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Register a new userCompact differences of composition operators on Bergman spaces induced by doubling weights
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Bounded and compact differences of two composition operators acting from the weighted Bergman space $A^p_omega$ to the Lebesgue space $L^q_ u$, where $0<q<p<infty$ and $omega$ belongs to the class $mathcal{D}$ of radial weights satisfying a two-sided doubling condition, are characterized. On the way to the proofs a new description of $q$-Carleson measures for $A^p_omega$, with $p>q$ and $omegainmathcal{D}$, involving pseudohyperbolic discs is established. This last-mentioned result generalizes the well-known characterization of $q$-Carleson measures for the classical weighted Bergman space $A^p_alpha$ with $-1<alpha<infty$ to the setting of doubling weights. The case $omegainwidehat{mathcal{D}}$ is also briefly discussed and an open problem concerning this case is posed.
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A radial weight $omega$ belongs to the class $widehat{mathcal{D}}$ if there exists $C=C(omega)ge 1$ such that $int_r^1 omega(s),dsle Cint_{frac{1+r}{2}}^1omega(s),ds$ for all $0le r<1$. Write $omegaincheck{mathcal{D}}$ if there exist constants $K=K(omega)>1$ and $C=C(omega)>1$ such that $widehat{omega}(r)ge Cwidehat{omega}left(1-frac{1-r}{K}right)$ for all $0le r<1$. In a recent paper, we have recently prove that these classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights. Classical results by Hardy and Littlewood, and Shields and Williams, show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $omegainwidehat{mathcal{D}}setminus check{mathcal{D}}$ and $0<ple 1$. In this paper we establish sharp estimates for the norm of the analytic Bergman space $A^p_omega$, with $omegainwidehat{mathcal{D}}setminus check{mathcal{D}}$ and $0<p<infty$, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.
Let $1le p<infty$, $0<q<infty$ and $ u$ be a two-sided doubling weight satisfying $$sup_{0le r<1}frac{(1-r)^q}{int_r^1 u(t),dt}int_0^rfrac{ u(s)}{(1-s)^q},ds<infty.$$ The weighted Besov space $mathcal{B}_{ u}^{p,q}$ consists of those $fin H^p$ such that $$int_0^1 left(int_{0}^{2pi} |f(re^{itheta})|^p,dthetaright)^{q/p} u(r),dr<infty.$$ Our main result gives a characterization for $fin mathcal{B}_{ u}^{p,q}$ depending only on $|f|$, $p$, $q$ and $ u$. As a consequence of the main result and inner-outer factorization, we obtain several interesting by-products. In particular, we show the following modification of a classical factorization by F. and R. Nevanlinna: If $fin mathcal{B}_{ u}^{p,q}$, then there exist $f_1,f_2in mathcal{B}_{ u}^{p,q} cap H^infty$ such that $f=f_1/f_2$. In addition, we give a sufficient and necessary condition guaranteeing that the product of $fin H^p$ and an inner function belongs to $mathcal{B}_{ u}^{p,q}$. Applying this result, we make some observations on zero sets of $mathcal{B}_{ u}^{p,p}$.
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.
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.
Suppose $ngeq 3$ and let $B$ be the open unit ball in $mathbb{R}^n$. Let $varphi: Bto B$ be a $C^2$ map whose Jacobian does not change sign, and let $psi$ be a $C^2$ function on $B$. We characterize bounded weighted composition operators $W_{varphi,psi}$ acting on harmonic Hardy spaces $h^p(B)$. In addition, we compute the operator norm of $W_{varphi,psi}$ on $h^p(B)$ when $varphi$ is a Mobius transformation of $B$.
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