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Authors
Atte Reijonen
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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 $mathcal{B}_{ u}^{p,p}$.
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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.
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.
We give a proof that every space of weighted square-integrable holomorphic functions admits an equivalent weight whose Bergman kernel has zeroes. Here the weights are equivalent in the sense that they determine the same space of holomorphic functions. Additionally, a family of radial weights in $L^1(mathbb{C})$ whose associated Bergman kernels have infinitely many zeroes is exhibited.
In a previous work we introduced Besov spaces $mathcal{B}^s_{p,q}$ defined on a measure spaces with a good grid, with $pin [1,infty)$, $qin [1,infty]$ and $0< s< 1/p$. Here we show that classical Besov spaces on compact homogeneous spaces are examples of such Besov spaces. On the other hand we show that even Besov spaces defined by a good grid made of partitions by intervals may differ from a classical Besov space, giving birth to exotic Besov spaces.
The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.
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