Radial Two Weight Inequality for Maximal Bergman Projection Induced by a Regular Weight

It is shown in quantitative terms that the maximal Bergman projection Pω+(f)(z)=∫Df(ζ)|Bzω(ζ)|ω(ζ)dA(ζ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {P}^{+}_{\omega}(f)(z)={\int}_{\mathbb{D}} f(\zeta)|{B}^{\omega}_{z}(\zeta)|\omega(\zeta) dA(\zeta), $$\end{document} is bounded from Lνp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}_{\nu }$\end{document} to Lηp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$L^{p}_{\eta }$\end{document} if and only if sup0<r<1∫0rη(s)∫s1ω(t)dtpds+11p∫r1ω(s)ν(s)1pp′ds1p′<∞,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \underset{0<r<1}{\sup}\left( {\int}_{0}^{r}\frac{\eta(s)}{\left( {\int}_{s}^{1}\omega(t) dt\right)^{p}} ds+1\right)^{\frac{1}{p}} \left( {\int}_{r}^{1}\left( \frac{\omega(s)}{\nu(s)^{\frac{1}{p}}}\right)^{p^{\prime}}ds\right)^{\frac{1}{p^{\prime}}}<\infty, $$\end{document} provided ω,ν,η are radial regular weights. A radial weight σ is regular if it satisfies σ(r)≍∫r1σ(t)dt/(1−r)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\sigma }(r){\asymp {\int \limits }}_{r}^{1}{\sigma }(t) dt/(1-r)$\end{document} for all 0 ≤ r < 1. It is also shown that under an appropriate additional hypothesis involving ω and η, the Bergman projection Pω and Pω+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${P}^{+}_{\omega }$\end{document} are simultaneously bounded.


Introduction and Main Results
A function ω : D Ñ r0, 8q, integrable over the unit disc D, is called a weight. It is radial if ωpzq " ωp|z|q for all z P D. For 0 ă p ă 8 and a weight ω, the Lebesgue space L p ω consists of complex-valued measurable functions f in D such that where dApzq " dx dy π denotes the element of the normalized Lebesgue area measure on D. The weighted Bergman space A p ω is the space of analytic functions in L p ω , and is equipped with the corresponding L p ω -norm. If the norm convergence in the Hilbert space A 2 ω implies the uniform convergence on compact subsets of D, the point evaluations are bounded linear functionals on A 2 ω . Therefore there exist reproducing Bergman kernels B ω z P A 2 ω such that The Hilbert space A 2 ω is a closed subspace of L 2 ω , and hence the orthogonal projection from L 2 ω to A 2 ω is given by P ω pf qpzq " ż D f pζ qB ω z pζ qωpζ q dApζ q, z P D.
The operator P ω is the Bergman projection. In this paper we will characterize the radial two-weight inequality for the maximal Bergman projection Pὼ pf qpzq " ş D f pζ q|B ω z pζ q|ωpζ q dApζ q under certain smoothness requirements on the three radial weights involved. The question of when (1.1) is satisfied is an open problem even in the very particular case ω " ν " η if no preliminary hypotheses is imposed on the radial weight.
Two weight inequalities for classical operators have attracted a considerable amount of attention in Complex and Harmonic Analysis, and are closely connected to other interesting questions in the area [2, 5-7, 10, 11]. The most commonly known result on Bergman projection is due to Bekollé and Bonami [3,4], and concerns the case when ν " η is an arbitrary weight and the inducing weight ω is standard, that is, of the form ωpzq " p1´|z| 2 q α for some α ą´1; see [1,11,13] for recent extensions of this result. In this classical case, the Bergman reproducing kernel B ω z pζ q is given by the neat formula p1´zζ q´p 2`αq . However, for a general radial weight ω such explicit formulas for the kernels do not necessarily exist, and that is one of the main obstacles in tackling (1.1). Moreover, kernels induced by radial weights may have zeros, and that of course does not make things any easier. Nonetheless, (1.1) has been recently characterized in the particular case ν " η provided ω and ν are regular weights [10].
For a radial weight ω, we assume throughout the paper that p ωpzq " ş 1 |z| ωpsq ds ą 0 for all z P D, for otherwise the Bergman space A p ω would contain all analytic functions in D. A radial weight ω belongs to the class p D if there exists a constant C " Cpωq ą 1 such that p ωprq ď C p ωp 1`r 2 q for all 0 ď r ă 1. Moreover, if there exist K " Kpωq ą 1 and C " Cpωq ą 1 such that ωprq -ωprqp1´rq for all 0 ď r ă 1. The class of regular weights is denoted by R, and R Ĺ D. For basic properties of these classes of weights and more, see [8,9,12] and the references therein.
The main result of this study is the following theorem, which provides a quantitative description of the boundedness of Pὼ : L p ν Ñ L p η in terms of a Muckenhoupt-type condition related to weighted Hardy operators.
The key tools in the proof of Theorem 1 are the precise estimates for the L p -means and A p ν -norms of the Bergman kernel B ω z obtained in [10, Theorem 1]. A special case of the said result is repeatedly used in the proof and it is stated for further reference as Theorem A below. For a function f analytic in D and 0 ă r ă 1, write Theorem A Let 0 ă p ă 8 and ω, ν P p D. Then the following assertions hold: It is worth noticing that the upper estimate holds for ω P p D and any radial weight ν [10, p. 106]. The argument used to establish the one weight inequality [10, Theorem 3] for regular weights does not carry over as such to the two weight case. The proof of the sufficiency in Theorem 1 is much more involved due to the presence of the second weight η.
The operators P ω and Pὼ are simultaneously bounded under a natural additional hypothesis. This is the content of the other main result of this study. Then the following statements are equivalent: Theorem 2 is a generalization of [10, Theorem 3] because the hypothesis (1.5) is satisfied for ν " η. Although, the conditions N p pω, ν, ηq ă 8 and M p pω, ν, ηq ă 8 are equivalent for many weights, for example the standard weights have this property, the condition N p pω, ν, ηq ă 8 may be essentially weaker than M p pω, ν, ηq ă 8 under the hypothesis of Theorem 1. Indeed, if we pick up an arbitrary ω P R, and define νpsq " It is readily seen that the methods used to prove Theorems 1 and 2 carry over to the case p " 1. In fact, the proof in this case turns out much more simple for obvious reasons. To be precise, one can show that in the case p " 1 the operators P ω and Pὼ are simultaneously bounded, and the uniform boundedness of the quantity ωprq νprq is the characterizing condition. Throughout the paper 1 p`1 p 1 " 1 for 1 ă p ă 8. Further, the letter C " Cp¨q will denote an absolute constant whose value depends on the parameters indicated in the parenthesis, and may change from one occurrence to another. We will use the notation a À b if there exists a constant C " Cp¨q ą 0 such that a ď Cb, and a Á b is understood in an analogous manner. In particular, if a À b and a Á b, then we will write ab.

Proof of Theorem 1
Throughout the proofs we will repeatedly use several basic properties of weights in the classes p D and q D, gathered in the following two lemmas. For a proof of the first lemma, see [8, Lemma 2.1]; the second one can be proved by similar arguments. For each radial weight ω and x ě 1, we write ω x " ş 1 0 s x ωpsq ds.

Lemma B
Let ω be a radial weight. Then the following statements are equivalent: (ii) There exist constants C " Cpωq ą 0 and β " βpωq ą 0 such that p ωprq ď Cˆ1´r 1´t˙β p ωptq, 0 ď r ď t ă 1; (iii) There exists a constant C " Cpωq ą 0 such that Lemma C Let ω be a radial weight. Then ω P q D if and only if there exist C " Cpωq ą 0 and γ " γ pωq ą 0 such that Lemma B (ii) shows that if ω P p D, then there exists β " βpωq ą 0 such that p ωprq p1´rq β is essentially increasing on r0, 1q. Similarly, by Lemma C, p ωprq p1´rq γ is essentially decreasing on r0, 1q for γ " γ pωq ą 0 sufficiently small if ω P q D.

Necessity
In this section we prove that M p pω, ν, ηq ă 8 is a necessary condition for Pὼ : L p ν Ñ L p η to be bounded under the hypotheses of Theorem 1, and establish the desired lower estimate for the operator norm. This is done in the following result under slightly weaker hypotheses than those of the theorem, by using an appropriate family of test functions depending on the weights ω and ν. Proof Assume that Pὼ : If ν vanishes on a set E Ă D of positive measure, then by choosing f " χ E the right side of (2.1) is zero. It follows that ω vanishes (almost everywhere) on E or else η " 0 (almost everywhere) on D. The latter option being unacceptable as p ηprq ą 0 for all 0 ď r ă 1, we deduce that ωdA is absolutely continuous with respect to νdA. Therefore ω{ν is well defined almost everywhere. Hence, for each n P N and 0 ď t ă 1, the function f n,t " min , 0 ă t ă 1, and the proposition is proved.
If ω P p D, then by Lemma B (ii) there exists βpωq ą 0 such that J ω prq Á p ωprq´1p1ṕ 1´rq β q for all 0 ď r ă 1. Therefore, under the hypotheses of Theorem 1, we have }Pὼ } L p ν ÑL p η Á M p pω, ν, ηq, and thus the necessity part is proved.

Sufficiency
The proof of the sufficiency of M p pω, ν, ηq ă 8 for Pὼ : L p ν Ñ L p η to be bounded is more involved than that of the necessity. We begin with the following technical lemma.
Proof Let α " αpω, ν, pq P p0, 1q to be appropriately fixed later. Then Hölder's inequality and Lemma C yield for all 0 ď r ă 1. The latter term is of the desired form. To deal with the first term, observe first that by Lemma C (ii) there exists a constant β " βpωq ą 0 such that p ωprq p1´rq β is essentially increasing on r0, 1q. Further, for each sufficiently small γ " γ pνq ą 0 the function p νprq p1´rq γ is essentially decreasing on r0, 1q by Lemma C. Pick up such a γ from the interval p0, pβq, and fix α P p1´γ pβ , 1q. Then Finally, by combining the above inequality with (2.3) we obtain the claim.
We are now ready to prove the sufficiency part of Theorem 1. To do this, assume M p pω, ν, ηq ă 8, and observe that then the function hpzq " νpzq