Bergman-type singular integral operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball (Q2903088)

The purposes of this paper are twofold. First, the authors extend the method of non-homogeneous harmonic analysis to handle Bergman-type singular integral operators. Second, they use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for Besov-Sobolev space of analytic functions \(B^{\sigma}_2\) on the complex ball of \({\mathbb C}^d.\)NEWLINENEWLINEThey consider singular integral operators that do not satisfy the standard estimates. Let \(X\) be a closed set in \({\mathbb R}^d\) and \(m \leq d.\) Assume that NEWLINE\[NEWLINE | K(x,y) | \leq \frac{C}{| x-y |^m} \quad \text{if}\quad x,y \in X, NEWLINE\]NEWLINE and NEWLINE\[NEWLINE | K(x,y) - K(x',y) | + | K(y,x) - K(y,x') | \leq \frac{ C {| x-x' |}^{\varepsilon}} {{| x-y |}^{m + \varepsilon}}, NEWLINE\]NEWLINE where \( | x-x' | \leq \frac{1}{2} | x-y | \) and \(x,x',y \in X.\) Furthermore assume that NEWLINE\[NEWLINE | K(x,y) | \leq \frac{1}{\max( d_H(x)^m, d_H(y)^m)} \quad \text{if}\quad x,y \in X, NEWLINE\]NEWLINE where \(d_H(x) := \text{dist}(x, {\mathbb R}^d \setminus H)\) and \(H\) being an open set in \({\mathbb R}^d\). Let \(\mu\) be a probability measure with compact support in \(X\) and suppose that all balls such that \(\mu(B(x,r)) > r^m\) lie in \(H\) and let NEWLINE\[NEWLINE T_{\mu,m}f(x) = \int_{{\mathbb R}^d} K(x,y)f(y) d\mu. NEWLINE\]NEWLINE They prove the following. If NEWLINE\[NEWLINE \| T_{\mu,m} \chi_Q \|_{L^2({\mathbb R}^d; \mu)}^2 + \| T_{\mu,m}^{*} \chi_Q \|_{L^2({\mathbb R}^d; \mu)}^2 \leq C \mu(Q) \quad \text{for all cubes }\;Q, NEWLINE\]NEWLINE then \(T_{\mu,m}\) is bounded on \(L^2({\mathbb R}^d; \mu).\)NEWLINENEWLINEThe balls such that \(\mu(B(x,r)) > r^m\) are called non-Ahlfors balls. Non-Ahlfors balls are enemies and their presence makes the estimate of singular integral operators basically impossible. The key hypothesis is that one can capture all the non-Ahlfors balls in some open set \(H\). The method of proof of this theorem is to use the tools of non-homogeneous harmonic analysis developed by Nazarov, Treil and Volberg. See, for example, [\textit{F. Nazarov, S. Treil} and \textit{A. Volberg}, Acta Math. 190, No. 2, 151--239 (2003; Zbl 1065.42014)] and [\textit{A. Volberg}, Calderón-Zygmund capacities and operators on nonhomogeneous spaces. CBMS Regional Conference Series in Mathematics 100. Providence, RI: American Mathematical Society (AMS). (2003; Zbl 1053.42022)].NEWLINENEWLINELet \({\mathbb B}_{2d}\) denote the unit ball in \({\mathbb C}^d\). Roughly speaking, the Besov-Sobolev space \(B_2^{\sigma}({\mathbb B}_{2d})\) is the collection of analytic functions on the unit ball such that the derivatives of order \(d/2 - \sigma\) belong to the classical Hardy space \(H^2({\mathbb B}_{2d})\). A non-negative measure \(\mu\) supported inside \({\mathbb B}_{2d}\) is called a \(B_2^{\sigma}({\mathbb B}_{2d})\)-Carleson measure if NEWLINE\[NEWLINE \int_{{\mathbb B}_{2d}} | f(z) |^2 d\mu(z) \leq C \| f \|_{B_2^{\sigma}({\mathbb B}_{2d})}^2 \quad \text{for all}\quad f \in B_2^{\sigma}({\mathbb B}_{2d}). NEWLINE\]NEWLINE The characterizations of \(B_2^{\sigma}({\mathbb B}_{2d})\)-Carleson measures were obtained by \textit{N. Arcozzi, R. Rochberg} and \textit{E. Sawyer} [Adv. Math. 218, No. 4, 1107--1180 (2008; Zbl 1167.32003)] when \(0 \leq \sigma \leq 1/2\), and by \textit{E. Tchoundja} [Ark. Mat. 46, No. 2, 377--406 (2008; Zbl 1159.32004)] when \(d/2 \leq \sigma\). By introducing the singular integral operators NEWLINE\[NEWLINE T_{\mu, 2\sigma}f (z) = \int_{{\mathbb B}_{2d}} \text{Re} \Big( \frac{1}{ ( 1 - \overline{w} \cdot z)^{2 \sigma}} \Big) f(w) d\mu(w), NEWLINE\]NEWLINE the authors characterize these measures for all values of \(\sigma\) at once. Their result is the following. Suppose that \(0 < \sigma\). Then the following conditions are equivalent:NEWLINENEWLINE(1) \quad \(\mu\) is a \(B_2^{\sigma}({\mathbb B}_{2d})\)-Carleson measure;NEWLINENEWLINE(2) \quad \(T_{\mu, 2\sigma}\) is bounded on \(L^2({\mathbb B}_{2d} ; \mu)\);NEWLINENEWLINE(3) \quad There is a constant \(C\) such thatNEWLINENEWLINE(i) \quad \( \| T_{\mu, 2\sigma} \chi_Q \|_{L^2(\mu)}^2 \leq C \mu (Q) \) for all \(\Delta \text{-cubes \;} Q\);NEWLINENEWLINE(ii) \quad \(\mu( B_{\Delta}(x,r)) \leq C r^{2\sigma}\) for all balls \(B_{\Delta}(x,r)\) that intersect \({\mathbb C}^d \setminus {\mathbb B}_{2d},\)NEWLINENEWLINE\noindent where the sets \(B_{\Delta}(x,r)\) are balls measured with respect to a naturally occurring quasi-metric \(\Delta\) in the problem.