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Register a new userBergman-type Singular Operators and the Characterization of Carleson Measures for Besov--Sobolev Spaces on the Complex Ball
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The purposes of this paper are two fold. First, we extend the method of non-homogeneous harmonic analysis of Nazarov, Treil and Volberg to handle Bergman--type singular integral operators. The canonical example of such an operator is the Beurling transform on the unit disc. Second, we use the methods developed in this paper to settle the important open question about characterizing the Carleson measures for the Besov--Sobolev space of analytic functions $B^sigma_2$ on the complex ball of $mathbb{C}^d$. In particular, we demonstrate that for any $sigma> 0$, the Carleson measures for the space are characterized by a T1 Condition. The method of proof of these results is an extension and another application of the work originated by Nazarov, Treil and the first author.
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We show that for an entire function $varphi$ belonging to the Fock space ${mathscr F}^2(mathbb{C}^n)$ on the complex Euclidean space $mathbb{C}^n$, the integral operator begin{eqnarray*} S_{varphi}F(z)=int_{mathbb{C}^n} F(w) e^{z cdotbar{w}} varphi(z- bar{w}),dlambda(w), zin mathbb{C}^n, end{eqnarray*} is bounded on ${mathscr F}^2(mathbb{C}^n)$ if and only if there exists a function $min L^{infty}(mathbb{R}^n)$ such that $$ varphi(z)=int_{mathbb{R}^n} m(x)e^{-2left(x-frac{i}{2} z right)cdot left(x-frac{i}{2} z right)} dx, zin mathbb{C}^n. $$ Here $dlambda(w)= pi^{-n}e^{-leftvert wrightvert^2}dw$ is the Gaussian measure on $mathbb C^n$. With this characterization we are able to obtain some fundamental results including the normaility, the algebraic property, spectrum and compactness of this operator $S_varphi$. Moreover, we obtain the reducing subspaces of $S_{varphi}$. In particular, in the case $n=1$, we give a complete solution to an open problem proposed by K. Zhu for the Fock space ${mathscr F}^2(mathbb{C})$ on the complex plane ${mathbb C}$ (Integr. Equ. Oper. Theory {bf 81} (2015), 451--454).
We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents.
We prove Carleson embeddings for Bergman-Orlicz spaces of the unit ball that extend the lower triangle estimates for the usual Bergman spaces.
We completely characterize the boundedness of the Volterra type integration operators $J_b$ acting from the weighted Bergman spaces $A^p_alpha$ to the Hardy spaces $H^q$ of the unit ball of $mathbb{C}^n$ for all $0<p,q<infty$. A partial solution to the case $n=1$ was previously obtained by Z. Wu in cite{Wu}. We solve the cases left open there and extend all the results to the setting of arbitrary complex dimension $n$. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, factorization tricks for tent spaces of sequences, as well as techniques and integral estimates related to Hardy and Bergman spaces.
We completely characterize the boundedness of the area operators from the Bergman spaces $A^p_alpha(mathbb{B}_ n)$ to the Lebesgue spaces $L^q(mathbb{S}_ n)$ for all $0<p,q<infty$. For the case $n=1$, some partial results were previously obtained by Wu. Especially, in the case $q<p$ and $q<s$, we obtain the new characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to $n$-complex dimension.
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