Some Hilbert spaces related with the Dirichlet space
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Arcozzi, N., Mozolyako, P., Perfekt, K.-M., Richter, S. and Sarfatti, G. (2016) Some Hilbert spaces related with the Dirichlet space. Concrete Operators, 3 (1). 0011. ISSN 2299- 3282 doi: https://doi.org/10.1515/conop-2016-0011 Available at http://centaur.reading.ac.uk/71561/
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Reading’s research outputs online Concr. Oper. 2016; 3: 94–101
Concrete Operators Open Access
Research Article
Nicola Arcozzi*, Pavel Mozolyako, Karl-Mikael Perfekt, Stefan Richter, and Giulia Sarfatti Some Hilbert spaces related with the Dirichlet space
DOI 10.1515/conop-2016-0011 Received December 23, 2015; accepted May 16, 2016.
Abstract: We study the reproducing kernel Hilbert space with kernel kd , where d is a positive integer and k is the reproducing kernel of the analytic Dirichlet space.
Keywords: Dirichlet space, Complete Nevanlinna Property, Hilbert-Schmidt operators, Carleson measures
MSC: 30H25, 47B35
1 Introduction
Consider the Dirichlet space D on the unit disc z C z < 1 of the complex plane. It can be defined as the f 2 W j j g Reproducing Kernel Hilbert Space (RKHS) having kernel
n 1 1 X1 .zw/ kz.w/ k.w; z/ log : D D zw 1 zw D n 1 n 0 D C d We are interested in the spaces Dd having kernel k , with d N. Dd can be thought of in terms of function spaces 2 d on polydiscs, following ideas of Aronszajn [4]. To explain this point of view, note that the tensor d-power D˝ d of the Dirichlet space has reproducing kernel kd .z1; ; zd w1; : : : ; wd / …j 1k.zj ; wj /. Hence, the space d I D D d of restrictions of functions in D to the diagonal z1 zd has the reproducing kernel k , and therefore ˝ D D coincides with Dd . We will provide several equivalent norms for the spaces Dd and their dual spaces in Theorem 1.1. Then we will discuss the properties of these spaces. More precisely, we will investigate:
– Dd and its dual space HSd in connection with Hankel operators of Hilbert-Schmidt class on the Dirichlet space D;
– the complete Nevanlinna-Pick property for Dd ; – the Carleson measures for these spaces.
Concerning the first item, the connection with Hilbert-Schmidt Hankel operators served as our original motivation for studying the spaces Dd .
*Corresponding Author: Nicola Arcozzi: Università di Bologna, Dipartimento di Matematica, Piazza di Porta S.Donato 5, Bologna, E-mail: [email protected] Pavel Mozolyako: Chebyshev Lab at St. Petersburg State University, 14th Line 29B, Vasilyevsky Island, St. Petersburg 199178, Russia, E-mail: [email protected] Karl-Mikael Perfekt: Department of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway, E-mail: [email protected] Stefan Richter: Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA, E-mail: [email protected] Giulia Sarfatti: Istituto Nazionale di Alta Matematica “F. Severi”, Città Universitaria, Piazzale Aldo Moro 5, 00185 Roma, and Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, place Jussieu, F-75252 Paris, France, E-mail: [email protected]
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Note that the spaces D live infinitely close to D in the scale of weighted Dirichlet spaces D , defined by the d Q s norms ZC 2 Z 2 ˇ it ˇ dt ˇ ˇ2 2 s dA.z/ ' ˇ'.e /ˇ ˇ'0.z/ˇ .1 z / ; 0 s < 1; Ds ˇ ˇ k k Q D 2 C j j Ä z <1 j j dA.z/ where is normalized area measure on the unit disc. d Notation: We use multiindex notation. If n .n1; : : : ; nd / belongs to N , then n n1 nd . We write D j j D C C A B if A and B are quantities that depend on a certain family of variables, and there exist independent constants 0 < c < C such that cA B CA. Ä Ä
Equivalent norms for the spaces Dd and their dual spaces HSd
Theorem 1.1. Let d be a positive integer and let
X 1 ad .k/ : D .n1 1/ : : : .nd 1/ n k j jD C C P k Then the norm of a function '.z/ k1 0 b'.k/z in Dd is D D !1=2 X1 1 2 ' D ad .k/ '.k/ Œ'd ; (1) k k d D jb j k 0 D where !1=2 X1 k 1 2 Œ'd C '.k/ : (2) D d 1 jb j k 0 log .k 2/ D C An equivalent Hilbert norm Œ' d Œ'd for ' in terms of the values of ' is given by j j 0 11=2 Z 2 2 1 dA.z/ Œ' d '.0/ @ '0.z/ A : (3) j j D j j C j j d 1 1 Á log 1 z 2 D j j
Define now the holomorphic space HSd by the norm: !1=2 ˇ ˇ2 X1 2 ˇ ˇ HSd .k 1/ ad .k/ b.k/ : (4) k k D C ˇ ˇ k 0 D
Then, HSd .Dd / is the dual space of Dd under the duality pairing of D. Moreover, Á !1=2 ˇ ˇ2 X1 d 1 ˇ ˇ HSd Œ HSd .k 1/ log .k 2/ b.k/ k k WD C C ˇ ˇ k 0 0 D 11=2 Z  à 2 2 d 1 1 dA.z/ Œ HSd @ .0/ 0.z/ log A : (5) j j WD j j C j j 1 z 2 D j j Furthermore, the norm can be written as
2 X 2 en : : : en ; D ; (6) k kHSd D jh 1 d i j .n1;:::;nd /
zn where en is the canonical orthonormal basis of D, en.z/ . n1 0 pn 1 f g D D C
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The remainder of this section is devoted to the proof of Theorem 1.1. The expression for ' Dd in (1) follows d k k by expanding .kz/ as a power series. The equivalence ' D Œ'd , as well as ' HS Œ'HS , are k k d k k d d consequences of the following lemma. We denote by c; C positive constants which are allowed to depend on d only, whose precise value can change from line to line.
Lemma 1.2. For each d N there are constants c; C > 0 such that for all k 0 we have 2 d 1 log .k 2/ cad .k/ C Cad .k/: Ä k 1 Ä C Consequently, if t .0; 1/, then 2  Ãd d 1  Ãd 1 1 X1 log .k 2/ 1 1 c log C t k C log : t 1 t Ä k 1 Ä t 1 t k 0 D C Proof of Lemma 1.2. We will prove the Lemma by induction on d N. It is obvious for d 1. Thus let d 2 2 D and suppose the lemma is true for d 1. Also we observe that there is a constant c > 0 such that for all k 0 and 0 n k we have Ä Ä c logd 2.k 2/ logd 2.n 2/ logd 2.k n 2/ 2 logd 2.k 2/: C Ä C C C Ä C Then for k 0 X 1 ad .k/ D .n1 1/ : : : .nd 1/ n1 nd k CC D C C k X 1 X 1 D n 1 .n2 1/ : : : .nd 1/ n 0 n2 nd k n D C CC D C C k d 2 X 1 log .k n 2/ C by the inductive assumption n 1 k n 1 n 0 D C C k d 2 d 2 1 X log .n 2/ log .k n 2/ C C C D 2 .n 1/.k n 1/ n 0 D C C k X 1 logd 2.k 2/ by the earlier observation C .n 1/.k n 1/ n 0 D C C d 2 k log .k 2/ X 1 1 C D k 2 n 1 C k n 1 n 0 C D C C logd 1.k 2/ C : k 1 C Next, we prove the equivalence Œ'HS Œ' HS which appears in (5). d j j d Lemma 1.3. Let d N. Then 2 1  Ãd 1 d 1 Z 1 1 log .k 2/ t k log dt C ; k d: t 1 t k 1 0 C
Given the Lemma, we expand
Z ˇ ˇ2 2 2 ˇ X1 k 1ˇ d 1 1 dA.z/ Œ b.0/ ˇ b.k/kz ˇ log j jHSd D j j C ˇ ˇ 1 z 2 ˇk 1 ˇ D D j j 1 2 Z 2 X1 2 ˇ ˇ d 1 1 k 1 b.0/ k ˇ b.k/ˇ log t dt D j j C ˇ ˇ 1 t k 1 D 0
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2 d 1 2 X1 2 ˇ ˇ log .k 2/ b.0/ k ˇ b.k/ˇ C j j C ˇ ˇ k 1 k 1 C Œ 2 ; D HSd obtaining the desired conclusion.
Proof of Lemma 1.3. The case d 1 is obvious, leaving us to consider d 2. We will also assume that k 2: D Then by Lemma 1.2 we have
1 1 Z Â Ãd 1 Z d 2 d 2 k 1 1 k X1 log .n 2/ n X1 log .n 2/ t log dt t C t dt C S1 S2; t 1 t n 1 D .n 1/.n k 1/ D C n 0 n 0 0 0 D C D C C C where
k 2 k 1 d 2 k 1 d 2 ZC d 2 X log .n 2/ 1 X log .n 2/ 1 log .t/ S1 C C dt D .n 1/.n k 1/ k 1 n 1 k 1 t n 0 n 0 D C C C C D C C 1 1 logd 1.k 2/ C D d 1 k 1 C and
j 1 d 2 d 2 k C 1 d 2 X1 log .n 2/ X1 log .n 1/ X1 X log .n 1/ S2 C C C D .n 1/.n k 1/ Ä n2 Ä n2 n k C C C n k 1 j 1 n kj D D C D D kj 1 1 X1 XC 1 X1 Z1 1 .j 1/d 2 logd 2 k logd 2.k 2/ .j 1/d 2 dx n2 x2 Ä C j Ä C C j 1 n k j 1 j D D k 1 D d 2 d 2 log .k 2/ X1 k 1 log .k 2/ X1 k 1 C .j 1/d 2 C C .j 1/d 2 C D k 1 C kj 1 Ä k 1 C .k 1/kj 1 j 1 j 1 C D C D d 2 d 1 ! log .k 2/ X1 3 log .k 2/ C .j 1/d 2 o C : Ä k 1 C 2j 1 D k 1 j 1 C D C
Now, the duality between Dd and HSd under the duality pairing given by the inner product of D is easily seen by 2 considering Πd and ΠHS . They are weighted ` norms and duality is established by means of the Cauchy-Schwarz d inequality.
Next we will prove that Œ'd Œ' d . This is equivalent to proving that the dual space of HSd , with respect j j to the Dirichlet inner product ; D, is the Hilbert space with the norm Œ d . h i 1 1 d j j 2 Let d N and set, for 0 < t < 1, wd .t/ t log 1 t and, for 0 < z < 1, Wd .z/ wd . z / and 2 D j j D j j Wd .0/ 1. D Lemma 1.4. Let d N. Then 2 1 1 Z Z 1 2 wd .t/dt dt " as " 0: wd .t/ ! 1 " 1 " 1 d Proof. Write w.t/ .log / , and note that it suffices to establish the lemma for w in place of wd . Let " > 0. Q D 1 t Q Then w is increasing in .0; 1/ and w.1 "k 1/ .k 1/d .log 1 /d , hence Q Q C D C "
k 1 1 1 " C Z X1 Z X1 w.t/dt w.t/dt w.1 "k 1/."k "k 1/ Q D Q Ä Q C C k 1 k k 1 1 " D 1 " D
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X1 1 1 1 .k 1/d .log /d "k.1 "/ ".log /d D C " " .1 "/d k 1 D For 1=w we just notice that it is decreasing and hence Q 1 Z 1 1 " dt " w.t/ w.1 "/ 1 d Ä D .log " / 1 " Q Q Thus as " 0 we have ! 1 1 Z Z 1 "2 w.t/dt dt O."2/: Ä Q w.t/ D 1 " 1 " Q
is is For 0 < h < 1 and s Œ ; / let Sh.e / be the Carleson square at e , i.e. 2 is it Sh.e / re 1 h < r < 1; t s < h : D f W j j g A positive function W on the unit disc is said to satisfy the Bekollé-Bonami condition (B2) if there exists c > 0 such that Z Z 1 W dA dA ch4 W Ä is is Sh.e / Sh.e / is for every Carleson square Sh.e /. If d N and if Wd .z/ is defined as above, then 2 1 1 Z Z Z Z 1 2 1 4 Wd dA dA h wd .t/dt dt h Wd D wd .t/ is is Sh.e / Sh.e / 1 h 1 h by Lemma 1.4, at least if 0 < h < 1=2. Observe that both Wd and 1=Wd are positive and integrable in the unit disc, hence it follows that the estimate holds for all 0 < h 1. Ä P k Thus Wd satisfies the condition (B2). Furthermore, note that if f .z/ 1 f .k/z is analytic in the open D k 0 O unit disc, then D Z 2 2 dA.z/ X1 2 f .z/ wd . z / wk f .k/ ; j j j j D j O j z <1 k 0 j j D R 1 k logd .k 2/ where wk t wd .t/dt C . D 0 k 1 A special case of Theorem 2.1 ofC Luecking’s paper [7] says that if W satisfies the condition (B2) by Bekollé and 2 2 1 Bonami [5], then one has a duality between the spaces La.W dA/ and La. W dA/ with respect to the pairing given R by z <1 f gdA. Thus, we have j j 2 2 ˇR dA.z/ ˇ ˇP g.k/ ˇ Z ˇ g.z/f .z/ ˇ ˇ k1 0 O pwkf .k/ˇ 2 1 ˇ z <1 ˇ ˇ .k 1/pwk O ˇ g.z/ dA sup j j sup D C W .z/ R f .z/ 2W .z/dA P 2 j j d f 0 z <1 d D f 0 k1 0 wk f .k/ z <1 ¤ j j j j ¤ D j O j j j X1 1 2 2 g.k/ D .k 1/ wk j O j k 0 D C This finishes the proof of (5). It remains to demonstrate (6). We defer its proof to the next section. By Theorem 1.1 we have the following chain of inclusions:
:::, HSd 1 , HSd , :::, HS2 , HS1 D D1 , D2 , :::, Dd , Dd 1 , ::: ! C ! ! ! ! D D ! ! ! ! C ! with duality w.r.t. D linking spaces with the same index. It might be interesting to compare this sequence with the sequence of Banach spaces related to the Dirichlet spaces studied in [3]. Note that for d 3 the reproducing kernel of HSd is continuous up to the boundary. Hence functions in HSd extend continuously to the closure of the unit disc, for d 3.
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Hilbert-Schmidt norms of Hankel-type operators
zn Let en be the canonical orthonormal basis of D, en.z/ . Equation (6) follows from the computation pn 1 f g D C
1 1 1 X X 2 X X n1 nd 2 en1 :::end ; z :::z ; jh ij D .n1 1/ ::: .nd 1/jh ij k 0 n k k 0 n k D j jD D j jD C C 2 X1 X 1 X1 X .k 1/ zk; 2 C .k/ 2 D .n1 1/ ::: .nd 1/jh ij D .n1 1/ ::: .nd 1/j O j k 0 n k k 0 n k D j jD C C D j jD C C d 1 X1 2 X1 log .k 2/ 2 .k 1/ad .k/ .k/ C .k/ : D C j O j k 1 j O j k 0 k 0 D D C
Polarizing this expression for HS , the inner product of HSd can be written k k d X 1; 2 HS 1; en : : : en D en : : : en ; 2 D: h i d D h 1 d i h 1 d i .n1;:::;nd /
Hence, for any ; D, 2 X X k; k HSd k; en1 : : : end D en1 : : : end ; k D en1 ./ : : : end ./en1 ./ : : : end ./ h i D d h i h i D d n N n N 2 2 !d X1 d d d ei ./ei ./ k./ k ; k D : D D D h i d i 0 D That is,
d Proposition 1.5. The map U k k extends to a unitary map HSd Dd . W 7! !
When d 2, HS2 contains those functions b for which the Hankel operator Hb D D, defined by D W ! Hbej ; ek ej ek; b D, belongs to the Hilbert-Schmidt class. h iD D h i Analogous interpretations can be given for d 3, but then function spaces on polydiscs are involved. We consider the case d 3, which is representative. Consider first the operator Tb D D D defined by D W ! ˝ D E Tbf; g h fgh; b D : ˝ D D D h i ˝ The formula uniquely defines an operator, whose action is
Tbf .z; w/ Tbf; kzkw D D D h i ˝ f kzkw; b D D h i n m X z w n m j f .j / ; b D D O n 1 m 1h C C i n;m;j C C X n m j 1 f .j /b.n m j / C C C znwm D O O C C .n 1/.m 1/ n;m;j C C
Then, the Hilbert-Schmidt norm of Tb is:
X ˇ ˇ2 X 2 2 ˇ Tbel ; emen ˇ el emen; b b : h iD D D jh iDj D k kHS3 l;m;n ˝ l;m;n
Similarly, we can consider Ub D D D defined by W ˝ ! D E Ub.f g/; h fgh; b D : ˝ D D h i
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The action of this operator is given by
X1 .l m n 1/bb.l m n/ n Ub.f g/.z/ fb .l/g.m/ C C C C C z : ˝ D b n 1 l;m;n 0 D C
The Hilbert-Schmidt norm of Ub is still b HS . k k 3
Carleson measures for the spaces Dd and HSd
The (B2) condition allows us to characterize the Carleson measures for the spaces Dd and HSd . Recall that a nonnegative Borel measure on the open unit disc is Carleson for the Hilbert function space H if the inequality Z f 2d C./ f 2 j j Ä k kH z <1 j j holds with a constant C./ which is independent of f . The characterization [2] shows that, since the (B2) condition holds, then
Theorem 1.6. For d N, a measure 0 on z < 1 is Carleson for Dd if and only if for a < 1 we have: 2 fj j g j j Z Â Ã d 1 1 2 2 dxdy log .1 z /.S.z/ S.a// C1./.S.a//; 1 z 2 j j \ .1 z 2/2 Ä S.a/ j j j j Q where S.a/ z 0 < 1 z < 1 a ; arg.za/ < 1 a is the Carleson box with vertex a and S.a/ z D f W j j j j j j j jg Q D f W 0 < 1 z < 2.1 a /; arg.za/ < 2.1 a / is its “dilation”. j j j j j j j j g 1 1 Á The characterization extends to HS2, with the weight log 1 z 2 . Since functions in HSd are continuous for d 3, all finite measures are Carleson measures for these spaces. j j Once we know the Carleson measures, we can characterize the multipliers for Dd in a standard way.
The complete Nevanlinna-Pick property for Dd
Next, we prove that the spaces Dd have the Complete Nevanlinna-Pick (CNP) Property. Much research has been done on kernels with the CNP property in the past twenty years, following seminal work of Sarason and Agler. See the monograph [1] for a comprehensive and very readable introduction to this topic. We give here a definition which is simple to state, although perhaps not the most conceptual. An irreducible kernel k X X C has the CNP W ! property if there is a positive definite function F X D and a nowhere vanishing function ı X C such that: W ! W ! ı.x/ı.y/ k.x; y/ D 1 F .x; y/ whenever x; y lie in X. The CNP property is a property of the kernel, not of the Hilbert space itself.
Theorem 1.7. There are norms on Dd such that the CNP property holds.
P k Proof. A kernel k D D C of the form k.w; z/ k1 0 ak.zw/ has the CNP property if a0 1 and the W ! D D D sequence an n1 0 is positive and log-convex: f g D an 1 an : an Ä an 1 C
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See [1], Theorem 7.33 and Lemma 7.38. Consider .x/ ˛ log log.x/ log.x/, with real ˛. Then, 00.x/ log2.x/ ˛ log.x/ ˛ D D , which is positive for x M˛, depending on ˛. Let now x2 log2.x/
d 1 d 1 log .Md .n 1// 1 log .n 1/ an C C (7) D log.Md / .n 1/ n 1 C n 1 C C C
Then, the sequence an n1 0 provides the coefficients for a kernel with the CNP property for the space Dd . f g D The CNP property has a number of consequences. For instance, we have that the space Dd and its multiplier algebra
M.Dd / have the same interpolating sequences. Recall that a sequence Z zn n1 0 is interpolating for a RKHS Dn f g D o H '.zn/ 1 H with reproducing kernel k if the weighted restriction map R ' H 1=2 maps H boundedly W 7! k .zn;zn/ n 0 2 D onto ` ; while Z is interpolating for the multiplier algebra M.H / if Q .zn/ n1 0 maps M.H / boundedly W 7! f g D onto `1. The reader is referred to [1] and to the second chapter of [8] for more on this topic. It is a reasonable guess that the universal interpolating sequences for Dd and for its multiplier space M.Dd / are characterized by a Carleson condition and a separation condition, as described in [8] (see the Conjecture at p. 33). See also [6], which contains the best known result on interpolation in general RKHS spaces with the CNP property.
Unfortunately we do not have an answer for the spaces Dd .
Acknowledgement: This work was supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). The first author was partially supported by GNAMPA of INdAM, the fifth author was partially supported by GNSAGA of INdAM and by FIRB “Differential Geometry and Geometric Function Theory” of the Italian MIUR.
References
[1] J. Agler, J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44 (American Mathematical Society, Providence, RI, 2002), xx+308 pp. ISBN: 0-8218-2898-3. [2] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 18 (2002), no. 2, 443-510. [3] N. Arcozzi, R. Rochberg, E. Sawyer, B. D. Wick, Function spaces related to the Dirichlet space, J. Lond. Math. Soc. (2) 83 (2011), no. 1, 1-18. [4] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, (1950) 337-404. [5] D. Bekollé, A. Bonami, Inégalités à poids pour le noyau de Bergman, C. R. Acad. Sci. Paris Sér. A-B 286 (1978), no. 18, A775- A778 (in French). [6] B. Boe, An interpolation theorem for Hilbert spaces with Nevanlinna-Pick kernel, Proc. Amer. Math. Soc. 133 (2005), no. 7, 2077- 2081. [7] D. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (1985), no. 2, 319- 336. [8] K. Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, 33 (American Mathematical Society, Providence, RI, 2004), xii+139 pp. ISBN: 0-8218-3554-8.
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