathematics
Professor Brian T. Street, Assistant Professor, l'vlathematics
Professor Andrei H. Caldararu, Professor, Mathematics Abstract
We study the size of the Szegéí kernel on the boundary of unbounded domains defined by convex polynomials. Given a convex polynomial b : JR.n -+ lR. such that its mixed terms are dominated by its pure terms, we consider the domain ítb = { z E en+! : Im[zn+d > b(Re[z¡J, ... , Re[zn])}.
Given two points (x, y, t) and (x', y', t') in é)ítb, define b( v) = b ( v + x~x') ~ Vb ( x~x}
V~ b ( x~x'); S(x, x') = b(x) + b(x') ~ 2b ( x~x'); and w = (t' ~ t) + 'Vb ( x~x') ·(y'~ y).
\Ve obtain the following estímate for the Szegéí kernel associated to the domain ítb :
1 IS((x,y,t):(x',y',t'))I;S , 2 Jsz + b(y ~ y')2 + w 2 j{ v : b(v) < Jsz + b(y ~ y')2 + w 2 }j where the constant depends on the degrees of the highest order pure terms of b and the dimension of the space, but is independent of the two given points.
This is a generalization of the one-dimensional result by Nagel [23]. 11 Acknowledgements
l would like to dedicate this thesis to my parents, my biggest role models. I would also like to thank my husband for al! his support. I would especially like to thank my advisor, Professor Nagel, foral! his guidance. Finally, I would like to acknowledge the support of the government of Chile and CONICYT in the form of the scholarship "Beca de Doctorado en el Extranjero por Gestión Propia 2008." lll Contents
Abstract i
Acknowledgements ii
1 Introduction 1
1.1 The Szegéí kernel 1
1.1.1 Definitions . 1
1.1.2 A survey of the literature 7
1.2 Convexity . . . . . 12
1.2.1 Loewner-John Ellipsoids 12
1.2.2 Coefficients of convex polynomials of one variable 14
1.2.3 An estimate for sets defined by convex functions . 15
1.3 l\Iain results 18
2 Coefficients of convex polynomials of severa! variables 23
3 Decay of B(r¡) 30
3.1 The dominant term 32
3.2 A polynomial bound for the remaining terms 35
3.3 Decay of the derivatives ... 48
4 A Geometric Bound for the Szego kernel 59
4.1 Construction of the factors fJ.¡ ... , fln . . . 61 iv
4.2 Proof of the Main Theorem .. 68
4.2.1 The bound in terms of .5 71
4.2.2 The bound in terms of b(y- y') 73
4.2.3 The bound in terms of w ..... 78
A An integral expression for the Szego kernel 105
Bibliography 110 1 Chapter 1
Introduction
1.1 The Szego kernel
1.1.1 Definitions
The Szegéí projection was first introduced by Gabor Szegéí in 1921 in the paper Über orthogonale Polynome, die zu einer gegebenen Kurve der komplexen Ebene gehoren [30].
Almost concurrently, Stefan Bergman introduced the Bergman projection as part of his doctoral dissertation. Since then, these projections have been extensively studied.
The Bergman projection associated to a domain O e en is the orthogonal projec-
2 2 tion of L (0) onto the space A2 (0) = {! E L (0) 1 f is holomorphic in 0}. If T is the
Bergman projection and fE L2 (0), then there exists a function B(·, ·) E C 00 (0 x O) such that
Tf(z) =}; B(z,w)f(w)dp, íl where dp is the volume measure and B(z, w) is holomorphic in z and antiholomorphic in w. This function E (· , ·) E C 00 (O x O) is called the Bergman kernel.
There are severa! (equivalen!) ways of defining the Szegéí kernel for bounclecl dornains 2
O ce en (see, e.g., [18]). The purpose of this thesis, however, is to obtain a bound for
1 the Szegií kernel on a class of unbounded domains O e en+ , n ~ 1, defined by convex polynomials. Given a convex polynomial b : IRn -> IR, consider the domain
For these domains, it is convenient to define the Szegií projection as in [13]. We can identify the boundary aob with en X IR so that a point (z, t) E en X IR corresponds to
(z, t + ib(Re[z¡], ... , Re[zn])) E 80&.
Let 0(0&) be the set of holomorphic functions in ob· Given F E 0(0&) and E > o, set
F,(z, t) = F(z, t + ib(Re[z1], ... , Re[zn]) + iE).
The Hardy space 1-¡2(f:?.&) is defined as
Now let p be a defining function for the domain, i.e., Ob = {x E en+! : p(x) We say that Lis tangential ifin addition L(p) =O (notice that writing L = (a1, ... , an+l) the condition L(p) =O can actually be written as< L, op >=O, where < ·, · > is the Hermitian inner product). 3 For a class of convex polynornials b : IRn --+ IR we will define the Szego projection subspace of functions fE L2(éJílb) that are annihilated in the sense of distributions by al! tangential Cauchy-Riernann operators on éJr:lb. To show that such a rnap exists, sorne work is required. vVe will present a brief outline of the basic facts leading up to this definition. For this derivation to hold, it does not suffice to simply require that the polynornials b : IR" --+ IR that define the dornains be convex. In particular, the result does not follow for convex polynornials that are flat along sorne directions (for exarnple, consider 2 the convex polynornial b(x1 , x 2 ) = xi in IR ). vVe introduce a growth condition via the following definition: Definition l. l. Let m 1 , ... , m, be positive integers. We will say that a polynornial p : IRn --+ IR is of "combined degree" (m1 , ... , mn) if it is of the forrn where each indexo= (o1 , ... , on) satisfies 2. -"L" + ... + "' = 1 if and only if there exists sorne j su eh that o = 2m ; 2mt 2mn 1 1 and the exponents of its pure terrns of highest arder are 2m1, •.. , 2mn respective/y. Example 1.2. The polynornial p(x1 , .r2 ) = xi + x 1x 2 + xix~ + xt + x·~ is of "cornbined degree" (2, 3). However, the polynornial j5(x 1, x2 ) = xl + x1x, + xlx~ + xi + x3 is not. 4 Throughout the rest of this work we will assume that where b: IRn--+ IRis a convex polynomial of "combined degree" (m1 , ... , mn). There are two key steps needed to justify that the Szegií kernel as given above is well-defined: that the space 1i2 (Db) is a closed subspace of L 2 (iJDb), and that 1i2 (Db) is equivalent to the space of functions on L2 (EJDb) that are annihilated in the sense of distributions by the tangential Cauchy-Riemann operators. The first follows as in Lemma A.5. in [13]. Lemma. Let F E 1i2 (Db)· Then there exists Fb E L 2 (EJDb) with the following proper ties. a) For almost every (z, t) E en x IR, lim,_.,0+ F(z, t + ib(Re[z¡], ... , Re[zn]) + iE) Fb(z, t + ib(Re[z1], ... , Re[zn])); b) IIFbllu(ao,) = IIFIIw(ob); e) lim,_,o+ IIF, - Fbllu(ao,) = O; d) There is a constant Co independent ofF with IINo[FJIIu(ao,) :S CoiiFIIH'(!l,)• where No[F](z, t) = sup00 IF(z, t + ib(Re[z¡J, ... , Re[zn]) + iE)I; e) The boundary function Fb is annihilated (in the sense of distributions) by all tangen tial Cauchy-Riemann operators on EJ[lb· f) For any compact subset K e [lb there is a positive constant C(K) independent ofF such that 5 sup IF(z)l :s; C(K)IIFIIH2 (n,)· zEK This result do es not require the growth condition imposed on the polynomials b. However, this condition is used in the proposition that follows. Define the partial Fourier transform F: U(JR2n+l) --+ L2(JR2n+!) by the integral F[fl( X,y, t) = !'( X,1J,T ) = J e-2rri(y·,¡+tr)J(x, y, t) dy dt. ll(n+l Then, as in Proposition 2.5 in [13], one can show the following: Proposition. Let fE U(80b)· Then a) the function f is annihilated by the tangential Cauchy-Riemann opemtors on en X lR in the distributional sense if and only if for alll :s; j :s; n the partial Fourier tmnsform F[f] = j satisfies CJ~ (e-2rr[,¡x-b(x)rl](x,1J,T)) =O J on JR2n+ 1 in the sense of distributions; b) if f is annihilated by the tangential Cauchy-Riemann opemtors in the distributional sense, then ](x, 1], T) = O almost everywhere when T.< O. In particular, if we set h,(x, 7], T) = e-Zrrn ](x, 7], T), then h, E L2 (lR 2n+l) for s:::: O; 6 e) if f is annihilated by the tangential Cauchy-Riemann operators in the distributional sense and if 1 F(z, Zn+l) = F(z, t + ib(Re[z¡], ... , Re[zn]) + is) = F- [h,] (x, y, t), then FE H'(D.b) and pb =f. The proof of this proposition is analogous to that in [13]. We have included an appendix where a detailed explanation can be found. lt follows from these proposítions that the set of functions f E L 2 (8D.) such that there exists F E 11_ 2 (0.) with pb = f is the set of functions that are annihilated in the sense of distributions by the tangential Cauchy- Riemann operators. Furthermore, starting from the projection onto the null space of the operators {a~, } on the weighted space L'(IR'n+l, érr!~·x-b(x)ridx dr¡ dT) we are able to retrieve the Szegéí projection. We show in the Appendix that for fE L2 (8íl), the Szegéí projection is given by II[/](x, y, t) = { f(x', y', t')S ((x, y, t); (x', y', t')) dx' dy' dt', }R,2n-+-1 where 00 27fry·[x+x' -i(y'-y)] S((x, y, t); (x', y', t')) = e-2rrr[b(x')+b(xi+i(t'-t)] e dr¡ dT érr!~·v-b(v)r] dv O [Rn J 1 (1 IR" l is the Szegéí kernel. 7 1.1.2 A survey of the literature For very simple domains, such as the unit ball, the Bergman and Szegií kernels can be computed explicitly. In fact, if D C C" is the unit ball, then the Bergman kernel is given by B( ) n! 1 z,<'; = 7rn (1- Z. ¿)n+l and the Szegií kernel is given by (n-1)! 1 S(z, <';) = 27r" (1- z. ¿)n A derivation of these formulas can be found, e.g., in [29]. Even for sorne more complex domains closed formulas have been obtained. Greiner and Stein [12] compute an explicit formula for the Szegéí kernel in domains of the type 2 2 Ok = {(z, z¡) E C : Im[z¡] > jzj k} for any positive integer k. They show that for 2 2 <'; = (z, t + i(lzl k + p)) and w = (w, s + i(lwl k + v)), with J.l, v >O, 2 2 1 [ ( i lzl k + lwl k J.l + S(t;,w) = 7r [s- t] + + ~v) - zw~]-' 4 2 2 2 1-k 2 2 1 [i lzl k + lwl k p + x 47r' 2[s- t] + 2 + -2-vl-, Díaz [8] shows that for these domains, the Szegií Projection is bounded in LP, for 1 < p Francsics and Hanges [10] generalized this result to domains in en. They compute an explicit formula for the Szeg/5 kernel in domains of the type O= {(z,~,w) E rcn+m+l Im[w] > llzll 2 + llt;WP}. They show that 8 n+l -1)'--n-! (A - z. z p S(z, ~, t; z', E,', t') = L ck , , , (11) k=! [(A-z·z')¡; -~·~Jm+k where A= ~[llziJ2 + llz'll2+ II~II 2P + II(IJ2P- i(t- t')]. More recently, Park obtains closed formulas for the Bergman kernel in domains of 2 4 4 the form {(z 1, z2) E IC : lz1 1;¡¡ + lz21'2 < 1} for any positive integers q1 and q2 in 3 4 4 4 [26] and for domains of the form {(z1,z2,z3) E IC : lztl + lz2l + lz3l < 1} and 3 4 4 2 {(z¡,Z2,z3) E IC : lz1 1 + lz21 + lz3 l < 1} in [27]. Furthermore, he shows [26] that 2 2 2 2 among the domains {(z 1 ,z2 ) E IC : lz 11P' + lz2 1P < 1} with p = (p1 ,p2 ) E 1\F, the Bergman kernel is represented in terms of closed forms if and only if p = (p 1 , 1), (1,p2 ) or (2, 2). Since for most domains it is not feasible to obtain closed formulas for the Szego kernel, one of the main questions in the field is whether one can obtain an estimate for these kernels in terms of the geometry of the domains defining them. M uch progress has been done in the case of bounded domains. In fact, if íl is a bounded strongly pseudoconvex domain, complete asymptotic expressions for the Bergman and Szego projections are known from the work of Fefferman [9] (see also [3]). Even though no such results are known if the domain is only weakly pseudoconvex, there has been much work showing that in special cases one can obtain estimates for these kernels in terms of the geometry of the domains. Machedon [19] shows that the magnitude of the Szego kernel for smooth bounded pseudoconvex domains of finite type in en with one degenera te eigenvalue is bounded by the reciproca! of the volume ofthe non-isotropic hall defined by the domain (a discussion 9 on balls defined by non-isotropic distances given in terms of vector fields can be found in [25]). MeNea! [20] obtains sharp upper bounds for the Bergman kernel for smoothly bounded convex domains O of finite type in en in terrns of the volumes of polydiscs fitting inside DO. i\lore precisely, if O is convex in sorne neighborhood U o[ a point p E DO of finite type lvl then there exists a defining function r for Un O such that the sets {r : r( z) < ¡¡} are convex on ¡¡ for sorne range -¡¡0 < ¡¡ < r¡0 , ¡¡0 > O. For q E O near p, he defines a set of coordina tes ( z1, ... , Zn) and distances r 1 ( q, e), ... , T n ( q, f) by extremizing the distance frorn q to the leve! set {z E U : r(z) = r(q) +e} and defines the polydisc P,(q) = {z E U: lz,¡ < r,(q,c), ... ,lznl < Tn(q,c)}. lv!c'\eal shows that there exists a constan! e so that for al! q¡, q, E U n 0 IK(q,, q,)l :S: IP,~¡)I' where K(q1 , q2 ) is the Bergrnan kernel and 5 = lr(q1 )1 + lr(q2 )1 +inf{ f >O : q2 E P,(q1 )}. By integrating this result along "normal directions", MeNea! and Stein [21] obtain a bound for the Szego kernel S(z, ·w) for these dornains in terms of the smallest "tent" in DO containing z and w. That is, they show that for smoothly bounded convex domains of finite type in en, there exists a constant e so that for all z, lL' E o X o\ {diagonal in DO}. e IS(z,w)l :S: IT(z,¡)l' Here T(z,¡) = R 1(7r(z)) nO; the projection": U --t DO is a smooth map such that if bE DO, r.(b) = b and r.-1(b) is a smooth curve, transversa!ly intersecting iXl at b; and "f = lr(z)l + lr(w)l + inf{c >O : w E T(z,c)}. Similar estimates are obtained in both 10 papers for the derivatives of these kernels. Although no general results have been obtained for the kernels defined over un bounded convex domains, some particular classes of domains have been studied. In the last section of [23], Nagel studies the Szegéí kernel on the boundary of domains of the 2 kind Sl = {z E IC : Im[z2] > 4>(Re[z1])}, where 4> is a subharmonic, non-harmonic polynomial with the property that b..P(z) = b. 1 that in this case the Szegéí kernel is bounded by IBI- , where IBI is the volume of the non-isotropic ball. That is, for ((x, y, t), (r, s, u)) E 80 x 80, IS((x, y, t); (r, s, u)) 1:S CIB((x, y, t); JW', where 8 is the non-isotropic distance between (x, y, t) and (r, s, u). Nagel, Rosay, Stein 2 and Wainger [24] generalize this result to domains of the form Sl = {(z1 , z2 ) E IC : Im[z2] > P(z¡)}, where Pis a subharmonic, non-harmonic polynomial in IC. Halfpap, Nagel and Wainger [13] study the singular behavior of the Bergman and 2 Szegéí kernel in domains ofthis kind, i.e., domains ofthe form Sl = {(z1 , z2 ) E IC 1 Im[z2] > b(Re[z¡J)}, but where bE C 00 belongs toa particular class of convex functions (a model 2 example of which is b(r) = exp( -lrl-") for lrl small, and b(r) = r m for lrllarge, with a, m > 0). They show that if!::. C 80 x 80 is the diagonal of the boundary, then if O < a < 1 the Bergman and Szegéí kernels extend smoothly to Sl x Sl \ !::., while if a :0:: 1, the kernels are singular at points on Sl x Sl \ !::.. The Szegéí kernel has also been studied in some non-pseudoconvex domains of this 2 type, i.e., domains ofthe kind Sl = {z E IC : Im[z2] > b(Re[z1])}, but where the function 11 b is not convex. In [5], Carracino studies a domain given by a particular choice of a non- convex b, and shows that there are singularities not only on the diagonal ( which is true in the pseudoconvex case), but also off the diagonal. In [11], Gilliam and Halfpap consider domains where b is a non-convex quartic polynomial with positive leading coefficient, and show that there are points off the diagonal of iJO x iJO at which the Szegéí kernel is infinite, as well as points on the diagonal at which it is finite. 2 Haslinger studies the Szegéí kernel in domains of the kind O= {z E e : Im[z2] > p(z¡)}, for functions p: e--+ IR+. In [14] he finds an integral expression for the Szegéí kernel in terms of the Bergman kernel and in [15] he computes an asymptotic expansion for the Szegéí kernel when p(z¡) = IRe[z¡J 1", where a > l He then uses this result to study the singularities on the diagonal of iJO x iJO. By definition the Szegéí projectíon ís bounded in L 2 lt ís a relevant question in the field, however, to study the LP boundedness of this operator for p fe 2. Estimates for the Szegéí kernel in terms of the volumes of non-isotropic balls can be used to obtain such bounds. Bonami and Lohoué [2] show that the Szegéí projection defined over domains of 2 2 thekindDa = {z E en: lz¡l;;¡+ ... +lznl"" < 1}where0 < "'J < 1foralljiswcaktype (1, 1) and strong type (p,p) for 1 < p 3-dimensional C 00 Cauchy-Riemann manifold of finite type and the associated first- ordcr differential operator lib has closed range, then the Szegéí projection extends to an operator bounded in LP(M) for 1 < p < oo. Mitchell [22] proves LP estimates for the Szegéí projection for a class of bounded symmetric domains; and l\Jcl\" eal and Stein [21] 12 generalize this result to smoothly bounded convex domains of finite type, showing that the Szegií projection maps L~(éKl) to L~(aO) for 1 < p < oo and sE!'\!. 1.2 Convexity There is much that can be said about convexity, but we will limit our discussion to a few classical results that we use repeatedly throughout our work. The proof of our main theorem relies heavily on the fact that given a compact convex set, one can find a maximal ellipsoid ( called a John ellipsoid) that can be inscribed in said body. We will begin this section by giving a brief summary of the history of this result. We will then go on to describe two lemmas by Bruna, N agel and \Vainger on their paper Convex Hypersurfaces and Fourier Transforms [4] which describe the size of convex polynomials and their derivatives relative to the absolute value of their coefficients. In the course of this thesis we will repeatedly use these two results together, since applying John's results to these two lemmas yields absolute bounds for the coefficients of compact convex polynomials of the form we study. On section 2 we generalize the result by Bruna, Nagel and \Vainger to severa! variables. \Ve will finish the discussion of convexity by deriving an estimate for sets defined by convex functions that will be relevant to our study. 1.2.1 Loewner-John Ellipsoids Since antiquity mathematicians have estimated the size of regions in terms of simple geometric shapes of similar size. In particular, on the mid 20'h century, the ellipsoids (i.e., the images under invertible linear transformations of the unit ball) that can be inscribed and circumscribed in compact convex bodies were extensively studied. \Ve 13 will restrict our discussion to the question of uniqueness of the inscribed ellipsoid of maximal volume (called a John ellipsoid) and the circumscribed ellipsoid of minimal volume ( called a Loewner ellipsoid), as well as the ratio between their volumes. For a more complete exposition of the development and applications of this problem see, e.g., [16]. In his paper Über die kleinste umbeschriebene und die grofJte einbesehriebene Ellipse cines konvexen Bereichs [1], Behrend shows that given a convex body in IR2 there exists a unique maximal inscribed ellipse and a unique minimal circumscribed ellipse. This result was generalized ton dimensions by Danzer, Laugwitz and Lenz [7] as well as by Zaguskin [31]. In 1948, in his paper Extremum problems with inequalities as subsidiary conditions [17], John used an applícation of Lagrange multipliers (or rather, an extension of La- grange multipliers where the constraints are given by inequalities) to study the volume of these ellipsoids. In particular, he showed that the ratio between these ellipsoids is independent of the given convex body, and depends only on the dimension of the space. :-!ore precisely, Jet K be a compact convex body in IRn, En be the unit ball and Q; be the minimal circumscribed ellipsoid to K. :\otice that there exists a t E IRn ancl an in vertible linear transformation T E IR'"n such that Q; = t +TEn. By John, it follows that 1 . t+-TEnc;:;K <;:;t+TEn. n Furthermorc, if K has a center of symrnetry (i.e., there exists a e E IRn such that 14 K = e - K = {e - y : y E K}), then the ratio can be improved to },¡. That is to say, for a symmetric compact convex body K, the containment can be sharpened to 1.2.2 Coefficients of convex polynomials of one variable Bruna, Nagel and \Vainger obtained estimates on [4] for convex polynomials of one variable in terms of the absolute values of their coefficients. \Ve generalize these results to polynomials of severa! variables in Section 2. The one-dimensional results (which can be found on Section 2, page 338, of the aforementioned paper) are as follows: Let C(m, T) denote the space of polynomials m P(t) = L, a1 ti J~O which satisfy: l. The degree of P is no bigger than m; 2. P(O) = ao =O; P'(O) = a1 =O; 3. P is convex for O ::; t ::; T. Lemma 2.1. There zs a constan! Cm, independent of T, so that if P E C(m, T), 15 m P(t) ::0 Cm L iaJiti ( 1.2) j=2 forO :S: t :S: T. In particular, since a0 = a 1 = O, m P(t) ::0 Cmtm¿ iaJI if o:s;t:s;1, and j=2 m P(t) ::0 Cmt2 L la1i if 1 :S: t :S: T. j=2 Remark 1.3. Notice that since Cm L.';2 ia1 iti :S: P(t) :S: L.';2 ia1 iti it follows that Cm :S: l. Lemma 2.2. There is a constant Cm, independent of T, so that if P E C(m, T), m P'(t) ::0 Cm L iaJitj-! j=2 for O :S: t :S: T. In particular, m P'(t) 2: Cm¡m-l L la1 1 if o:s;t:s;1, and j=2 m P'(t) 2: Cmt L iaJI if 1 :S: t :S: T. j=2 1.2.3 An estímate for sets defined by convex functions The thircl arre! last topic we woulcl like to cover in this overview of convexity is the following estimate, which plays a crucial role in the proof of one of our lemmas. 16 Proposition 1.4. Jf f : IRn --+IR is a convex function such that f(O) = O and V' /(0) = O then 1 = e-f(w)dw"" l{w f(w)::; 1}1. }J>.nr In the proof of the above estimate, we will use the following inequality: Claim 1.5. Let f : IRn --+ IR be a convex function such that f(O) = O. Given x > O, let Ax = {w E IRn: f(w)::; x}. Then for any constan! O::; ,\::; 1, vol(A.\x) 2 ,\nvol(Ax). Pmof By convexity of /, for any vectors w and u in IRn, and any constant O::;,\::; 1, f(,\w + (1- .\)u)::; .\f(w) + (1- .\)f(u). In particular, taking u= O, and since by hypothesis f(O) =O, it follows that f(.\w)::; .\f(w). Thus, if w E Ax, then f(.\w)::; ,\f(w)::; .\x. That is, ,\ · Ax s;;: A.\x· It follows that D The proof of Proposition 1.4 is as follows: 17 Proof Without loss of generality we can assume f =jo O. Notice that under these hypothesis f(v)?: O Vv E IRn. In fact, by the Fundamental Theorem of Calculus we can write f(u+v)- f(u) =- fo'vJ(u+vt) · v (:t(1-t)) dt. And integrating by parts, we have that lo 1 \lf(u+tv)·vdt= -\lf(u+tv)·v(1-t) 1' 0 f' d +lo dt[\lf(u+tv)·v](1-t)dt. But d n - [\lf(u + tv) · v] = L ¡,1(u + tv)v,v1. dt i.J~l It follows that f(u + v)- f(u) = \7 f(u) · v + l t f,Jv,v,(1- t) dt. (1.3) o i.j=l Letting u = O we have by convexity that (1.4) A lower bound for I can be easily obtained, since r e-f(w) dw?: r e-f(w) dw?: ~ l{w J(w) :S 1}[. }JI.n l{w,f(w)<;l} e 18 To obtain an upper bound we can write But But by Claim 1.5, and taking A= and x = j 1, it follows that 1! 1 + l{w: f(w) :S j + 1}1 :S (j + 1)nl{w f(w) :S 1}1. Hence, by equation (1.5), we have that I :S l{w: f(w) :S 1}1 (1 + ~e-i(j + 1)n). Since the sum converges we get the desired upper bound. D 1.3 Main results The purpose of this thesis is to study the size of the Szegéí kernel on the boundary of certain convex domains in cn+l vVe consider domains of the kind for convex polynomial functions b : !Rn -+ lR of "combined degree" (refer to definition on page 3). vVe generalize Nagel's one-dimensional size estimate for the Szegéí kernel 19 established in [23] to severa! variables. Throughout this paper we willlet z1 = x1 + iy1, and we will use boldfont;; to denote vectors x = (x 1 , ... ,xn) in lll:.". It can be shown, as in [23], that the Szegéí kernel for the domains under consideration can be written as an integral formula. Proposition A, l. The Szego kernel on ílb is given by 2rrr¡· [x+x' -i(y' -y)] S((x,y,t);(x',y',t'))= e~2n[b(x')+b(x)+i(t'~t)] e dr¡ dT, oe é~l~·v~b(v)r] dv O (1Rn J l l Jln ( 1.6) where (x, y, t) and (x', y', t') are points in iJíl6 . \Ve have included an Appendix with the proof of this proposition. Our estimates will all follow from a study of this integral expression for the Szegéí kernel. Perhaps one of the most striking differences between the one-dimensional case studied m [23] and the n-dimensional case 1 study io the fact that in the former it is enough to assume that b is a convex polynomial, whereas in the latter that assumpt.ion is not. enough. Convexit.y alone willnot. ensure that f!Rn e't·v~rb(v) dv converges in lll:.n for n ;::: 2 (in fact, for n = 2 consider the polynomial b(x1 , x2) = xi). The "combined degree" condition ensures that the class of convex polynomials under consideration satisfy that the set R = {v E lll:.n : b( v) ::; 1} is compact. This guarantees the convergence of the denorninator integral. \Ve devote the second section to a study of the coefficients of convex polynornials in severa! variables. In the one-variable case it was shown by Bruna, Nagel and \Vainger 20 (as we explained in detail in the previous subsection) that the absolute value of the coefficients of a convex polynomial with no constant or linear terms can be bounded by the value of the polynomial at l. lt is not possible to obtain such a bound in more variables, since the polynomial might be growing in sorne directions but not along others. However, we show that the absolute value of the coefficients can be bounded by the average of the polynomial over a circle of arbitrary positive radius. \Ne obtain the following result: Proposition 2.1. Let f(M) = {(a1, ... , an) E Nn : 2 ::; lnl ::; M}. Let S(M) be the set of convex polynomials g( v) = I:aEr(M) Ca va. Then for any fixed a > O, there exists a positive constant C(M, a) that depends only on M and the constant a such that if g E S(M), L leal ::; C(M, a)¡ g(u) du. aEr(M) lal~a Remark 1.6. N atice that with f(M) defined as above, polynomials of the form g(v) = I:aEr(M) CaVa are such that the degree of g is no bigger than M; g(O) = O; and \7 g(O) = O. In Sections 3 and 4 we present the proof of our main result. That is, we derive an estímate for the size of the Szego kernel in terms of the geometry of the domain that defines it. Our goal is to o btain a geometric bound for the Szego kernel from a st udy of the integral expression in equation (1.6). One of the main technical difficulties comes from handling the denominator integral. After sorne manipulation, this denominator integral can be expressed as a function that we will denote as e( 1J). In the following lemma we show that e( 1J) is a Schwartz function. 21 Lemma 3.1. Let g : JR.n --+ lR. be a convex polynomial such that i) g(O) =O; ii) 'Vg(O) =O; iii) there exists a constant O< A< 1 such that {v lvl ~A} e {v g(v) < 1} e {v : lvl ~ 1}; and iv) there exist positive integers m 1 , .•• ,mn such that the "combined degree" of gis (m¡, ... , mn)· Then 1 O(ry) = [¡ e' v~g(vl dvr' is a Schwarlz function. Moreover, its decay depends only on the constant A and the exponents {nl¡ 1 ••• : mn}· 'We devote an en tire section to the study of the decay of O( 7J), rather than incorporat- ing it in the proof of the main theorem, which is discussed in Section 4. The estimates for convex polynomials of severa! variables obtained in Section 2 will play a crucial role in the proof of this lemma. The bound obtained for the coefficients of the polynomials combined with the compactness requirement (given by hypothesis (iii)) ensure that the decay of O(ry) is independent of the coefficients of g. Our main res u! t is the following: Main Theorem. Let b : JR.n --+ lR. be a convex polynomial of "combined degree" (m 1 , ... , mn)· Define ll& = {z E en+!: Im[zn+l] > b(Re[z1], ... , Re[zn])} and let (x, y, t) and ( x', y', t') be any two points in 8!1&. Define 22 -b(v)=b ( v+ x + x') -Vb (x + x') ·v-b (x + x') ; 2 2 2 S(x,x') = b(x) + b(x')- 2b x + x') ; ( 2 and w = (t'- t) +\lb ( x ~ x') ·(y'- y). We obtain the following estímate for the Szeg6 kernel associated to the doma in Ob : 1 IS((x,y,t);(x',y',t'))l ;S 12· 2 2 2 2 2 2 Js + b(y- y') + w 1{ v : b(v) < Js + b(y- y') + w } Here the consta ni depends on the exponents { m 1 , ... , mn} and the dimension of the space, but is independent of the two given points. Remar k l. 7. N atice that since b is convex, S(x, x') ~ O. It is convenient to think of c5 as a measure of the curvature between x and x'. The more curved the domain is, the larger the value of S. The proof of the main theorem is discussed in Section 4. The proof is, at its core, an application of John's ellipsoids. vVe introduce a change of variables in the integral expression for the Szegií kernel comprised of factors defined by the length of the axes of the unique maximal inscribed ellipsoid associated to a symmetrization of the convex body R = { v : b( V) < n. 23 Chapter 2 Coeffi.cients of convex polynomials of severa! variables In this section we obtain bounds for the absolute value of the coefficients of convex polynomials of severa! variables. \Ve begin by showing that for polynomials whose constant and linear terms are zero, the sum of the absolute value of the coefficicnts can be bounded ( up to a universal constant) by the average of the polynomial on a sphere of arbitrary positive radius. \Ve then consider convex polynomials 9 : IRn --+ IR with no constant or linear terms such that there exist sorne positive constants A, B so that the set {v : 9( v) :S: 1} contains the ball of radius A and is contained in the ball of radius B. For these, we show that the sum of the absolutc value of the coefficients is bounded by a universal constant. l\Ioreover, we show that the absolute val u e of the coefficients is bounded (up toa universal constant) by the value of 9 at any point on the boundary of the ball of radius A. These results are necded for the proof of the geometric bound for the Szegií kernel on the domains we study. In particular, we will use these results in Section 3 to show that the decay of the function ()(r¡) (which arises naturally from a study of the denominator integral of the expression for the Szegií kernel obtained in Proposition A.l) does not depend on the coefficients of the polynomial which defines it. 24 Proposition 2.1. Let r(M) = {(a1, ... ,an) E Nn : 2:::; lal:::; M}. Let S( M) be the set of convex polynomials g( v) = LoEr(M) c0 v". Then for any fixed a > O, there exists a positive constant C(M, a) that depends only on M and the constant a such that if g E S( M), (2.1) Remar k 2.2. This is a generalization of the result in one variable by Bruna, Nagel and Wainger in [4} (Lemma 2.1}. Proof. Let a > O be a fixed positive constant and let lf(M)I denote the cardinality of the set of índices f(M). We identify the space S(M) with JRir(M)I vía the identification 0 g(v) = L C0 V E S(M) +-+ (c1 , ... ,c¡qM)I) E JRI'(M)I, aEr(M) where we have ordered al! the a E r( NI) so that e1 eorresponds to the coefficient e0 for the j'h element a with this ordering. Let I:M = {g(v) = L eav" E S(M) L leal= 1}. (2.2) oEr(M) oEr(M) We claim that I:M is a compact subset of S(lvf). In fact, let { c,..}nEN in JRI'(M)I be a sequence of tuples associated to a sequence of polynomials {qn}nEN in I:M. Since {cn}nEN is a sequence contained in the compact set BM = {(e1 , ... ,e¡qM)I) E JRI'(m)l : L!SJSir(M)I le; 1 = 1}, it has a convergent subsequence { cnJn,EN . Let e be the limit of 25 this subsequence, and let q be the polynomial associated to this tuple. \Ne claim that q is an element of LAf. In fact, the identification preserves the degree of the polynomial and the fact that there are no constant or linear terms. Also, since e is an elernent of BM, it satisfies that LIS:J<'If(M)IIc1 1 =l. Thus, is suffices to show that q is convex. This follows easily, since given any polynomial qn, associated to an element of the convergent subsequence {cnJn,El\1, we have that qn,(ax + (1- a)y) ~ aqn,(x) + (1- a)qn,(Y) for all O ~ a ~ 1 and for all points x, y in !Rn. Thus, and sine e qn, ( ax + (1 - a )y) -+ q(ax + (1- a)y); aqn1 (x)-+ aq(x); and (1- a)qn,(Y)-+ (1- a)q(y), the convexity of q follows immediately. Let ¡¡(g) = L leal· aH( M) Notice that these functions are continuous on S(Af), and that \Ve claim that Moreover, on LAJ at least one of the coefficients of g must be different from zero, so g can not be the zero polynomial. Thus g must be positive almost everywherc. In particular, the average over the circle of radius a must be strictly positive. Therefore, and sin ce ¡ (g) is continuous as a function of g, it attains a mínimum in LAJ, and this mínimum is strictly positive. Thus, and since n(g) = 1 on LAI, there exists a constant e> o such that for any g E LM, 26 r(g);::: C = Cn(g). That is, ___!_____() r g(a) da:::: Cn(g) =e L leal, Wn a liai=a aEr(M) as desired. Consider now a polynomial g(v) = LaEriM) eav" E 5(1\!I), but which is not neces sarily in Í::M. Then Jet h(v) = LaEr(M) bav"' where b - Co " - LBEr(M) lea 1 so that LaEr(M) lbal = 1 and h(v) E Í::M. It follows from the previous case that ___!_____() r h (a) da :::: c. Wn a JIO'I=a That is, _1_ { g(a) da> C. wn(a) llal=a LaEr lepl - This gives the desired inequality. D Corollary 2.3. Let g(v) = Laeav" be a eonvex polynomial sueh that g(O) =O and V' g(O) = O. Suppose there exist two positive eonstants A and B such that {v: lvl :S A}<;;; {v: g(v) :S 1} <;;; {v: lvl :S B}. (2.3) Then there exists a constant that depends only on A and the degree of g such that 27 (2.4) Moreover, for any point x = (x 1 , ... , xn) on the boundary of the circle of mdius A, there exist constants e, > O, e2 > O that depend only on A, B and the degree of g such that (2.5) a Remark 2.4. The bound given by equation (2.4) can be obtained using just the left containment, i.e, the existence of a constant A> O such that {v : lvl :S A} <;;: {v : g(v) :S 1}. The second bound, howevel', requires the existence of both an inner andan outer· ball. Proof. Thc first result follows immediately from the previous claim. In fact, we showed that ~leal :0:: e jui~A g(a) da. But by (2.3) we have that g(a) :S 1 foral! a such that lal =A. The result follows. Observe that the bound g( x) 2 e2 L..: a lea 1 will be an immediate consequence of the above bound on the coefficients once we show that g(x) 2 e,. The proof of equation (2.5) requires the use of Lemma 2.1 on the paper Convex hypersurfaces and Fourier Tmnsjol'ms by Bruna, Nagel and vVainger [4]. The lemma states that given a convex polynomial of one variable of degree lvf of the form M p(t) = L a1 ti j=2 28 there exists a constant CM > O that depends only on J'vf such that ~I ~~ CM L la1ltl :S: p(t) :S: L la1lt1 Vt 2': O. (2.6) j=2 In particular, this result implies that for any ,\ > 1 and t 2': O, (2.7) Given a point x on the boundary of the circle centered at the origin of radius A, we will let p(t) = g(tx). K otice that this defines a convex polynomial of one variable for which the bounds in equation (2.6) apply. Taking t = 1 and ,\ = ~ (where A and B are the radius of the inner and outer ball respectively) in equation (2.7), we have that CM p(1) 2': ),M p (B)A . That is, g(x) 2': CM BMAM g (Bx)A . (2.8) Since x is a point on the boundary of the inner circle, the point ~x is on the border of the outer circle (the circle of radius B). Thus, by hypothesis, it follows that 29 This together with equation (2.8) implies the desired result. D Remar k 2.5. N atice that th.e convexity of g implies that g(x):::: C¡ ::::e, L leal· Q for any point x such that lxl ::C: A. 30 Chapter 3 Decay of B(TJ) In this section we study the decay ofthe function e(r¡) = [ j e"·v-g(v) dv] -l, where gis Rn a convex polynomial. This function arises naturally from the study of the denominator integral of the integral expression obtained for the Szegéí kernel in Appendix A. We show that e(r¡) is a Schwartz function. This implies, in particular, that j¡¡¡_n e(r¡) dr¡ converges. We use this fact in the proof of the geometric bound for the Szegéí kernel in Section 4. Lemma 3.1. Let (3.1) be a convex polynomial in lll!.n such that i) g(O) = O; ii) V g(O) = O; iii) there exists a constant O < A < 1 su eh that { v lv 1s; A} e;; {v g( v) s; 1} e;; {v: lvl s; 1}; and iv) there exist positive integers m 1 , ... , mn such that the "combined degree" of g is (m1 , ... ,mn) (refer to definition on page 3). 31 Then is a Schwarlz function. Moreover, its decay depends only on the constan! A and the exponents {m1, ... , mn}· Remark 3.2. Notice that under these assumptions, the coefficients of the polynomial g( V) = I:aEr Ca v" satis/y I:aer 1 Ca 1 ::; e, where e depends only on the constan! A, on the degree of the polynomial and on the dimension of the space. This was shown in eorollary 2. 3 on page 26. Let I = f2.n ery·v-g(v) dv. \Ve will show that I grows at an exponential rate. \Ve can \\Tite ¡ = eh(vo) { eh(v)-h(vo) dv, }?_n (3.2) where h( v) = r¡ · v - g( v) and v 0 is the point wherc h( v) attains its maximum (notice that r¡ = \lg(v0 )). \Ve will show that the dominant term, e"(vo), grows at an exponen tia! rate in r¡. This tcrm will provide the clesired decay for ¡-I \Ve will then show that J e"(v)-h(vo) dv does not decrease too fast, that is, that it does not annul the growth of the clominant term. Notice that h(v0 ) = L(r¡) = supv {r¡ · v- g(v)} is the Legendre Transform of g. 32 3.1 The dominant term We begin by studying the growth of the term eh(vo), where h( v) = r¡ · v - g( v) and v0 is the point where h( v) attains its maximum. vVe show that eh(vo) grows exponentially as a function of r¡. Moreover, we claim that the growth is independent of the choice of g, but rather depends only on the constant A, on the "combined degree" of g and on the dimension of the space. More precisely, we show that there exist positive constants e, e which depend only on the "combined degree" of g, the dimension of the space and the constant A, such that (3.3) We begin by showing that the polynomial g is dominated, independently of its coef- ficients, by its pure terms of highest order. Claim 3.3. lf g( v) is as in the statement of Lemma 3.1, then there exists a constan! e > O that depends only on the constan! A, on the "combined degree" of the polynomial and on the dimension of the space such that Proof. Let Notice that for any vE !P?.n, 33 (3.4) In fact, if any of the components of v are zero, the statement is trivial. Otherwise, dividing equation (3.4) by vr' ... v~n' it suffices to show that -".L r(v)'m' 1 < ~-;;-:-- - vf' That is, it suffices to show that This last inequality is trivial, since each factor in the right hand size is larger than l. Furthermore, since g is a convex function such that g(O) = V' g(O) = O, we have that g 2: O. This was discussed on page 17, equation (1.4). Thus, g(v) = lg(v)l = L cavr' · · · v~" a:::r < L lcallvr' · · · v~" 1 (3.5) aEr < L leal 1 r( V).,';;;¡-+ +~ 1 aEr 1\loreovcr, recall that sincc gis of "combincd degree" (m1 , ... , mn), any indexa E r satisfies that -+0: l ... +-<1.O:n 2m1 2mn- 34 Hence, (3.6) Thus, and since ¿ leal ::; e, it follows from equations (3.5) and (3.6) that aEr g(v)::; L lcnl(l + r(v))::; e(l + r(v)). This finishes the proof of Claim 3.3 D Since this estimate does not depend on the coefficients of g, it is now easy to obtain a lower bound for h( v 0 ) in terms of 17 which does not depend on the choice of g. Claim 3.4. The Legendre Transform of g( v) where v E !Rn is large for large values of 1171· More precise/y, where e and C are positive constants that depend only on the constant A, on the "com- bined degree" of g and on the dimension of the space. Proof It follows from the previous claim that L(r¡) =sup {r¡· v- g(v)} V 2 2 ~ sup {r¡· V- e- elv,l m¡- ... - elvnl mn} V 2 1 2 =-e+ sup {1)¡ V¡ - elv,l m } + ... + sup { 'lnVn - elvnl mn} U¡ ~ 35 - 2k But given w E !Re, the Legendre Transform of ~ lwl'k is Bl7)1"-1, where - -1 1 (2k-1) B = B"- 2k . Thus, 1 1 1 1 1 where C = min { B 1'm~ - (>';~~ ) , ... , B~m-n - (>';,;;~')} , and B1 = C2rn1 D This finishes the proof that the dorninant term, eh(vo), grows at an exponen tia! rate in r¡, independently of the coefficients of g. 1\Iore precisely, we have shown that (3.7) 3.2 A polynomial bound for the remaining terms It suffices now to show that J = fRn eh(v)-h(vo) dv is not too small to obtain the desired decay for ¡-I Recall that J = r e'¡-(v-vo)+g(vo)-g(v) dv. fgn In order to estirnate this integral, we will approximate it by an area by using Claim 1.5. Recall that if f : !Ren --) !Re is a convex function such that f(O) = O and \7 f(O) = O then 36 e-f(w) dw""' l{w : J(w) ~ 1}1. }~nr This was discussed in the introduction (page 16). 'vVe must start by rewriting J as stated in the above result. Since r¡ = \7g(v0 ), and making the change of variables w = v- v 0 , we can write J = r e-f(w) dw, }~n where f(w) = -Vg(vo) · w- g(vo) + g(vo + w). (3.8) Clearly f(O) = O and V j(O) = O. Also, since g is convex, so is f. Thus, it follows by Claim 1.5 that J""' l{w: J(w) ~ 1}1. (3.9) That is, I = e'I·V-g(v) dv""' eh(vo)l{w: J(w) ~ 1}1. }n<.nr (3.10) Our goal is to show that as 1711 grows, this area decreases slower than the rate of growth we obtained for eh(vo) We begin by obtaining an upper bound for f that is independent of the choice of g, but rather that depends only on its "combined degree", on the dimension of the space and on the constant A. To do so we will write f as an integral in terms of the quadratic form associated to the Hessian of g. In Claim 3.5 37 wc obtain an upper bound for this quadratic form in terms of a polynomial that is independent of the coefficients of 9· In Claim 3.6 we use this estímate to obtain the desired bound for f. Claim 3.5. The quadmtic form associated to the H essian of 9 is bounded in terms of a polynomial that depends on the "combined degree" of 9, but which is otherwise independent of the choice of 9· More precise/y, n 2 L 9;1 (v)u.·,w1 ;S (1 + r(v)) lwl , i.j=l 2 2 &2 where r(v) = v m' + ... + vnm, and 9iJ(v) =--'!:..!La. (v). 1 V~ 8 'LJ Proof. Let L be the Hessian matrix of 9· That is, 911 912 9ln 921 922 L= 92n so that w T L w = ¿~J~l 9;1w;u.·1 . Sin ce L is symmetric, it has n linear! y inclepenclent eigenvectors. Let u;, i = 1, ... , n be the eigenvectors of L, ancl A;, i = 1, ... , n be the corresponding eigenvalues. Since 9 is convex, the matrix L is positive clefinite, so its eigenvalues are positive. Let P = Tr(L)I, where Tr(L) = ).. 1 + ... + An is the trace of the matrix L and I is the identity matrix. 38 Let Q = P - L. We claim that Q is positive definite, and hence that L < P as quadratic forms. In fact, notice that for i = 1, ... , n Qui =Fui- Lui = Tr(L)ui- >.,u,= (Tr(L)- >.,) Ui = L .\Jui. l:SJSn j-:f;i Thus, for i = 1, ... , n, ui is an eigenvector of Q, with eigenvalue /Li = L.: Áj. l:"S;jSn j,fi l\iotice that for i = 1, ... , n, /Li is positive. Also, since P is a diagonal matrix and L is symmetric, Q is symmetric. Thus, since Q is a symmetric matrix whose eigenvalues are al! positive, Q is positive definite. Hence, since 9ll + · · · + 9nn o o o 911 + · · · + 9nn o P = Tr(L)I = o o 911 + · · · + 9nn andwTLw < wTPw,itfollowsthat n O S: L.: 9i; ( v )w,w1 i,j=l S: (911(v) + · · · + 9nn(v))lwl 2 S: (l911(v)l + · · · + l9nn(v)l) lwl 2 39 Notice that each 9JJ(v) is a polynomial of the form L,c~vi3 where the indexes (3 B satisfy (3¡ f3n 1 -+ .. ·+--<1--<1. 2m, 2mn - m 1 In particular, this implies that -'!L+··+_2n_ ( ) r ( V ) 2m t 2mn ::; 1 + r V . P,Ioreover, recall that, as was shown on equation (3.4) on page 33, lt follows that :S L lepl (1 + r(v)). i3 Furthermore, the coefficients c13 are constant multiples of the ca, where the factors are bounded by a factorial of the degree of the polynomial. Since L. aa lea 1 :S C, the coefficients c13 are bounded by a constant that do es not depend on the choice of g, but rather on the "combined degree" of g. Then, and by the previous inequality, we have that l9n(v)l + · ·. + l9nn(v)l :S C(1 + r(v)). 40 for a universal constant C. That is, n 2 L 9iJ(v)w,w1 ;S (1 + r(v))[w[ . i,j=l D Using this result it is now possible to obtain an upper bound for f which is indepen- dent of the choice of g. We do so in the following claim. Claim 3.6. If and f(w) = -'Vg(vo) · w- g(vo) + g(vo + w) then, 2 j(w) ;S lwl (1 + r(v0 ) + r(w)), where the constan! depends only on the constan! A, the "combined degree" of g and the dimension of the space. Proof. We begin by rewriting f as an integral in terms of the quadratic form associated to the Hessian, so that we can apply our previous estimate. 41 vVe can write {! n g(vo + w)- g(vo) = \7g(vo) · w +Jo i~! g;1(vo + tw)w;w1 (1- t) dt. This was shown in detail on page 17 (equation (1.3)). lt follows that f(w) = [ t 9iJ(Vo + tw)wiu·1(1- t)dt. ( 3.11) 0 z,j=l In particular, by convexity of g we have that f 2: O. vVe can now use the bound for the Hessian obtained in Claim 3.5. lt follows that f(w) ;S [ (1 + r(vo + tw))lwl 2(1- t) dt. (3.12) vVe would like to get rid of the dependence on t of the term r( v 0 + tw ). To do so, we use a triangle inequality. vVe claim that given vectors u and v, the function T satisfies r(u + v) S C[r(u) + r(v)], where the constante depends OII the exponents {m¡, ... ' rnn}- In fact, by convexity it follows that for j = 1, ... , n Thus, 42 Applying this inequality to r(v0 + tw), and since O :S: t :S: 1, we have that r(v0 + tw) ;S r(v0 ) + r(tw) :S: r(v0 ) + r(w). Hence, it follows from equation (3.12) that 2 f(w) ;S lwl (1 + r(v0 ) + r(w)). D 1 Recall that we are studying the decay of J- , where I :::o é(vo) 1{ w : f( w) :S: 1} 1· Thus far we have shown that the dominant term, eh(vo), grows at an exponential rate in r¡ (where by definition r¡ = \i'g(vo)). Our goal is to show that 1{ w : f(w) :S: 1}1 is not too small. More precisely, that it does not decay exponentially, so that it does not annul the growth of the dominant term. In the next three claims we show that there is a polynomial P and a constant C depending only on the degrees { rn 1 , ... , rnn} so that l{w: f(w) :S: 1W1 :S: C(l + P(lr¡l))~. In Claim 3.7 we show that l{w : f(w) :S: 1}1- 1 is bounded by a polynomial in terms of r(v0 ), and in Claim 3.8 we compare the sizes of lvol and 1'71· In Claim 3.9 we conclude that r(v0 ) grows at most ata polynomial rate in 1'71· 2 Claim 3.7. If f(w) ;S lwl (1 + r(v0 ) + r(w)), then l{w: f(w) :S: 1}1 :2: (1 + r(v0 Wli. 43 Proof. Let C be such that f( w) S: Clwl'(l + r( v 0 ) + r( w)) and Jet 2 T(w) = Clwl (1 + r(v0 ) + r(w)). Then, l{w: f(w) 'S 1}1 ::> l{w: T(w) 'S 1}1. Let 2: = {u : T(u) = 1} ancllet m= min{ lul : u E 2:}. Choose WT such that T(wT) = 1 and lwTI =m. Then the set l{w : T(w) S: 1}1 is bounded from below by the volume of the ball of radius lwT 1· That is, where Cn = (:~ ) . r 2 +1 Let a = 1 + r(v0 ). Our goal is to show that lwTin 2: a-'}. lf lwTI' ::> ,¿a, then lwTin 2: a-'i, as desired. Otherwise, we have that lwTI' < ,¿a. Since 1 = T( WT) = ClwTI 2 (1 + r(vo) + r(wT)) it follows that 2 2 1 alwTI + lwTI r(wT) = C" But since lwTI' < ,¿a, it follows that 1 2 < lwTI r(wT)· (3.13) 20 Also, since a::> 1, we have that lwTI' < ,2,. Thus, wf1 < ,b for every 1 'S j 'S n. Hence, 44 1 )m¡ ( 1 )ffin r( WT) < ( 2C + · · · + 2C Using this in equation (3.13) we have that 1 2 ( ( 1 ) ffi¡ ( 1 ) ffin) 2C < lwTI 2C + ... + 2C . Thus, and since a 2: 1, it follows that z A lwTI >A 2: -, a where A= 2b ((zb)m¡ + .. · + ( 2b)mnrl is a strictly positive constant. Therefore, This finishes the proof of Claim 3. 7. o It follows from the estimate for the dominant term as well as from this bound for l{w: f(w) :S 1}1 that where the constants only depend on A, the "combined degree" of g and the dimension of the space. vVe must show that r( v 0 ) is not too large as a function of r¡ in order to obtain the desired decay. 4.5 Claim 3.8. There exist positive constants /31 and (32 such that The constants depend only on m 1 , ... , mn and the dimension of the space. Proof Recall that by definition \7 g( v 0 ) = 7], and that by hypothesis {v: g(v) :S 1} <;;; {v: lvl :S 1}. The statement is trivial if lvol :S L so we will assume that lvol > l. Let G(t) = g (¡'!¡). Then G'(t) = 'Vg G::~) ·e~:~). Thus, since Vg(O) =O by hypothesis, G'(O) = 'Vg(O) ·(~)=O. Also, notice that since gis convex, sois G. Thus, G'(t) >O if t >O. By Cauchy-Schwarz, IG'(t)l :S lvg G:: ) 1· 1 Evaluating at t = lvol we have that IG'(Ivol)l :S IVg(vo)l = I7JI. But since lvol > O, it follows that 46 G'(lvol) = IG'(Ivol)l ::0: 1711· (3.14) It suffices now to obtain a polynomiallower bound for G'(lv0 1) in terms of lvol· Nagel, Bruna and Wainger proved in [4] (Lemma 2.2) that given a convex polynomial of one variable of the form m P(t) = L aiti j=2 there exists a constant Cm such that for t 2': O, m P'(t) 2': Cm L lajltj-l j=2 In particular, if t 2': 1, m P'(t) 2': Cmt L lajl· (3.15) j=2 Notice that G(t) is a convex polynomial of one variable such that G(O) = G'(O) =O, so we can use the aforementioned result. vVrite m G(t) = L;aiti. j=2 Since we are considering lvol > 1, it follows from equations (3.14) and (3.15) that m lvol L lail ~ G'(lvol) ::0: 1711· (3.16) j=2 47 It suffices now to obtain a lower bound for L,'J'= 2 ia11, which must be independent of the choice of g. To do so, we use the fact that {v: g(v) :S: 1} ~ {v: lvl :S: 1}. In particular, if 1 v 1 1, it must follow that g(v) ~ l. Thus, evaluating at t = 1, it follows that But G(1) = L,'J'= 2 aj :S: "L'; 2 IaJI· This implies that m 1 :s: :L la1l· J~2 Using this last bound on equation (3.16) yields !vol ;S 1771· This finishes the proof of Claim 3.8. D Claim 3.9. (1 + r(v0)P)'l' is at most of polynomial growth in 1771· Proof The proof is trivial. In fact, since !vol :S: ;31 1771 + ¡32 , it is clear that r(v0 ) 1 2 1 2 v5;" + ... + v6;;'n :S: lvol m + ... + lvol mn is bounded from above by a polynomial in 1771· vVe will cal! this polynomial P(l771). D Recall that we had shown that 48 lt follows from this last claim that there exists a polynomial P(lr¡l), which does not depend on the choice of g, such that This finishes the proofthat B(r¡) decays atan exponential rate. Moreover, this decay is independent of the coefficients of the polynomial g that defines it. We must now show that the same is true of all the derivatives of O(r¡). 3.3 Decay of the derivatives Claim 3.10. The derivatives of O(r¡) consist of sums of terms of the form (3.17) where i1,1, ... , in,eo a,, ... , ar, dE N anda¡ + ... + ar + 1 = d. Remar k 3.11. The fact that a 1 + ... + ar - d < O is crucial. As befare ( equation ( 3. 2)), we can factor out a term eh(vo) for each of these integmls. That is, we will factor out h( l)a¡ + ... +ac-d h( ) ( e vo = e- vo . This term will provide the desired decay. Proof. We will prove it by induction. We have that for any j = 1, ... , n, ae - r eryv-g(v)v· dv -(r¡)- JJRn J OT)j - [fJRn eryv-g(v) dvf is of this form. 49 Suppose 8,/k.~ry~" (r¡) is of this form. Then, for any j consists of sums of terms of the form !!_ (e [fiRn eryv-g(v)v;'·' · · · v~"' dv]a' · · · [fRn eryv-g(v)v;'·' ···V~"·' dv]"') ÜriJ [fRn eryv-g(v) dv]d , Let /,( r¡) = [!Rn eryv-g(v)v;'·' ... V~"·' dv r and '1( r¡) = [fiRn eryv-g(v) dvl d. By the quotient rule we have that !!_ (e [IRn eryv-g(v)v;l,l ... v~"·'dv r ... [fll!,n eryv-g(v)v;'·' ... V~"·' dv r) Ür7J [JIR" eryv-g(v) dv]d consists of sums of terms of the form e¡,··· /,-t %8 '/,+t · · · j, (3.18) í "' and C(f¡ .. · fr )(ti;) (3.19) '12 But and U{ (r¡) = d [ r er¡v-g(v) dv]d-1 [ r eryv-g(v)Vj dv]. Ür¡j }JI.n }íRn 50 Thus, a generic term of the form given in equation (3.18) is given by The sum of the exponents of the numerator is a 1 + a2 + ... + a,_1 +(a,- 1) + 1 + a,+1 + ... + ar = a 1 + ... + ar = d - l. Thus this term has the desired form. In the same way, a generic term of the form given in equation (3.19) is given by 11 1 ([J e~v-g(v)v; ·• · v~n dv] a¡ • • • [J. e~v-g(v)v;'·" · · · v~n r dv] "") ([¿ e~v-g(v) dvrr X ( ~ eryv-g(v) dvr! ~ eryv-g(v)VJ dv]) . The sum of the exponents of the numerator is a 1 + a2 + ... + ar + (d - 1) + 1 = 2d- l. Since the exponent of the denominator is 2d, this term also has the desired form. This completes the proof of Claim 3.10. D By the previous claim, in order to understand the decay of the derivatives of f.!(r¡) we need to study integrals of the form fJ>.n eryv-g(v)vl' · · · v~n dv. 51 Claim 3.12. { eryv-g(vJv¡• ... v~n dv ;S eh(vo) H¡[jvoj], }[ H¡ [lvol] 8 is a constant that depends only on the "combined degree" of g and the dimension of the space; s = s 1 + · · · + sn: andE= 4max{m1 , ... , mn}· Proof. As before (equation (3.2)), we can write (3.20) where h(v) = r¡ · v- g(v) ami v 0 is the point where h(v) attains its maximum: and where f(w) is as in equation (3.8). That is, f(w) = g(vo + w)- g(vo)- \7g(vo) · w. lt follows that 52 Let s = S¡ + · · · + Sn and J, = fJF J, = r lwl'e-f(w) dw + r lwl'e-f(w) dw = J,, + Js,. (3.21) J{wEJF Then J,, ::; r e-f(w) dw ~ l{w : J(w)::; 1}1. (3.22) }JF Given v 0 , we can estimate the size of J,, by splitting the integral into the two following regions: J,, = J lwl'e-f(w) dw + j lwl'e-f(w) dw, (3.23) lwl>l lwl>l lwi<:Aivol 8 lwl>.llvol 8 for sorne large constant ), yet to be determined and B = 4 max{ m 1 , ... , mn}. Then lwl>lJ lwl<:.\lvol 8 In order to estimate f¡w,lwl>l,lwl>.llvoiBJ lwl'e-f(w) dw, we will find a lower bound in this region for f(w) in terms of lwl 2 and we will then bound the integral by a constant. Recall that f(w) = g(vo + w)- g(v0 )- \7g(v0 ) · w. 53 Thus (and since f 2': 0), f(w) 2': lg(vo + w)l-lg(vo)I-IVg(vo) · wl. (3.25) 2 'vVe will show that g(v0 + w) is bounded from below by a constant multiple of lwl . It will then suffice to show that the remaining terms in the above expression can be dominated by this bound. Let F(t) = (t(v0 + w)), 9 lvo+wl where tER Then F(t) is a convex polynomial in one variable, such that F(O) = F'(O) = O. We will write M F(t) = L,a,tl. j=2 By Lemma 2.1 of the paper by Bruna Nagel and Wainger [4], we know that there exists a constant CM > O that depends only on the degree of F su eh that for all t 2': 1, M F(t) 2: eMe L. laJI· (3.26) j=2 Thus, M F(lvo + wl) 2': CMivo + wl 2 L la,¡. j=2 Fnrtherrnore, we claim that L,~~ 2 la/ 2': 1 so that F(lvo + wl) 2': CMivo + wl 2 In fact, by hypothesis 54 {v: g(v) :e; 1} <;;;; {v: lvl :e; 1}. It follows that 1 Vo +w) F( ) = 9 ( lvo + wl · Thus, since the vector ~~~!: 1 is on the unit sphere and g is convex, vo+w) F(1) = 9 (lvo + wl ;::: l. Therefore, M M L laJI :2: :LaJ = F(1) :2: l. J~2 J~2 Notice that (3.27) 1 Also, since in the region we are considering lwl > .\lvoiB (i.e., -lvol > _lwl;'') and ),II lwl > 1, and since B = 4max{m1, ... ,mn} > 1, it follows that lvo+wl :2: lwl-lvol > lwl- -,-:2:lwli lwl- -,lwl :2: lwl ( 1--,1 ) . (3.28) ,\8 ,\8 ,\8 Thus, by equations (3.27) and (3.28) it follows that 55 8 ) , it Choosing A > ( K1 follows that (3.29) We woulcl now like to obtain an upper bouncl for lg(v0 )1. Recall that by Claim 3.3, for any v E lR.n wc have that g( v) :S C(1 + r( v) ). Thus, ancl since max{2m1, ... , 2mn} = ~ < B, lg(vo)l :S C(1 + r(vo)) = C(1 + vJ;n• + ... + v5;:'") :S C(1 + lvol 2m1 + · · · + lvol 2m") 8 8 :S C(1 + (1 + lv0 l ) + ... + (1 + lvol )) 8 = C(1 + n + nlvol ) < C ( 1+ n + nl;'l) . But since lwl :S lwl 2 + 1, it follows that For A > 'if7, it follows that (3.30) lt now suffices to obtain an upper bouncl for IVg(v0 ) · wl. l\otice that for each 1 :S j :S n, the j'h entry of V g is a polynomial whose exponents satisfy 56 2a¡ 2 That is, a1 + ... + B IV'g(vo)l ~ e¡(1 + lvol'). Hence, in the region under consideration we have lt follows that IV'g(vo) · wl ~ IV'g(vo)llwl ~el ( 1 + l:l~) lwl =el (lwl + l:l~). But given x > O, O ~ j < d, and any constant A > O, it follows that (3.31) Hence, lwl ~ ~lwl 2 + A2 for an arbitrary constant A> O. Also, lwl~ ~ lwl 2 +l. Thus, 2 Then for A > 16c, and >. > it follows that C.u (!oC,)C,..,! 57 2 2 1 ) CMiwl 1'V g ( V o ) . w ::; e, A + -1 + ----"'-'----'- (3.32) 1 ( .\2 8 Therefore, by equations (3.29),(3.30) and (3.32), and taking 8 2 A>max{( vÍ2 ) ,8Cn,(16C¡) } vÍ2 -1 CM CM it follows that (3.33) where E is a constant that depends on the "combined degree" of g and the dimension of the space, but is otherwise independent. Recall that our goal is to obtain an upper bound for lwl'e-f(w) dw. lwl>lJ lwl>.\lvol 8 Using the lower bound for f(w) obtained in eqnation(3.33) we have that I, < J lwl>l lwl>.\lvol 8 Since CM is a strictly positive constant, the above integral converges. This finishes the proof of Claim 3.12. o It follows from Claim 3.10, equation (3.10) and Claim 3.12 that the derivatives of e( 1)) are bounded from above by a su m of terms of thc form 58 eh(voll{w: J(w)::; 1}1d. Moreover, by Claim 3.12 these terms can be bounded by terms of the form q(lvol) eh(voll{w: J(w)::; 1}1k' where q: lR?.--+ lR?. is a polynomial, and k E [1, d] x Z. By Claim 3.8 it follows that q(lvol) is bounded by a polynomial in lr¡l. Furthermore, by Claim 3.7, l{w: f(w)::; 1}1-k ;S (1 +r( v0 )) >,n. By Claim 3.9 this latter bound is at most of polynomial growth in Ir¡ l. Thus, the derivatives of B(r¡) are bounded by sums of terms of the form e-h(volq(lr¡l), where q grows at a polynomial rate. Final! y, by equation (3. 7) e-h(vo) decays at an exponential rate in Ir¡ l. This finishes the proof that the derivatives of B( r¡) decay exponentially. Thus, e is a Schwartz function. :VIoreover, it follows from the previous computations that its decay is independent of the coefficients of g. D 59 Chapter 4 A Geometric Bound for the Szego kernel In this section we present the proof of our main result. \Ve consider convex unbounded domains of the kind Ob = {z E en+!: Im[zn+d > b(Re[zt], ... , Re[znlJ}, where b: !Rn-+ IR are convex polynomials of "combined degree" (m1 , ... , mn) (refer to definition on pagc 3). Let Ob be one such domain, and Jet (x, y, t) and (x', y', t') be any two points in éJCh. Define S(x,x') = b(x) + b(x')- 2b x+ x') : ( 2 and w=(t'-t)+'Vb x+ x') ·(y'-y). ( 2 We obtain the following estimate for the Szegéí kernel associated to the domain ílb : 60 Here the constant depends on the exponents { m 1 , ... , mn} and the dimension of the space, but is independent of the two given points. Notice that since b(v) = b (V+ X~ x') - Vb (X~ x') .v- b (X~ x') we can write b(v) = f(v)- L(v), where f(v) = b ( v + x~x') and Lis the tangent hyperplane to f at v =O. Thus, given any M> O, l{v : b(v)::; M}l is depicted in the following figure: L Figure 4.1. We obtain these bounds by estimating the integral expression for the Szegó kernel obtained in Appendix A. That is, we study 61 1 ex 2nry·[x+x'-i(y -y)] S((x, y, t); (x', y', t')) = e-2nr[b(x')+b(x)+i(t'-t)] [1 e dr¡ JdT. ¡O ll!.n j én[•¡-v-b(v)T] dv ll!.n The proof is in essence an application of John ellipsoids. Recall that by John [17], given a symmetric convex compact region, there exists a maximal inscribed ellipsoid IC in that region (centered at the center of symmetry) sueh that y'n IC contains the region, where n is the dimension of the space. The key step of our proof consists in introducing factors JL¡ ( x, x', T), ... , Jln(x, x', T) via a change of variable so that These factors are chosen to be the length of the axis of the John ellipsoid associated to a symmetrization of the convex region { v : b( v) :=; ~} . 4.1 Construction of the factors f..l1 ... , f..ln \Ve will begin by cliscussing how to construct these factors Jlt ... , Jln· Let R = {V : b( V) s; ~} . :\otice that since b is convex, so is b, and the region R is convex. In order to be able to use John's bounds, we need to show that the set R is also compact. \Ve do so in the following claim. 62 Claim 4.2. Let b: !Rn--+ IR be a eonvex polyrwmial of "eombined degree" (m1 , ... ,mn)· Let b(v) =b(v+ x~x') -~b(x~x') ·v-b(x~x'). Then Jor any M > O, the set { v : b( v) :S M} is eompact. Proof Notiee that if bis a convex polynomial of "combined degree" (m1, ... , mn), sois b. Also, since b(O) =O and ~b(O) =O, it follows that b 2: O (this was already shown in equation (1.4) on page 17). Observe that bis not the zero polynomial, since by definition of "combined degree", b is of strictly positive degree. Fix M> O. Since bis continuous and {v : b(v) :S M} C !Rn, it suffices to show that this set is bounded. Suppose that this is not so. We claim that if { v : b( v) :S M} is unbounded, then there exists a x E !Rn such that Ve> O, b(cx) :S lvf. In particular, since b(O) =O and bis convex, Ve> O, b(cx) =O. In fact, if the set is unbounded, then there exists a sequence { v,}iEN in !Rn such that lv,¡ > i and b(v¡) :S M. Define y, = ¡;;:¡. Let x be any limit point of {y,}iEN· Notice that since {y,};EN is a sequence in the unit ball, which is compact, it has a convergent subsequence, and so there exists at least one suitable x. Let e be given and let N E N so that N > c. Then b( cy;) :S M for all i 2: N. In fact, cy, = :, • v;, but the sequence { v;}iEN was chosen so that ;, < 1, so :, :S ;, < l. 1 1 1 1 1 1 1 1 Thus, it follows from the convexity of b that b( cy;) :S b( v,) :S M. 63 Furthermore, if {Y;,} is a subsequence of {Y;} which converges to x, then, and since b is continuous, b( cy, ) converges to b( ex). But sin ce b( cy, ) ::::= M for all i ~ N, it 1 1 1 follows that b(ex) ::::= JI,J for all e> O. Moreover, from the above and the convexity of b, it follows that b(cx) =O. In fact, supposc there exists O < r ::::= M such t.hat b(cx) = r. Then by convexity, b(>.cx) > ,\b( ex) = ,\r for all ). > l. In particular, taking ). = M:! it follows that b ( M,+ 1ex) > M:1r =M+ l. This contradicts the fact that b(cx) ::::= l'vf for all e> O. Now let x = (x 1 , ... ,xn) be as above, and define w(t) = t · (x 1 , ... ,xn)· Then since b(cx) =O for all e> O, it follows that b(w(t)) =O, and so the polynomial of one variable b o w is the zero polynomial. 1 On the other hand, if Ca ( tx¡)" • • • ( txn)"n is a term of b( w( t)), then the corresponding term in b o w is of degree jaj = a 1 + · · · + an, and its coefficient is caxf' · · · x·~n. Thus, the maximal degree terms in b correspond to the maximal degree terms in bo w. But since bis of "combined degree," its highest order term (or terms) corresponds to one of the pure terms. That is, the highest degree term of b( w( t)) is of the form ca(tx;) 2m' (or a sum of terms of this form). It is easy to check that the coefficients of the highest degree pure terms of b are positive. This follows from the fact that b(O, ... , O, tx;, O, ... , O) : m;n--+ m;+ Since for these terms the a; are even, it follows that the coefficient of the highest degree term of bo w is positive. Thus, b o w is not the zero polynomial. This yields the desired contradiction. It follows that {v b( v) :'S: M} is bounded, and therefore com pact. 64 D Recall that by construction b(O) = O, so the region R contains the origin. We would now like to show that there exists an ellipsoid IC centered at the origin such that for sorne independent positive constant C. If this were the case, we could choose fL¡ to be the length of the largest semi-axis of IC, ¡L 2 to be the length of the second largest semi-axis of IC, etc. It would follow that fL¡···fLn "'Vol(IC) "'Vol(R). That is, we would have found factors fL¡, ... , fLn such that The existence of such an ellipsoid would follow immediately by John if the region were symmetric (with the origin as center of symmetry). However, we have made no symmetry assumptions on our domain. Nevertheless, we can show the following: Claim 4.3. Let L be any line through the origin (recall the the origin is contained in the set R). This line L will intersect R in two points. Let d1 be the shortest distance along L from the origin to the boundary of R, and let d2 be the largest distance. Then there exist constants m, M depending only on the degree of the polynomial b such that 65 d2 O < m ::; d¡ ::; M < +oo. Proof Let h( v) = rb( v ). Along the line L, the polynomial h( v) is a polynomial of one variable which we will cal! hL(t). This polynomial satisfies hL(O) = h~(O) =O. Write N hL(t) = L Cjtl. j=2 Then there exists sorne 2 ::; k ::; N such that N hL(d¡) ::; L lc1 ld{ ::; (N- 1)lckld~ j=2 On the other hand, it follows from Lemma 2.1 on [4] ( refer to page 15) that there exists a constant O < eN ::; 1 such that N hL(d2):::: eN :L icJidi. j=2 For k as above, it follows that N eNickld~::; eN L lcJid;::; hL(d2)· j=2 :VIoreover, hL(d 1 ) = hL(d2 ) = 1, since d1 and d2 where chosen as the distances where L intersects the boundary of the regio u R = { v : h( v) ::; 1}. Thus, Therefore, 66 1 (Nc~ lr On the other hand, since d1 is the shortest distance along L from R to the origin and d2 is the largest, it follows that d1 ::; d2 . 1 Choosing M = max29<;N { ( ~-;, ) k} and m = 1 it follows that d, m<-<- d¡ - M. This finishes the proof of Claim 4.3. o \Ve have shown that even though the region R is not symmetric, the ratio between rays passing through the origin is bounded by universal constants that only depend on the degree of b. In the following lemma we will show that this is enough to guarantee the existence of an ellipsoid centered at the origin contained in R and such that a dilation by a universal constant contains R. Lemma 4.4. Let R = {v : b( v) ::; ~} and R = {x -x E R}. Let (!: be the maximal inscribed ellipsoid in the region R n k Then where A! is as in Claim 4.3, and n is the dimension of the space. 67 Proaf. By definition, the set R n R is symmetric. 1\loreover, sin ce R and R are compact and convex, their intersection is also compact and convex. It follows from John that there exists an ellipsoid lE centered at the origin such that It is clear that lE <;;; R. We would like to show that there is a dilation of lE which contains R. Let x be any point in R. Then, if ~x E R, it follows by definition that 1.· E foiE. N ow suppose that ~x rf. R. Let L be the line that goes through the origin and x. Using the notation of the previous claim, we have that jxj :S: d2 . vVe would like to find a constant p > O sueh that ~ px E R. Given M as in Claim 4.3, let O < p :S: ,{¡- Then 1 ~ pxj :S: pd2 :S: pMd1 :S: d1. But since d1 is the minimum distance from the boundary of R to the origin along line L, it follows that ~px E R. It follows that givcn any point x E R the point ~ ,{¡x is also contained in R. That is, ;Ix E R n R <;;; foiE. Thus, This finishes the proof of Lemma 4.4. D It follows from the previous lemma that 68 Thus, choosing jJ 1 to be the length of the largest semi-axis of ylnMQ;, 1-'2 to be the length of the second largest semi-axis of ylnMQ;, etc., we have that /-'1 · · · J.'n ~ 1{V : b( V) ~ ~} 1· ( 4.1) Remar k 4.5. The reason we ha ve chosen the components of ¡.t to be the lengths of the axes of yln.lvJQ; rather than those of (5 will become apparent in the proof of the Main Theorem. This normalization will be used to construct a function satisfying hypothesis (iii) in Lemma 3.1. 4.2 Proof of the Main Theorem vVe are now ready to present the proof of the l'vlain Theorem. For the reader's conve- nience, we have divided the proof into three subsections, corresponding to a bound in terms of b, a bound in terms of b(y - y') and a bound in terms of w. We finish by combining all three bounds to obtain the estimate stated in the Main Theorem. lt will be convenient in the course of the proof of all three bounds to rearrange the terms of the integral expression for the Szegéí kernel obtained in Theorem A.l as follows: l'v!aking the change of variables v -+ v + ~ + t to get rid of the term e 2 ~ry·(x+x') in the original expression obtained for the Szegií kernel given by 00 hry·[x+x'-i(y'-y)[ ) S((x, y, t); (x', y', t')) = e-2rrr[b(x')+b(x)+i(t'-t)[ (1 e d1] dT o IR" { é~[ry·v-b(v)r[ dv 1 fR.n it follows that 69 oe hi~·(y-y') S= -2n[b(x')+b(x)+t(t'-t)J1 e d1J dr 8 lo lll.» r /~[~·v-rb(v+·~·')] dv . }J 1 forrn [B(17)]- , where e is the function studied in Section 3. In particular, the exponent of our integral rnust be of the form 1) · v- g(v), where g(O) =O and 'V'g(O) =O. Since 'V [•b (v + x ~ x')] j(v~o} = 'Vb (X~ x') T we can make the change of variables 1)--+ 1) + 'Vb ( x~x') T. Then, ex:; -21l"T[b(x')+b(x)+i(t'-t)] 4r.T b( x~x') 27riTVb( x-;_x') ·(y-y') S= ¡ e e e o where -b(v)=b ( v+ x+x') -'Vb (x+x') ·v-b (x+x') 2 2 2 and the term b ( x~x') has been added so that b(O) =O. Letting 5(x,x') = b(x) + b(x')- 2b x+ x') ( 2 and 70 w=(t'-t)+'Vb x+ x') ·(y'-y) ( 2 it follows that ( 4.2) 1\'otice that since bis convex, Tb is also convex. l'v!oreover, if bis of "combined degree" (m1 , ... , mn), sois Tb. Thus, Tb is a convex polynomial satisfying conditions (i), (ii) and (iv) of Lemma 3.1. It remains to renormalize Tb so that it also satisfies condition (iii). With p. 1 , ... ,¡Ln chosen as in equation (4.1), let g(v) = Tb(¡.w). ( 4.3) N otice that sin ce and letting A= ( foNI)- 1 condition (iii) of Lemma 3.1 is satisfied. That is, {v lvl :":A}<;;; {v g(v) :": 1} <;;; {v lvl :": 1}. ( 4.4) By making the change of variables v -7 Ji. V so as to introduce the factors p. 1 , ... , /Ln in the denominator integral of equation (4.2), as well as the change of variables 1) -7 ~ we have that 71 (4.5) 2 8 The term e- "" will provide the necessary decay to obtain the bound in terms of 5. However, to obtain the bound in terms of b(y- y') and the bound in terms of w we will 2 need to use the oscillation of the terms e rriry·( •·/) and e-Zniw respectively. 4.2.1 The bound in terms of b Proposition 4.6. Let b: JFtn -e> 1Ft be a convex polynomial of "combined degree "(m1, ... , mn)· Let (x, y, t) and (x', y', t') be any two points in é!íh. Define b(v) = b (V+ X~ x') _'Jb (X~ x') .V_ b (X~ X'); and 5(x,x') = b(x) + b(x')- 2b x+ x') . (4.6) ( 2 Then, IS((x,y,t);(x',y',t'))l ;S l{v: < }12 5 b(~) 5 where the constant may depend on the "combined degree" of b and the dimension of the space, but is independent of the two given points. Remark 4.7. N atice that since bis convex, o(x, x') 2 O. 72 Proof It follows from equation ( 4.5) that ce e-2trn) ¡ 1 ISI < --..-----=---- dr¡ dr. -i o J.LI ... ¡.t'f, Rn hn eh[ry·v-Tb(¡w)[ dv But Thus, it follows that (4.7) We showed in Lemma 3.1 that e(r¡) = [¡ é~[ryv-g(v)[ dv] -! decays at an exponential rate, where the decay is independent of the coefficients of g. In particular, the decay does not depend on r. Thus, f11.n e(r¡) dr¡ converges. Hence, ISI ;S 1oo-j{_v__:_e_-2_rrr_'-~2 dr. b(v) :s; ~ }j \Ve can wri te Thus, 73 In order to get rid of the dependence on j of 1 { v b( v) :S 2/H }1 we can write e-2JT2) 21 -21r21 21 ISI ;S e_ ( 4.8) I: - 2 + I: 2 -oc l{v b(v) :S 5}1 :S 2"(j+l) v : b(v) :S : 1{ 21 1 }1· Thus, it follows that Since both sums converge, we obtain the desircd estimate. That is, 1 ISI;S5I{v This finishes the proof Proposition 4.6. D 4.2.2 The bound in terms of b(y- y') Proposition 4.8. Let b: ¡p;n--+ lR be a convex polynomial of "combined degree" (m 1, ... , mn). Let (x, y, t) and (x', y', t') be any two points in é!ílb. Define 74 -b(v)=b (v+ x+ x') -Vb (x + x') ·v-b (x + x') . 2 2 2 Then 1 IS ((x, y, t); (x', y', t'))l ;S b(y- y')l{v : b(v) < b(y- y')}l2 where the constant may depend on the "combined degree" of b and the dimension of the space, but is independent of the two given points. Proof We had shown in equation ( 4.5) on page 71 that Thus, and since c5 ::> O, oc 1 ISI ::; 2 ¡o /J¡ ... p~ But by Lemma 3.1, is Schwartz. l'vloreover, its decay is independent of .,- and the coefficients of b. The same is true of its Fourier transform, íJ. vVe can write 1 ISI :S: 2 21íJ (y- y') 1 d-r. looo /J¡ · · · Pn J.l Recall that the factors Pi were chosen so that 75 Hence, oe 1 A y- y' ISI ;::; 2 e dT. lO 1 {V : b( V) :S n1 1 ( JL ) 1 Let u = y- y'. vVe can split the integral into integrals defined over smaller intervals in the following way: 1 1 oc- = ------,2 18 (Y- y') 1 dT f (:(:: 218 (u) 1 dT. lo 1{ V b(v) :S n1 JL J~-oc .Jtr~l 1{ V : b(v) :S n1 JL But if b(u) 1 b(u) --<-<-- 21+1 - T - 2i ' then 1{ v : b(v) s; ~~~: }1 :S 1{ v : b(v) :S~ }1 It follows that 1 ISI ;S f 1~(:: 218 (u) 1 dT J~-oo ,;~, 1{ v : b(v) :S ;;~( }1 JL Splitting the sum for positive and negative values of j, we can write 76 ~ ISI < 2.:= { '1"' 1 ¡e (u) 1 dT "' -oo~j ~ 2.:= rb(u) + O~j~oo j.Jl_ l{v : b(v;::; ~ }121e (:)1 dT. b(u) But for j - -1 1 l{v: b(v)::; ~~~:}¡ ::; l{v: b(v) :Sb(u)w . On the other hand, for j ? O, we can use Claim 1.5 on page 16 to show that - -1 1 1 1{ v : b(v)::; ~~~: }1 ::; 2n(j+ ) 1{ v : b(v)::; b(u) W . Hence, ISI < 2.:= 1 r:(::l'(u)l -x'S,j + 2.:= ---__:2::....2n_(J_.+_1J_---o2 r'~:: le (U)I dT. O~j~cc l{v b(v)::; b(uJ}I }~~) 1'" The term 21 that will com from the bounds of integration will be enough to obtain the convergence of the first sum. Thus, for the first sum it suffices to bound ¡e¡ by a universal constant. lt follows that 77 1 1 L , t~=: lé (:!!:) 1dr < L {~:: dr -xo75J vVe would like to obtain a similar bound for the second sum. The main obstacle is obtaining decay in j to counteract the growth of the term 22n(j+l), thus ensuring the convergence of the series. Our goal is to show that ¡':l:;l'(u)l 1 J.J!_ e ¡¡ dr ;S 2Kib(u)' b(u) for sorne arbitrarily large constant K. l\'otice that in this interval, 1 1 ---- < ~. rb(u) - 21 Al so, é (u) 1 < 1 11 l ~ll+lv(~)IIN for any polynomial p. Thus, it suffices to show that there exists sorne polynomial p such that rb(u) S: 1 + lv (~) 1, In fact, it would follow that e-'(u)l < 1 <-. 1 l 11 ~ lrb(u)["' - 2NJ' so that 78 2 22n(j+l) 1:(:: 1. (u) 1 2 b(v)::;b(u)}l ,~:, g IL dT For sufficiently large N, the series converges, and we would obtain the desired estimate. We will now find a polynomial p such that Tb(u) ::; 1 + IP (~) 1· Letting s = ~, we can write the above requirement as Tb(JLs)::; 1 + IP (s) 1· Recall that by equation (4.4), there exists a universal constant A < 1 such that {v: lvl::; A}<:;; {v: Tb(JLv)::; 1} <:;; {v: lvl::; 1}. (4.9) Then the polynomial g(s) = Tb(JLs) satisfies al! the hypothesis of Lemma 3.1. In par- ticular, Claim 3.3 on page 32 holds. Thus, there exists a universal constant such that This finishes the proof of Proposition 4.8. o 4.2.3 The bound in terms of w Proposition 4.9. Let b : !Ri.n -+ IR:. be a convex polynomial of "combined degree" (m1 , ... , mn)· Let (x, y, t) and (x', y', t') be any two points in é!Db. Define b(v) = b (V+ X~ x') -\lb (X~ x') .v- b (X~ x') 79 and w=(t'-t)+V'b x + x') ·(y'-y). ( 2 Then, jS((x,y,t);(x',y',t'))j ;S jwjj{v: b~v) < jwj}j 2 ' where the constant may depend on the "combined degree" of b and the dimension of the space, but is independent of the two given points. The derivation of this last bound is rather long and technical. Before giving all the technical details, however, we shall begin by briefly outlining thc main ideas behind the proof. It follows from equation ( 4.2) on page 70 that 1 S= fo'~ e-iuT F(T) dT + j~ e-iuT F(T) dT. (4.10) lul where u= 211 [(t'- t) + V'b (x:_;x') ·(y'- y)] and ( 4.11) The integral for O ::; T ::; : in equation ( 4.10) yields the desired estimate by using 1 1 similar techniques as those detailed in the proof of the previous two bounds. Thus, the main difficulty lies in estimating the integral for : ::; T ::; oo. In particular, we must 1 1 show that the integral converges. To do so, we will take advantage of the oscillation of the term e-iUT. lntegrating the latter by parts N times, for an arbitrary positive integer N, we obtain formally an equation of the form 80 vVe then show that after introducing the factors J1 as in the two previous bounds, every derivative of F(r) yields a factor of ~times a bounded function, so that 0C -iUT e (N) rr luiN F (r) dr 1'U[ Finally, using Claim 1.5, we show that this last integral is bounded by an expression of the form 1 1oo lul2nT2n ------.,2 N dr, luiNj{v:b(v):o;lul}j : T 1 1 yielding the desired estimate for large enough values of N. Before presenting a rigorous proof of Proposition 4.9, we discuss three technical results that will be used in the course of the proof. In Claim 4.10 we obtain an upper 00 bound for f0 e-iur F(r) dr in terms of the N+ 1'h derivative ofF. The method we use is analogous to integration by parts, but does not yield boundary terms, making the computation slightly simpler. In Claim 4.11 we compute the N'h derivative of F. In Claim 4.12 we show that, after introducing the factors Jl, the N'h derivative of F is dominated by ~ times a bounded function. Claim 4.10. Let tE IR and 81 00 where F E C (IR). Then given N > O there exist positive coefficients c1 , ... , CN+I such that II(t)l ~ ,~¡ c1 lio{¡ e-itr F (r + ~~) drl +eN+! IJ_,_oo e-itr 1,{¡ · · ·l-!f¡ p(N+!)(r +S¡+ ... + SN+J) ds¡ · · · dsN+l dr¡. ltl o o (4.12) Proof. \Ve can write "'. X· I(t) = l'" e-itr F(r) dr + j.z_ e-itr F(r) dr =S+ L. (4.13) o lti Introducing a factor of e' >gn(t)rr, we can split L as follows: Writing F (r + ¡\'¡) = [F (r + ¡\'¡)- F(r)] + F(r), we have that L = ~ e-itr F(r) dr- foco e-itr [F (r + : ) - F(r)] dr- f~ e-itr F(r) dr) v; 1 1 = ~ ( _ {" e-itr F(r) dr- f' e-itr [F (r + : ) - F(r)] dr). 1 1 82 Using this Jast expression in equation (4.13), it follows that Now Jet F1 (r) = F (r + ¡T¡)- F(r) and Jet Then, by the same argument, it follows that After N times of repeating this process, we wouJd have that i}.,¡(t) = fo"'" e-itTFN(r)dr =Hlam e-itTFN(r)dr- f' e-itT [FN (r+ l:l) -FN(r)] dr), Letting for 1 ::; j ::; N- 1, it follows that 83 1 I(t) = :l [S(t)- J1 (t)J = ~ [s(t)- ~ [Sr(t)- h(t)J] = ~ [s(t)- ~ [sr(t)- ~ [s2(t) · · ·- ~ [SN-r(t)- JN(t)J]]]. That is, i\"otice that after expanding and rearranging terms, we can write for 1 ::; k::; N It follows that 1 1 1 I(t) = ~S(t) + ~ ~--l (k- ) ( - )i+l [m e-itT [F (T + (j + )") - F (T + J")] dT 2 ~ f:;f, j 2k+l lo itl itl + ~ (-1)i+l (N) {m e-itr [F (T + (j + 1)") - F(T + J71")] dT f:;f, 2'Y+ 1 J lo itl itl (-1)N+lj00 . + N+! -'- e-ztrpN+J(T)dT. 2 1'1 Changing the arder of summation, it follows that 84 That is, 1 ~ (-1)J ( N ) (Ti:: -itrF ( J7r) d + ;Sí zN+l j - 1 lo e T + ltl T N ( - 1)j (")1'V -"'ltl -itT ( )'Jr . ) ( - 1)N+l oc -itT +Ea zN+! j lo e F T +TtT dT + zN+! /-'h e FN+J(T) dT. Let 1 " "N (k-!) 1 "N (k-!) 1 ( N ) 1 (N) 1 f 1 . N CJ = L...k=J 1_ 1 "2"f+T + L...k=J+l 1 2k+I + \j-I 2N+T + 1 2N+T or ::;: J :=:;: - 1; CN = ~~~; and Then it follows that 85 (4.14) It suffices now to show that where (4.15) Using the identity it follows that for any integer AI > O and for any function h, f( _1 )M+J (lvf) h(j) j~O J = h(lvi) + (-1)Mh(O) + :~~(-1)M+J ( (Mj-1) + c~I ~n) h(j) 2 1 = h(M)- h(M -1) + ·I= (-l)M+J(M.~ )h(j) j~l J 1 + :~\ -1)''f+J+l (M j- ) h(j + 1) + (-1)Af+l h(1) + (-1 )M h(O). That is, 86 1 t( -1)M+j (M) h(j) = "'f\ -1)"-HJ+l (M:- ) [h(j + 1)- h(j)J. (4.16) j~O J J~O J Let h(j) = fo7¡ F' ( T +S+ m) ds and M= N. Notice that we can write (4.17) It follows from equations (3.8), (4.16) and (4.17) that Now Jet h(j) = Jf'i fofir F" (r + s1 + s2 +m) ds 1 ds 2 and M= N -1, and use equa tion (4.16) once more. Since . l h( .) 111 ( h( J + 1 - J =lorlh lorlh lorm F T +S¡ + s2 + s3 + TtfJ") ds 1 ds 2 ds 3 i t follows that Repeating this process N - 2 more times, we obtain rr rr {m {"' (N+!) ( ) FN+l (T ) =lo ···lo F T + s1 + ... + sN+l ds 1 • • · dsN+!· 87 Thus, it follows from equation (4.14) that This finishes the proof of Claim 4.10. o Claim 4.11. The N'h der·ivative of (4.18) consists of s-urns of terms of the form where ¡(T) = [¡ e4K[" V-Tb(v)j dVr a 1, ... , ak, k, dE l\1; O S: k S: N; a 1 + ... + ak = d- 1; and a 1 + 2a2 + ... + kak =k. Proof We will begin by showing by induction that the k'h derivative of 88 consists of sums of terms of the form where a1 + ... + ak = d- 1; and a1 + 2a2 + ... + kak =k. ?\otice that is of this form. Suppose J(k)(T) is of this form. We will show that J(k+l)(T) is of this form. Let and "f(T) = [j e4~[ryv-Tb(v)j dVr Then, by the quotient rule, fT [J(k) ( T)] consists of sums of terms of the form (4.19) 89 or ( 4.20) But and ~('¡) = ~ 47rd [ { e4rr[ry·v-Tb(v)l dv]d-J { b(v)érr[ryv-Tb(v)l dv dí }rJ.n }Rn = ~ 47rd'¡ ddl { b(v)érr[ryv-Tb(v)! dv. J.Rn Tlmo, a generic tenn of the fonn given in equation (4.19) is given by dr¡. (4.21) 4 Let hj = gj for j # S, s+ 1; h, = l hn b( V )'e rr[ryv-Tb(v)l dv ]"' 'where a, = a,~ 1; and h,+l = [ hn b(v)s+lérr[ry·v-Tb(v)l dv]"'+'' where as+! =as+!+ l. Then equation (4.21) can be written as For this term to have the desired form, the exponents must satisfy a1 + ... + a,_ 1 +a,+ 90 kak = k+ l. The former holds, since by inductive hypothesis a1 + ... + a,_ 1 +a,+ a,+ 1 + a,+ 2 + ... +ak = a1 + ... +a, -1 +a,+ 1 + ... +ak = a1 + ... +ak = d-1. The latter also holds, since by inductive hypothesis, a1 + ... + (s -1)a,_1 +sa,+ (s+ 1)a,+1 + (s+2)aH2 + .. . +kak = a1 + ... +(s-1)a,_1 +sa,-s+(s+1)a,+1 +s+1+(s+2)a,+2 + .. . +kak = a1 + ... + kak + 1 = k + l. In the same way, a generic term of the form given in equation ( 4.20) is given by (4.22) - 1 Let h, = 9J for j # 1, and h¡ = [ kn b(v)érr[ry·v-Th(v)] dvr ' where a¡= a¡+ 1 so that equation ( 4.22) can be written as ehiry·(y-y') · · · h 1 hk e d+! dr¡. ¡JF For this term to have the desired form, the exponents must satisfy a1 + a2 + ... + ak = d, and a1 + 2a2 + ... + kak = k + l. The former holds, since by inductive hypothesis, a¡ + az + ... + a k = a1 + ... + a k + 1 = ( d - 1) + 1 = d. The latter also holds, sin ce by inductive hypothesis a¡ + 2az + ... + kak = a¡ + 2az + ... + kak + 1 = k + l. It follows that for any k E 1\1, the k'h derivative of J(T) consists of sums of terms of the form 91 where a 1 + ... + ak = d- 1; and a 1 + 2a2 + ... + kak =k. Since F(T) = e-2n 5J(T), the N'h derivative ofF is given by F(N)(T) =E G) (e-2no(-k) j(k)(T). But ( e-ZrrrJ) (N-k) = Ci5N-ke-2ns Thus, the N'h deriva ti ve of F consists of sums of mul tiples of terms of the form 1 e2Ki"(y-y') [ [ b(v)érr["v-rb(v)J dv]" .•• [ [ b(v)ké*v-rb(v)J dv]"' sN-ke-2rrT6 }R_n }~n d7J. 1Rn [ kn e4rr["v-rb(v)J dvl where a 1 + ... + ak = d- 1 and a 1 + 2a2 + ... + kak = k. Final! y, writing for 1 :S s :S k, [jb(vJ'e•rrJ,,v-rb(v)J dvr = T;", [j(Tb(v))'é"I'JV-Tb(v)J dvr yields the desired expression. This finishes the proof of Claim 4.11. D Claim 4.12. Let 2Ki"·(y-y')J"( ) !"(-) A p. ( ) _ ( -')N+l-k -2no e " 1 T · · · k ' d I..J.V+!kT-Tu e () r¡, ( 4.23) . . 1~'T 1[Rn where 92 --yi'(T) = [j é~[>¡v-rb(¡w)J dv r a 1 , ... , ab k, dE N; O:':: k:':: N+ 1; a 1 + ... + ak = d- 1; and a 1 + 2a2 + ... + kak =k. Then there exists a constant C that depends only on N, k, the "combined degree" of b and the dimension of the space such that Proof It is easy to check that ( TJ)N+!-ke-'"rb is bounded. In fact, for x > O and A> O, define h(x) = xAe-x. Then h'(x) = xA-!e-x(A- x). It follows that h(x) attains its maximum at x =A. That is, h(x) :':: AAe-A It follows that (N+ 1- k)N+l-ke-(N+!-k) (27r)N+l k Thus, it suffices to show that is bounded. We will begin by studying integrals of the form t!:(T) = [j(Tb(¡wlJ"é~l'l"-Tb(¡w)J dvr By Claim 3.3 on page 32 there exists a universal constant such that Thus, we must study the behavior of integrals of the form 93 L, = [ p,(v)eryv-g(vl dv, }iRn for polynomials p,: !Rn-+ IR with non-negative coefficients and g(v) = Tb(¡.tv). As usual (see, e.g., equation (3.2) on page 31), we can write L, =eh(vol [ h(v)-h(volp,(v) dv, }JRn 8 where h( v) = r¡ · v - g( v) and v 0 is the point where h( v) attains its maximum. l'vlaking the change of variables v = w + v 0 , it follows that where f(w) is as in equation (3.8). That is, f(w) = g(vo + w)- g(v0 )- \lg(vo) · w. Since the coefficients of p are non-negative, it follows that p,(w, + Vo¡, ... , Wn + Von) <::: p,(lw[ + [vo[, ... , [w[ + [vo[). This last polynomial is now a polynomial of just one variable, ancl after expanding and rcgrouping al! the terms, it consists of sums of terms of the form [w[1'[v0 ['' for indices i., and j,. Hence, there exist positive coefficients ci,,J, such that Let 94 This integral is identical to the one studied in equation (3.21) in Clairn 3.12 on page 52. Thus, where 8 is a constant that depends only on m 1 , ... , mn and the dirnension of the space; and B = 4rnax{m1 , ... ,mn}· It follows that 8 L, :S é(vo) L C;,.j,lvol'' [l{w : f(w)::::; 1}1 (1 + lvol 1' ) +e]. Zs ,Js That is, for sorne polynornials cjJ, : lR --+ JR+, 1/Js : lR --+ JR+ On the other hand, by equation (3.10) Jén[>¡v-rb(¡w)[ dv ""eh(vo)l{w : J(w)::::; 1}1. Therefore, there exists sorne polynornial % : lR --+ JR+ such that But a 1 + ... + ak = d- 1, so 95 Moreover, by Claim 3.7, l{w: f(w) :S: 1}1;::; (1 + r(voW'i. where r( v) = v~m, + ... + v;;mn. Thus, and sin ce j - d < O, d !( ( ))n(d-j) 1) By Claims 3.9 and 3.8, L:1;;0 1+r v0 -2-q1(lv0 is at most ofpolynomial growth in I7JI· On the other hand, by equation (3.7), e-h(vo) decays exponentially in I7JI· Hencc, r fi(r) ... Jt(r) dr¡ }J' We are now ready to present the proof of Proposition 4.9. Proaf. We had shown in equation ( 4.2) on page 70 that oo 1 2 . 2Ki~·(y-y') _ -2TITÓ -27ri(t -t)T 7rlT 'Vb(•+•')-,- . (y-y ')1 e d d S - e e e k r¡ T. O Jll Let (4.24) so that Then by Claim 4.10 it follows that 1 1 00 -iUT íuTK íuTK N+l + N+l JK e pl )(T+s1 + ... +sN+¡)ds1 2 ¡;;¡ lo ···lo We will begin by obtaining the desired bound for the terms of the form We can write -2 o( 1-"-) e2niry·(y-y') e 1T T+ iul 1 dr¡ dT. 4n[ry·v-(r+¡';;\ )b(v)] d Rn e V n;:nJ 97 Making the change of variables s = T + f2¡ as well as r¡ --+ ~ and v --+ ¡w, where /-L = (¡;, 1 , ... , J.l,n) are eh osen as in equation ( 4.1) on page 68, it follows that (j-1-l)7r -lu-1 -h!s 8 ¡ 1 IIJ(u)l :<: 2 2 . dr¡ds. "¡ · · · ¡1!!... r ¡;,n Rn ~ e converges. Also, by convexity of b, 8 2 O. Thus, {j-l)7r -lu-1 1 II1 (u)l ~ 2 2 ds. ¡1!!... f-11 ' ' · J-ln lul Since the factors ¡;,1 were chosen so that it follows that (j7u'ic 1 IIJ(u)l ~ l{ _ '}l 2 dT. ¡E V : b( V) :<: - lu1 T But since we are considering T :<: (Jtu~)rr, then b(v) )} b(v) {v: :<: x()~ 1 ~ {v: :<: ~}. Thus, 98 (j H hr ~lul- 1 7r < dr = . IIi(u)l ¡ 2 2 ~ f;i¡ l{v: b(v) ~ n(j~l)}l lull{v: b(v) ~ n(j~l)}l Let w = 2';. = [(t'- t) + V'b (x~x') ·(y'- y)]. Then, 1 IIj(w)l ;S l{ _ 21>&} . lwl v : b(v) ~ (j+J) 12 Since {v: b(v) ~ lwl} <;:: {v: b(v) ~ 2lwl} it follows that 1 IIo(w)I;S _ 2· lwll{v: b(v) ~ lwl}l The same bound is obtained trivially for j = l. For j > 1, we can use Claim 1.5 on page 16 to show that Hence, Since the sum over j is finite, it follows that 1 ISI :S 2 lwll{v: b(v) ~ lwl}l ( 4.25) 1 oo -iuT {R {iür F(iV+l) ( ) d d d + 2N+l 1 j' e lo ' ... lo ' . T +S¡+ ... + SN+! S¡ . . . SN+! T 1 . ¡;;¡ vVe must now bound the term 99 00 -"- -"- -iuT :u¡ :u1 1\'+1 f~ e lo ... r F( )(T+s¡+ ... +sN+l)ds¡ 1 lul O Jo We showed in Claim 4.11 that the (N+ 1)'h derivative of consists of sums of tcrms of the form where ¡(T) = [j e4~[ryv-rb(v)] dvr a1 , ... , ak, k, dE N; O::; k::; N+ 1; a1 + ... + ak = d- 1; and a1 + 2a2 + ... + kak =k. \Ve can write these terms as .f,,t.N+I.k(T), where 7 2Kiry·(y-y')J (T) ... f ( ) , ( ) _ ( -)N+l-k -2nó e 1 k T d '-"N+! k T - T 0 e ( ) 'IJ· , ¡ Í T iR" Thus, we must study integrals of the form · · · dsN+! dT. \Vith J.l¡, ... , l"n chosen as in equation (4.1) on page 68 wc can make the changes of variable r¡ -t '1. and v -t ¡.w. We obtain an integral of the form " 100 oo (u,....,¡ '· ',.-n" )"' + ... +ak 1-"-lv.' 1-"-lul ¡.¡, J = N l( )d ¡ . . . t,.N+l,k (T +S¡+··.+ SN+l) ds¡ · · · dsN+l dT, 1'1" T + f.-Ll · · · 1-Ln + O O 1TUT where now hi"·(y-y')¡~( ) f~( ) t,.~' (T) = (Ti5)N+l-ke-2n8 e " 1 T ... k T dr¡· ( 4.26) N+!,k 1IR" ¡~'(T) ' and ¡"(T) = [j é~Jryv-rb(p.v)idvr But by Claim 4.12, lt>.~+l,k(T)I:::; C. Tt follows that Also, a 1 + ... + ak = d - l. Thus, 1 100 1 IJI ;S 1 IN+! N+l( )2 dT. U , T ¡_t¡ • • · 1-'n ¡;;¡ Since 11- was chosen so that and taking w = 2':c as befare, it follows that 101 IJI ;S 1N+l 1"' 1 2 dT. lwl" '2'j"WT' 7 N+~I{v : b(v) <- l}lT :\otice that on the interval under consideration, (2lwiT)~ 1 :::; l. l_;sing Claim 1.5, it follows that Also, as mentioned earlier, l{v: b(v):::; 2lu'l}l::>l{v: b(v):::; lwl}l It follows that Thus, taking N ::> 2n + 1, lwl2n 1"' T2n IJI;SI l'-'+l l{ }l2dT W, 21~, TN+l V ; b(v):::; lwl 1 2n~N~oc ""'lwiN+1~2n l{v: b(v):::; lw1}12, T 21~1· That is, It follows from equation (4.25) that 102 ISI < 1 2 . ~ lwll{v: b(v) :S: lw1}1 This finishes the proof of our third and last bound. o vVe have shown that IS ((x, y, t); (x', y', t'))l ~ min {A, B, C}, where 1 A= - ; 81{v: b(v) < 8}12 B= 1 . b(y- y')l{v: b(v) < b(y- y')}l 2 ' and e= _1 lwll{v: b(v) < lwl}l2 Tlms, to conclude the proof of the l'vlain Theorem, it suffices to show that 1 . min{A,B,C}~ 2 Jsz + b(y- y') 2 + w2 1{ v : b(v) < J(¡z + b(y- y') 2 + w 2 }1 Without loss of generality, suppose that J :s; b(y- y') :s; lwl. Then, and since b is non-negative, 103 {v: b(v) 1 1 1 ___-.,--::__ ___ < - < --~---- lwll{v: b(v) < lwl}l'- b(y- y')j{v: b(v) < b(y- y')W- Jj{v: b(v) < 8}12' so 1 ISI < ~ jwjj{v : b(v) < jlL·j}j2 C.Ioreover, since S::; b(y- y')::; jwj, then )82 + b(y- y')2 + w2 ::; ~- That is. 1 vÍ3 -<---¡==~~=== lwl - J6 2 + b(y- y')2 + w 2 . and But by Claim 1.5, Therefore, 1 ISI < ~ jwjj{v: b(v) < jwj}j2 ( y'3)2n+l ::; , . 2 2 2 .jJ + b(y- y') + w'l{ V : b(v) < .jJ + b(y- y')2 + w 2 }1 104 This finishes the proof of the lvlain Theorem. 105 Appendix A An integral expression for the Szego kernel In this appendix we derive an integral expression for the Szegéí kernel for a class of unbounded domains defined by convex polynomials. Proposition A.l. The Szego kernel on the boundary of domains of the kind D = { z E en+! : Im[zn+ll > b(Re[z¡], ... , Re[zn])} where b : ffi'.n ---+ lR'. is a convex function of "combined degree" is given by e2íT1J·[x+x' -i(y' -y)] S((x, y, t); (x', y', t')) = e-2rrr[b(x')+b(x)+i(t'-t)J dr¡ dT, érr[~·v-b(v)rJ dv O ~ (1~n J l 1 ~n (A.l) where (x, y, t) and (x', y', t') are any two points on 80. Let D = {z E en+!: Im[zn+l] > b(Re[z1], ... , Re[zn])} and (A.2) = b (Z1 + Z1 . Zn + Zn) _ Zn+l- Zn+! 2 , ... ' 2 2i 106 be a defining function for our domain. Recall that the Szegií Projection is the orthog- onal projection Ih : U(80) -+ H 2 (80), where H 2 (80) = {! E L2 (80) : L(f) = O as a distribution, for all tangential Cauchy-Riemann operators L }. We begin by find- ing a base for the tangential Cauchy-Riemann operators. vVe can Jet j = 1, ... ,n. For these operators to be tangential they must satisfy Z1 (p) = O. Thus, 8 z1 = 2 (!!.._- i~(x)--) j = 1, ... ,n OZj OXj OZn+t are a basis for the space of tangential Cauchy-Riemann operators for our domain in We can identify 80 with en X JR:. via the map (z¡, ... , Zn, t) E en X JR:. B (z¡, ... , Zn, t + ib(Re[z¡], ... , Re[zn])) E 80. Our operators zj are operators in en+ l. The restriction of these operators to en X JR:. is (A.3) Lemma A.2. Let and define the partial Fourier transform F[fl( x,y, t) = f-( x,r¡,T ) = J e-2ni(y·ry+tr)J(x, y, t) dy dt. 107 Then is an isometry, and 1 1 Z1[f] = F M- ~ MF[j] j = 1, .. . n. 0 J Proof It is easy to check that M is an isometry in this weighted L2 space, in fact 2 IIM[g]lli'le'~[ryz-b(z)T) = J jg(x,7J,T)e-2"[ry·x-b(x)r]l e4rr[ryx-b(x)ridxd7]dT R_2n+ 1 = j lg(x, 7], T)l 2 dx d7] dT. Jt2n+l Also, 1 z1[JJ = Z1 F (]) 2 1 +t) (8/(x,7J,T) , 8b , ) = e "' yry r "· . - 2Irr7Jf(x, 1), T) + ~(x)2n f(x, 1), T) d7]dT J u,r1 ux1 ]Rn+l = J C2rri(yr¡+tr) 8 2,.[ryx-b(x)r]O~. ( e-2rr[r¡x-b(x)r]/(x, 1), T)) d7] dT Rn+l ) =F1M-1 _!!_MF[f]. OXj o Since F and M are isometries, instead of projecting onto the null space of the tangential Cauchy-Riemann operators we can project onto the null space of the operators { 8~ 1 }. That is, we project onto functions /(x, 7), T) E U(!R2"+1 , e4rr[ry·x-b(x)ridx d7] Remar k A.3. i) N atice that for these functions j to be in U(!R2n+l, é•[ry·x-b(x)Tidx d1) dr) one must have that J IN ( Je4n[ry·x-b(x)Tj dx) d1) dr < OO. ]Rn+ l IR.n The inner integral j é•[ry·x-b(x)TJ dx diverges if r ~ O beca use of the growth hypothesis 'Rn on b. Thus we set fí](x, 1), r) =O if r ~O. ii) N atice that under these hypothesis the constant functions belong to L2 (!Rn, é•[ry·x-b(x)Tidx ). Since the basis for the null space of the operators { &~,} is just the constant function, the projection fi for r > O is given by g(x', 1), r)e"rr(ry·x'-b(x')Tj dx' < g, 1 > 1 r fi[g] }F.n < 1,1 > r e4n[ryv-b(v)Tj dv }Rn e4n[ry·x' -b(x')Tj ) dx'. kn é•[ry·v-b(v)Tj dv The Szegií Projection, then, is given by II[f](x, y, t) = F 1 M-1fíMF[f](x', y', t') 1 = F-! M-! fie-2rr[ry·x' -b(x )Tj r e-2ni(y'·ry+t'T) !( x'' y'' t') dy' dt' }Rn+l -2ni(y'·ry+t'T)j( 1 1 t') 4n[ry·x'-b(x')Tj = F-1 M-! e-2n['1·x'-b(x')Tj e X' y' e dy' dt' dx' 4n[ry·v-b(v)Tj ¡ ¡ e dv R.n Rn+l RnJ 109 Therefore, the Szeg6 kernel is given by oc 2u¡·[x+x'-i(y'-y)[ ) e-2rrr[b(x')+b(x)+i(t'-t)J (1 e dr¡ dT. ¡ IRn { e [1] F. BEHREND, Über die kleinste umbeschriebene und die grojJte einbeschriebene El lipse eines konvexen Bereichs, l\lath. Ann., 115 (1938), pp. 379-411. [2] A. BONAMI AND N. LOHOUÉ, Projecteurs de Bergman et Szego pour une classe de domaines faiblement pseudo-convexes et estimations L", Compositio Math., 46 (1982), pp. 159-226. [3] L. BOUTET DE MüNVEL AND J. SJOSTRAND, Sur la singularité des noyaux de Bergman et de Szego, in Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123-164. Astérisque, No. 34-35. [4] J. BRUNA, A. NAGEL, AND S. WAI!\GER, Convex hyper.surface.s and Fnu.rier tmnsforms, Ann. of :VIath. (2), 127 (1988), pp. 333-365. [5] C. CARRACINO, Estimates for the Szego kernel on a model non-pseudoconvex do main, Illinois J. Math., 51 (2007), pp. 1363-1396. [6] :VI. CHRIST, Regularity properties of the (Jb equation on weakly pseudoconvex CR manifolds of dimension 3, J. Amer. Math. Soc., 1 (1988), pp. 587-646. [7] L. DANZER, D. LAUGWITZ, AND H. LENZ, Über das Lownersche Ellipsoid und sein A nalogon un ter den einem Eikorper einbeschriebenen Ellipsoiden, Arch. l\Iath. (Base!), 8 (1957), pp. 214-219. [8] K. P. DIAZ, The Szego kernel as a singular integral kernel on a family of weakly pseudoconvex domains, Trans. Amer. Math. Soc., 304 (1987), pp. 141-170. 111 [9] C. FEFFERMAN, The Bergman kernel and biholomorphic mappings of pseudoconvex dornains, Invent. i\lath., 26 (1974), pp. 1-65. [10] G. FRANCSICS AND l'i. HANGES, Explicit formulas Jor· the Szego kernel on certain weakly pseudoconvex domains, Proc. Amer. i\lath. Soc., 123 (1995), pp. 3161-3168. [11] M. GJLLIAM AND J. HALFPAP, The Szego kernel for certain non-pseudoconvex 2 domains in IC , Illinois J. i\lath., 55 (2011), pp. 871-894 (2013). [12] P. GREINER AND E. STEIN, On the solvability of sorne differential operators of type Db, Proc. Internat. Conf. (Cortona, Italy, 1976-1977), (1978), pp. 106-16.5. [13] J. HALFPAP, A. NAGEL, AND S. WAINGER, The Bergman and Szego kernels near points of infinite type, Pacific J. :\lath., 246 (2010), pp. 75-128. 2 [14] F. HASLINGER, Szego kernels for certain unboundcd domains in C , Rcv. Roumainc Math. Purcs Appl., 39 (1994), pp. 939-950. Travaux de la Conférence Internationale d'Analyse Cornplexe et du 7e Séminaire Rournano-Finlandais (1993). [15] --, Singularities of the Szego kernel for certain weakly pseudoconvex domains in 2 C , J. Funct. Anal., 129 (1995), pp. 406-427. [16] J\I. llENK, Lowner-John ellipsoids, Doc. Math., (2012), pp. 95-106. [17] F. J OHN, Extremum problems with inequalities as subsidiary conditions, in Stud ies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publishers, Inc., :\'ew York, N. Y., 1948, pp. 187-204. [18] S. G. KRANTZ, Function theory of severa! complex variables, AMS Chelsea Pub lishing, Providence, RI, 2001. Reprint of the 1992 edition. 112 [19] M. MACHEDON, Szego kemels on pseudoconvex domains with one degenerate eigen value, Ann. of Math. (2), 128 (1988), pp. 619-640. [20] J. D. McNEAL, Estimates on the Bergman kemels of convex domains, Adv. Math., 109 (1994), pp. 108-139. [21] J. D. M eNEAL ANDE. M. S TE IN, The Szego projection on convex domains, Math. z., 224 (1997), pp. 519-553. [22] J. l'vl!TCHELL, Two-sided LP-estimates (p > 1) for the Szego kernel in matrix spaces with application toa mapping theorem, Complex Variables Theory Appl., 18 (1992), pp. 73-77. [23] A. NAGEL, Vector fields and nonisotropic metrics, in Beijing lectures in harmonic analysis (Beijing, 1984), vol. 112 of Ann. of l'vlath. Stud., Princeton Univ. Press, Princeton, KJ, 1986, pp. 241-306. [24] A. KAGEL, J.-P. ROSAY, E. M. STEIN, AND S. WAINGER, Estimates for the Bergman and Szego kernels in certain weakly pseudoconvex domains, Bu!!. Amer. Math. Soc. (N.S.), 18 (1988), pp. 55-59. [25] A. NAGEL, E. M. STEIN, AND S. WAINGER, Balls and metrics defined by vector fields. J. Basic properties, Acta Math., 155 (1985), pp. 103-147. [26] J.-D. PARK, New formulas of the Bergman kemels for complex ellipsoids in rC 2 , Proc. Amer. Math. Soc., 136 (2008), pp. 4211-4221. [27] --, Explicit formulas of the Bergman kernel for 3-dimensional complex ellipsoids, J. Math. Anal. Appl., 400 (2013), pp. 664-674. 113 [28] D. H. PHONG ANO E. M. STEIN, Estimates for the Bergman and Szego projections on strongly pseudo-convex domains, Duke lvlath. J., 44 (1977), pp. 695-704. [29] E. M. STEIN, Boundary behavior of holomorphic functions of seveml complex vari ables, Princeton l'niversity Press, Princeton, :\.J., 1972. Mathematical Notes, No. 11. [30] G. SZEGÓ, Über orthogonale Polynome, die zu ciner gegebenen Kurve der kom plexen Ebene gehoren, lvlath. Z., 9 (1921), pp. 218-270. [31] V. L. ZAGUSKIN, Circumscr-ibed and inscribed ellipsoids of extrema/ volume, Uspehi l\lat. Nauk, 13 (1958), pp. 89-93.