Some Hilbert spaces related with the Dirichlet space

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1 Concr. Oper. 26; 3 94 Concrete Operators Open Access Research Article Nicola Arcozzi*, Pavel Mozolyako, Karl-Mikael Perfekt, Stefan Richter, and Giulia Sarfatti Some Hilbert spaces related with the irichlet space OI.55/conop-26- Received ecember 23, 25; accepted May 6, 26. Abstract We study the reproducing kernel Hilbert space with kernel k d, where d is a positive integer and k is the reproducing kernel of the analytic irichlet space. Keywords irichlet space, Complete Nevanlinna Property, Hilbert-Schmidt operators, Carleson measures MSC 3H25, 47B35 Introduction Consider the irichlet space on the unit disc fz 2 C W jzj < g of the complex plane. It can be defined as the Reproducing Kernel Hilbert Space (RKHS) having kernel k z.w/ k.w; z/ zw log zw n.zw/ n n C We are interested in the spaces d having kernel k d, with d 2 N. d can be thought of in terms of function spaces on polydiscs, following ideas of Aronszajn [4]. To explain this point of view, note that the tensor d-power d of the irichlet space has reproducing kernel k d.z ; ; z d I w ; ; w d / d j k.z j ; w j /. Hence, the space of restrictions of functions in d to the diagonal z z d has the reproducing kernel k d, and therefore coincides with d. We will provide several equivalent norms for the spaces d and their dual spaces in Theorem.. Then we will discuss the properties of these spaces. More precisely, we will investigate d and its dual space HS d in connection with Hankel operators of Hilbert-Schmidt class on the irichlet space ; the complete Nevanlinna-Pick property for d ; 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 d. *Corresponding Author Nicola Arcozzi Università di Bologna, ipartimento di Matematica, Piazza di Porta S.onato 5, Bologna, nicola.arcozzi@unibo.it Pavel Mozolyako Chebyshev Lab at St. Petersburg State University, 4th Line 29B, Vasilyevsky Island, St. Petersburg 9978, Russia, pmzlcroak@gmail.com Karl-Mikael Perfekt epartment of Mathematical Sciences, Norwegian University of Science and Technology (NTNU), NO-749 Trondheim, Norway, karl-mikael.perfekt@math.ntnu.no Stefan Richter epartment of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA, richter@math.utk.edu Giulia Sarfatti Istituto Nazionale di Alta Matematica F. Severi, Città Universitaria, Piazzale Aldo Moro 5, 85 Roma, and Institut de Mathématiques de Jussieu, Université Pierre et Marie Curie, 4, place Jussieu, F Paris, France, giulia.sarfatti@imj-prg.fr 26 Arcozzi et al., published by e Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-Noerivs 3. License. ownload ate /9/6 54 PM

2 Some Hilbert spaces related with the irichlet space 95 Note that the spaces d live infinitely close to in the scale of weighted irichlet spaces s Q, defined by the norms C k'k 2 ˇ ˇ'.e Q it ˇ dt / ˇ2 s 2 C ˇ ˇ'.z/ˇˇ2. jzj 2 / s da.z/ ; s < ; where da.z/ jzj< is normalized area measure on the unit disc. Notation We use multiindex notation. If n.n ; ; n d / belongs to N d, then jnj n C C n d. We write A B if A and B are quantities that depend on a certain family of variables, and there exist independent constants < c < C such that ca B CA. Equivalent norms for the spaces d and their dual spaces HS d Theorem.. Let d be a positive integer and let a d.k/ jnjk Then the norm of a function '.z/ P k b'.k/zk in d is.n C /.n d C / k'k d! =2 a d.k/ jb'.k/j 2 Œ' d ; () k where Œ' d! =2 k C log d.k C 2/ jb'.k/j2 (2) k An equivalent Hilbert norm jœ' j d Œ' d for ' in terms of the values of ' is given by jœ' j d j'./j 2 efine now the holomorphic space HS d by the norm j'.z/j 2 log d jzj 2 da.z/ A =2 (3) k k HSd =2.k C / 2 ˇ ˇ a d.k/ ˇb.k/ ˇ2! (4) k Then, HS d. d / is the dual space of d under the duality pairing of. Moreover, k k HSd Œ HSd W Furthermore, the norm can be written as =2.k C / log d ˇ ˇ.k C 2/ ˇb.k/ ˇ2! k jœ j HSd 2 C j.z/j 2 log d k k 2 HS d.n ;;n d / where fe n g n is the canonical orthonormal basis of, e n.z/ da.z/ A jzj 2 =2 (5) jhe n e nd ; i j 2 ; (6) p zn nc. ownload ate /9/6 54 PM

3 96 N. Arcozzi et al. The remainder of this section is devoted to the proof of Theorem.. The expression for k'k d in () follows by expanding.k z / d as a power series. The equivalence k'k d Œ' d, as well as k'k HSd Œ' HSd, are 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.2. For each d 2 N there are constants c; C > such that for all k we have Consequently, if t 2.; /, then c t log d t ca d.k/ logd.k C 2/ k C k Ca d.k/ log d.k C 2/ t k C k C t log d t Proof of Lemma.2. We will prove the Lemma by induction on d 2 N. It is obvious for d. Thus let d 2 and suppose the lemma is true for d. Also we observe that there is a constant c > such that for all k and n k we have Then for k c log d 2.k C 2/ C log d 2.k n C 2/ 2 log d 2.k C 2/ a d.k/ 2 n CCn d k k n k n n C.n C /.n d C / n 2 CCn d k log d 2.k n C 2/ n C k n C k n log d 2.k C 2/ logd 2.k C 2/ k C 2 logd.k C 2/ k C n.n 2 C /.n d C / C log d 2.k n C 2/.n C /.k n C / k n k n.n C /.k n C / n C C k n C Next, we prove the equivalence Œ' HSd jœ' j HSd which appears in (5). Lemma.3. Let d 2 N. Then t k t log t by the inductive assumption by the earlier observation d dt logd.k C 2/ ; k d k C Given the Lemma, we expand jœ j 2 HS d jb./j 2 C jb./j 2 C b.k/kz k ˇ k k ˇˇˇb.k/ 2 ˇ k ˇ2 ˇˇˇˇˇ 2 log d da.z/ jzj 2 log d t t k dt ownload ate /9/6 54 PM

4 Some Hilbert spaces related with the irichlet space 97 obtaining the desired conclusion. jb./j 2 C Œ 2 HS d ; k ˇˇˇb.k/ 2 ˇ log ˇ2 d.k C 2/ k C k Proof of Lemma.3. The case d is obvious, leaving us to consider d 2. We will also assume that k 2 Then by Lemma.2 we have t k t log d t dt t k n t n dt n C n.n C /.n C k C / S C S 2 ; where and S k n.n C /.n C k C / k k C n C k C n log d.k C 2/ d k C kc2 log d 2.t/ dt t S 2 nk j.n C /.n C k C / nkc log d 2.n C / n 2 k j C.j C / d 2 log d 2 k n 2 logd 2.k C 2/ logd 2.k C 2/ k C logd 2.k C 2/ k C j nk j j k j C nk j.j C / d 2 j log d 2.n C / n 2 k j x 2 dx.j C / d 2 k C k j logd 2.k C 2/.j C / d 2 k C k C.k /k j j!.j C / d o logd.k C 2/ j k C j Now, the duality between d and HS d under the duality pairing given by the inner product of is easily seen by considering Œ d and Œ HSd. They are weighted `2 norms and duality is established by means of the Cauchy-Schwarz inequality. Next we will prove that Œ' d jœ' j d. This is equivalent to proving that the dual space of HS d, with respect to the irichlet inner product h ; i, is the Hilbert space with the norm jœ j d. Let d 2 N and set, for < t <, w d.t/ t log t d and, for < jzj <, Wd.z/ w d.jzj 2 / and W d./. Lemma.4. Let d 2 N. Then " w d.t/dt " w d.t/ dt "2 as "! Proof. Write Qw.t/.log t /d, and note that it suffices to establish the lemma for Qw in place of w d. Let " >. Then Qw is increasing in.; / and Qw. " kc /.k C / d.log " /d, hence " Qw.t/dt k " kc " k Qw.t/dt k Qw. " kc /." k " kc / ownload ate /9/6 54 PM

5 98 N. Arcozzi et al..k C / d.log " /d " k. "/ ".log " /d. "/ d k For = Qw we just notice that it is decreasing and hence Thus as "! we have " " 2 Qw.t/ dt Qw. "/ " ".log " /d " Qw.t/dt " Qw.t/ dt O."2 / For < h < and s 2 Œ ; / let S h.e is / be the Carleson square at e is, i.e. S h.e is / fre it W h < r < ; jt sj < hg A positive function W on the unit disc is said to satisfy the Bekollé-Bonami condition (B2) if there exists c > such that W da da ch4 W S h.e is / S h.e is / for every Carleson square S h.e is /. If d 2 N and if W d.z/ is defined as above, then S h.e is / W d da S h.e is / W d da h 2 h w d.t/dt h dt h4 w d.t/ by Lemma.4, at least if < h < =2. Observe that both W d and =W d are positive and integrable in the unit disc, hence it follows that the estimate holds for all < h. Thus W d satisfies the condition (B2). Furthermore, note that if f.z/ P O k f.k/z k is analytic in the open unit disc, then jf.z/j 2 w d.jzj 2 / da.z/ w k jf O.k/j 2 ; jzj< where w k R t k w d.t/dt logd.kc2/. kc A special case of Theorem 2. of Luecking s paper [7] says that if W satisfies the condition (B2) by Bekollé and Bonami [5], then one has a duality between the spaces L 2 a.w da/ and L2 a. da/ with respect to the pairing given W by R jzj< f gda. Thus, we have jzj< jg.z/j 2 da sup W d.z/ f k ˇ ˇR k da.z/ ˇ jzj< g.z/f.z/ ˇ2 Rjzj< jf.z/j2 W d.z/da sup f.k C / 2 w k j Og.k/j 2 ˇ ˇP k Og.k/ p.kc/ p wk w k P k w kj O f.k/j 2 This finishes the proof of (5). It remains to demonstrate (6). We defer its proof to the next section. By Theorem. we have the following chain of inclusions f O ˇ.k/,! HS dc,! HS d,!,! HS 2,! HS,! 2,!,! d,! dc,! with duality w.r.t. linking spaces with the same index. It might be interesting to compare this sequence with the sequence of Banach spaces related to the irichlet spaces studied in [3]. Note that for d 3 the reproducing kernel of HS d is continuous up to the boundary. Hence functions in HS d extend continuously to the closure of the unit disc, for d 3. ˇ2 ownload ate /9/6 54 PM

6 Some Hilbert spaces related with the irichlet space 99 Hilbert-Schmidt norms of Hankel-type operators Let fe n g be the canonical orthonormal basis of, e n.z/ k jnjk jhe n e nd ; ij 2 k jnjk k jnjk.n C /.n d C / jhzk ; ij 2 p zn nc. Equation (6) follows from the computation.n C /.n d C / jhzn z n d ; ij 2 k jnjk.k C /a d.k/j O.k/j 2 k.k C / 2.n C /.n d C / j O.k/j 2 Polarizing this expression for k k HSd, the inner product of HS d can be written h ; 2 i HSd h ; e n e nd i he n e nd ; 2 i.n ;;n d / k log d.k C 2/ j O.k/j 2 k C Hence, for any ; 2, hk ; k i HSd hk ; e n e nd i he n e nd ; k i e n./ e nd./e n./ e nd./ n2n d n2n d! d e i./e i./ k./ d hk d ; kd i d That is, i Proposition.5. The map U W k 7! k d extends to a unitary map HS d! d. When d 2, HS 2 contains those functions b for which the Hankel operator H b W!, defined by hh b e j ; e k i he j e k ; bi, belongs to the Hilbert-Schmidt class. 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 T b W! defined by E T b f; g h hfgh; bi The formula uniquely defines an operator, whose action is T b f.z; w/ ht b f; k z k w i hf k z k w ; bi f O z n w m.j / n C m C hncmcj ; bi n;m;j n;m;j f O.j / b.n O C m C j / n C m C j C.n C /.m C / zn w m Then, the Hilbert-Schmidt norm of T b is ˇ ˇhTb e l ; e m e n i ˇˇ2 jhe l e m e n ; bi j 2 kbkhs 2 3 l;m;n l;m;n Similarly, we can consider U b W! defined by E U b.f g/; h hfgh; bi ownload ate /9/6 54 PM

7 N. Arcozzi et al. The action of this operator is given by U b.f g/.z/ l;m;n The Hilbert-Schmidt norm of U b is still kbk HS3. bf.l/bg.m/.l C m C n C /b b.l C m C n/ z n n C Carleson measures for the spaces d and HS d The (B2) condition allows us to characterize the Carleson measures for the spaces d and HS d. Recall that a nonnegative Borel measure on the open unit disc is Carleson for the Hilbert function space H if the inequality jf j 2 d C./kf kh 2 jzj< holds with a constant C./ which is independent of f. The characterization [2] shows that, since the (B2) condition holds, then Theorem.6. For d 2 N, a measure on fjzj < g is Carleson for d if and only if for jaj < we have log d. jzj 2 /.S.z/ \ S.a// 2 dxdy jzj 2. jzj 2 / C./.S.a//; 2 QS.a/ where S.a/ fz W < jzj < jaj; j arg.za/j < jajg is the Carleson box with vertex a and QS.a/ fz W < jzj < 2. jaj/; j arg.za/j < 2. jaj/g is its dilation. The characterization extends to HS 2, with the weight log. Since functions in HS d are continuous for d 3, all finite measures are Carleson measures for these spaces. Once we know the Carleson measures, we can characterize the multipliers for d in a standard way. jzj 2 The complete Nevanlinna-Pick property for d Next, we prove that the spaces d 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 [] 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 W! C has the CNP property if there is a positive definite function F W! and a nowhere vanishing function ı W! C such that k.x; y/ ı.x/ı.y/ F.x; y/ whenever x; y lie in. The CNP property is a property of the kernel, not of the Hilbert space itself. Theorem.7. There are norms on d such that the CNP property holds. Proof. A kernel k W! C of the form k.w; z/ P k a k.zw/ k has the CNP property if a and the sequence fa n g n is positive and log-convex an a n a n a nc ownload ate /9/6 54 PM

8 Some Hilbert spaces related with the irichlet space See [], Theorem 7.33 and Lemma Consider.x/ log log.x/, which is positive for x M, depending on. Let now log 2.x/ log.x/ x 2 log 2.x/ log.x/, with real. Then,.x/ a n logd.m d.n C // log.m d /.n C / n C C logd.n C / n C (7) Then, the sequence fa n g n provides the coefficients for a kernel with the CNP property for the space d. The CNP property has a number of consequences. For instance, we have that the space d and its multiplier algebra M. d / have the same interpolating sequences. Recall that a sequence fz n g n is interpolating for a RKHS n H with reproducing kernel k H if the weighted restriction map R W ' 7! maps H boundedly '.z n / k H.z n ;z n / =2 o n onto `2; while is interpolating for the multiplier algebra M.H / if Q W 7! f.z n /g n maps M.H / boundedly onto `. The reader is referred to [] and to the second chapter of [8] for more on this topic. It is a reasonable guess that the universal interpolating sequences for d and for its multiplier space M. d / 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 d. Acknowledgement This work was supported by the LABE MILYON (ANR--LAB-7) of Université de Lyon, within the program Investissements d Avenir (ANR--IE-7) 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 ifferential Geometry and Geometric Function Theory of the Italian MIUR. References [] J. Agler, J. E. McCarthy, Pick interpolation and Hilbert function spaces, Graduate Studies in Mathematics, 44 (American Mathematical Society, Providence, RI, 22), xx+38 pp. ISBN [2] N. Arcozzi, R. Rochberg, E. Sawyer, Carleson measures for analytic Besov spaces, Rev. Mat. Iberoamericana 8 (22), no. 2, [3] N. Arcozzi, R. Rochberg, E. Sawyer, B.. Wick, Function spaces related to the irichlet space, J. Lond. Math. Soc. (2) 83 (2), no., -8. [4] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 68, (95) [5]. Bekollé, A. Bonami, Inégalités à poids pour le noyau de Bergman, C. R. Acad. Sci. Paris Sér. A-B 286 (978), no. 8, A775- A778 (in French). [6] B. Boe, An interpolation theorem for Hilbert spaces with Nevanlinna-Pick kernel, Proc. Amer. Math. Soc. 33 (25), no. 7, [7]. H. Luecking, Representation and duality in weighted spaces of analytic functions, Indiana Univ. Math. J. 34 (985), no. 2, [8] K. Seip, Interpolation and sampling in spaces of analytic functions, University Lecture Series, 33 (American Mathematical Society, Providence, RI, 24), xii+39 pp. ISBN ownload ate /9/6 54 PM