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Estimates for the Szegó Kernel on Unbounded Convex Domains

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Estimates for the Szegó Kernel on Unbounded Convex Domains 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) <O} where pE C'''(en+l) is such that \1 p #O when p =O. A Cauchy-Riemann operator is an operator of the form 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¡], ..
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