DUALITY OF WEIGHTED BERGMAN SPACES WITH SMALL EXPONENTS

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1 Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 42, 2017, UALITY OF EIGHTE BERGMAN SPACES ITH SMALL EXPONENTS Antti Perälä and Jouni Rättyä University of Eastern Finland, eartment of Physics and Mathematics P. O. Box 111, FI Joensuu, Finland; University of Eastern Finland, eartment of Physics and Mathematics P. O. Box 111, FI Joensuu, Finland; Abstract. It is shown that for each radial doubling weight ω and 0 < < 1, the dual of the weighted Bergman sace A ω can be identified with the Bloch sace under the A2 -airing, where satisfies a two sided doubling roerty and deends on both ω and. This result generalizes the duality relation (A α B, studied by Shairo, Coifman, Rochberg and Zhu, concerning the classical weighted Bergman saces. 1. Introduction and the result LetH( denote the sace of analytic functions in the unit disc. An integrable function ω: [0, is called a weight. It is radial if ω(z ω( z for all z. For 0 < < and a radial weight ω, the weighted Bergman sace A ω consists of f H( such that f f(z ω(zda(z <. A ω As usual, we write A α if ω(z (1 z 2 α for 1 < α <. hen 1, the sace A ω is a Banach sace, while for < 1 the exression above defines only a quasi-norm, under which A ω is comlete. In this work, we will study the continuous dual of A ω, denoted by (A ω, consisting of all linear mas F: A ω C such that F(f C f A ω with C > 0 indeendent of f. Note that (A ω is always a Banach sace, even for < 1, when the sace A ω itself is not. It is classical that in the case of standard weights the dual of A α is isomorhic to A α when > 1, and (A 1 α B. Here B denotes the Bloch sace that consists of f H( such that f B su f (z (1 z 2 + f(0 <. z For > 1 and through a similar argument for 1, the duality can be roven roughly as follows. Take element F (A α, and extend it by the Hahn Banach theorem to an element of the dual of L α. Now, due to the existence of an isometric isomorhism (L α L α with 1/ + 1/ 1, we find g F L ω reresenting F. Finally, we roject g F to A ω and use the self-adjointness to get the desired duality. Now, if < 1, using the above aroach one immediately runs into trouble. First, A α is not locally convex, so the Hahn Banach theorem fails to hold altogether. Even htts://doi.org/ /aasfm Mathematics Subject Classification: Primary 30H20; Secondary 46A16. ey words: Bergman rojection, Bergman sace, Bloch sace, Carleson measure, doubling weight, duality, reroducing kernel estimate. This research was suorted in art by Academy of Finland roject no and Ministerio de Economía y Cometitivivad, Sain, roject MTM P.

2 622 Antti Perälä and Jouni Rättyä if one blindly assumes that Hahn Banach holds, another obstruction resents itself, namely (L α {0}. This clearly cannot be the way to roceed, as (A α is easily seen to contain non-zero elements, such as the oint-evaluations. In [10] Zhu showed that for 0 < < 1, the duality (A α B still holds under an aroriately chosen dual airing deending on α. In fact, as Zhu mentions, this result was obtained for the Bergman sace A in 1976 by Shairo [9], and for the weighted Bergman sace A α by Coifman and Rochberg [1] a few years later. The Hardy sace case was first dealt with by Romberg in his 1960 Ph thesis [8]. In this work, we will obtain analogous duality result for a class of Bergman saces generated by much more general weights. To state the result and to lace it in a more general context, some notation and background is needed. rite ω 1 if ω(z ω(tdt satisfies the doubling roerty ω(r C ω ( 1+r 2 z for all r [0,1. All the standard weights (1 z 2 α belong to, but this doubling roerty is also satisfied by the weights (1 z 1( log 1 z e β for β > 1 that induce essentially smaller Bergman saces than the standard weights. If ω and there exists (ω > 1 and C C(ω > 1 such that ω(r C ω ( 1 1 r for all 0 r < 1, then we denote ω. For basic roerties of these weights and more, see [3], [6] and [7]. If ω, then (A ω A ω for 1 < <, and (A1 ω B by [5, Corollary 7]. The reason why the roof of the first-mentioned result does not carry over the whole class is that the maximal Bergman rojection (f(z f(ζ Bz ω (ζ ω(ζda(ζ, P + ω where Bz ω stands for the reroducing kernel of the Hilbert sace A 2 ω, is not necessarily bounded on L ω for all ω by [5, Theorem 5]. Therefore as far as we know, the existing literature does not offer a descrition of the dual of A ω when ω. It is also worth mentioning that it is not known if the Bergman rojection P ω is bounded on L ω if ω \ and 1 < < exceting of course the trivial case 2. In this work we describe the dual of A ω for 0 < < 1 and ω. To define the aroriate dual airing we define for ω and 0 < 1, the weight (z,ω (z ( 1 1 ω(z 1 (1 z ω(z ω(z1 1 (1 z 1 1, z. Then Ŵ(z 1 ω(z1 (1 z 1, and, in articular, 1,ω ω. One could equally well consider any of the summands aearing in the definition of only, but because of the identity for Ŵ we kee this definition since it slightly simlifies some calculations that we will face later. For r (0,1, define f r by f r (z f(rz for all z. ith these rearations we can state the main result of this study as follows. Theorem 1. Let 0 < < 1 and ω. Then (A ω B, with equivalence of norms, under the airing f,g A 2 lim f r (zg(z(zda(z, f A ω, g B. Comaring with Zhu s argument [10], we are led to deal with the following three main obstructions: (1 L -estimates for functions resembling the Bergman kernel; (2 Shar embedding A ω A1 ;

3 uality of weighted Bergman saces with small exonents 623 (3 Fractional differential oerators. In the case of the standard weights, (1 is taken care of by the classical Forelli-Rudin estimates [2], while the requirement (2 can be understood in terms of Carleson measures. It turns out, that these lie in the core of the theory of more general weights, and indeed can be understood. As far as we know, there is no direct analog for the fractional differential oerators induced by more general weights; for standard weights the order of differentiation comes out rather transarently. Surrisingly enough, we can avoid fractional derivatives altogether, and this actually simlifies the roof substantially. 2. Towards Theorem 1 Zhu s argument goes through exressing f A ω with the hel of the reroducing roerty. Our reasoning could be considered a somewhat more streamlined version of this roof. It relies in articular on [5, Theorem 1(ii] which lays the role of the Forelli-Rudin estimates and says that, for each ω,ν, 0 < < andn N {0}, the reroducing kernel of A 2 ω satisfies the estimate (2.1 (B ω z (n A ν z 0 ν(t ω(t (1 t (n+1 dt, z 1. This result was used in [5] to show that for each ω, the saces (A 1 ω and B are isomorhic via the airing f,g A 2 ω lim f r (zg(zω(zda(z, f A 1 ω, g B. The above result can be used to rove the surjectivity of P ω : L ( B. The roof is quite standard, but not readily available in the literature, so we record it here for the convenience of the reader. To find an exact reimage of g B under P ω is more laborious, see [7] for details. Here we just note that if ω satisfies the ointwise regularity condition ω(z ω(z(1 z, then for each g B, the function belongs to L and satisfies P ω (h g. h(z g(0+ ω(z z ω(z (2zg (z+g(z g(0 Lemma 2. Let ω. Then P ω : L B is bounded and onto. Proof. The fact that P ω : L B is bounded for each ω is contained in [5, Theorem 5(ii], but since the roof is short we sketch it here for the convenience of the reader. If h L, then (P ω (h (z h(ζ(bω ζ (zω(ζda(ζ, and hence (2.1 gives (P ω (h (z h L (B ω ζ (z ω(ζda(ζ h L 1 ζ, ζ 1. It follows that P ω : L B is bounded. To see that P ω : L B is onto, let g B. Then it induces an element in (A 1 ω by the formula f lim f r (zg(zω(zda(z.

4 624 Antti Perälä and Jouni Rättyä On the other hand, the Hahn Banach theorem and the well-known duality (L 1 ω L guarantee the existence of ϕ L such that lim f r (zg(zω(zda(z f(zϕ(zω(z da(z lim f r (zϕ(zω(zda(z for all f A 1 ω. Now, note that P ω(f r f r and that P ω is self-adjoint. e have lim f r (zg(zω(zda(z lim f r (z(p ω (ϕ(zω(zda(z. But P ω (ϕ B by the first art of the roof, thus g P ω (ϕ B and reresents the zero functional. It follows that g P ω (ϕ, which comletes the roof. ith these rearations we are ready to rove our main result. 3. Proof of Theorem 1 Let F (A ω. Since f r f in A ω, we have F(f r F(f. The weight induces a reroducing formula through its kernel Bz by f r (z f r (ζbz (ζ(ζda(ζ, z, and hence (for instance, by aroximating f r by olynomials one obtains ( F(f r f r (ζf z Bz (ζda(ζ. (ζ Here the subindex in F z indicates the variable of the function with resect to which F oerates. Of course, by using any orthonormal basis {e n } of A 2, the kernel can be exressed as Bz (ζ n e n(ζe n (z, and hence one can exlicitly write ( F(f r f r (ζ e n (ζf(e n (ζda(ζ. Anyhow, the function satisfies g(ζ F z (B z (ζ n F(f r n e n (ζf(e n f r (ζg(ζ(ζda(ζ f r,g A 2. Moreover, using the linearity of F on the difference quotient along with the fact that for a fixed ζ, z Bz (ζ is analytic in a disc containing, it is easy to see that g H( and g (ζ F z ((B z (ζ, ζ. Therefore, since Bz (ζ B ζ (z, (2.1 yields ( g (ζ d dζ F(B ζ d F dζ B ζ F d F z ζ (B ζ (z ω(zda(z F ζ ζ ω(t F dt 0 Ŵ(t F (1 t 2 ζ 0 dζ B ζ A ω (B ζ (z ω(zda(z dt F (1 t 1+ (1 ζ, ζ 1,

5 uality of weighted Bergman saces with small exonents 625 and it follows that g B. e next show that each g B induces a bounded linear functional on A ω via,g A 2. To see this, we first show that. A radial weight belongs to if and only if there exist C C( > 0, α α( > 0 and β β( α such that (3.1 C 1 ( 1 r 1 t αŵ(t Ŵ(r C ( 1 r 1 t βŵ(t, 0 r t < 1. It is easy to see that the right-hand inequality is equivalent to [6], and an analogous argument shows that the left-hand inequality is satisfied if and only if has the doubling roerty Ŵ(r CŴ ( 1 1 r for some C, > 1. Now that Ŵ(r ω(r 1 1 (1 r 1, (3.1 alied to ω imlies Ŵ(r C 1 ( β 1 r +1 1 Ŵ(t, 0 r t < 1, 1 t for some C C(ω > 0, and hence. Moreover, for each > 1, ( Ŵ(r ω 1 1 r Ŵ ( 1 1 r ( ( r 1 1, 0 < r < 1, 1 1( 1 r 1 ( 1 1 r 1 1 and thus. SinceP : L B is bounded and onto by Lemma 2, the oen maing theorem ensures the existence of M M( > 0 such that for each g B there is h L for which g P (h and h L M g B. Fubini s theorem shows f r (zg(z(zda(z f r (zp (h(z(zda(z f r (z h(ζbz (ζ(ζda(ζ(zda(z ( h(ζ f r (zbz (ζ(zda(z (ζda(ζ ( h(ζ f r (zbζ (ζda(ζ (z(zda(z f r (ζh(ζ(ζda(ζ, and hence, as the L 1 -mean M 1 (r,f of f H( is increasing in r, f,g A 2 lim f r (ζ h(ζ (ζda(ζ h L f A 1 M g B f A 1. Now that (S ω(s 1 for each Carleson square S by the identity Ŵ(r ω(r 1 1 (1 r 1, the measure da is a 1-Carleson measure for A ω by [4, Theorem 1], and hence we deduce f,g A 2 g B f A w. The exlicit use of the surjectivity of P : L B can be avoided by arguing in the following way. First, use Green s theorem to the moduli of monomials or calculate

6 626 Antti Perälä and Jouni Rättyä directly to get (3.2 f,g A 2 f,g A 2 + f(0 g(0 lim f r (z g (z (zda(z+ f A ω g B g B lim f (z r (z 1 z da(z+ f A g ω B, where 1 (z log s ω(ssds. Since the Cauchy integral formula and Fubini s z z ( theorem give M 1 (r,g (1 r 4M 1+r 1,g for all g H( and 0 < r < 1, and 2 (r Ŵ(r(1 r for each radial weight and all 1 r < 1, a change of variable 2 and the hyothesis yield f,g A 2 g B f (z Ŵ(zdA(z+ f A g ω B (3.3 g B f(z Ŵ(z 1 z da(z+ f A ω g B. The assertion now follows similarly as above once we have shown that Ŵ(z(1 z 1 da(z is a 1-Carleson measure for A ω. But since, by (3.1 there exist C C(ω > 0 and α α( > 0 such that 1 r Ŵ(s Ŵ(r ds C 1 s (1 r α 1 r ds Ŵ(r, 0 r < 1, (1 s 1 α and the desired roerty follows by [4, Theorem 1]. One may also comlete the roof directly from (3.2 by alying a result concerning differentiation oerators [4, Theorem 2]. References [1] Coifman, R., and R. Rochberg: Reresentation theorems for holomorhic and harmonic functions in L. - In: Reresentation theorems for Hardy saces, Astérisque 77, 1980, [2] Forelli, F., and. Rudin: Projections on saces of holomorhic functions in balls. - Indiana Univ. Math. J. 24, 1974/75, [3] Peláez, J. A., and J. Rättyä: eighted Bergman saces induced by raidly increasing weights. - Mem. Amer. Math. Soc. 227:1066, [4] Peláez, J. A., and J. Rättyä: Embedding theorems for Bergman saces via harmonic analysis. - Math. Ann. 362:1-2, 2015, [5] Peláez, J.A., and J. Rättyä: Two weight inequality for Bergman rojection. - J. Math. Pures Al. 105, 2016, [6] Peláez, J. A., and J. Rättyä: On the boundedness of Bergman rojection. - Advanced Courses of Mathematical Analysis VI, 2016, [7] Peláez, J.A., and J. Rättyä: eighted Bergman rojection on L. - Prerint. [8] Romberg, B..: The H saces with 0 < < 1. - Ph.. Thesis, University of Rochester, [9] Shairo, J.: Mackey toologies, reroducing kernels, and diagonal mas on the Hardy and Bergman saces. - uke Math. J. 43:1, 1976, [10] Zhu,.: Bergman and Hardy saces with small exonents. - Pacific J. Math. 162:1, 1994, Received 5 Setember 2016 Acceted 17 November 2016