Double integral estimates for Besov type spaces and their applications

For $0<p<\infty$, we give a complete description of nonnegative radial weight functions $\omega$ on the open unit disk $\mathbb{D}$ such that $$\int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2}\omega(z)dA(z)<\infty$$ if and only if $$\int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(\zeta)|^p}{|1-\overline{\zeta}z|^{4+\tau+\sigma}}(1-|z|^2)^{\tau}(1-|\zeta|^2)^{\sigma}\omega(\zeta)dA(z)A(\zeta)<\infty$$ for all analytic functions $f$ in $\mathbb{D}$, where $\tau$ and $\sigma$ are some real numbers. As applications, we give some geometric descriptions of functions in Besove type spaces $B_p(\omega)$ with doubling weights, and characterize the boundedness and compactness of Hankel type operators related to Besov type spaces with radial B\'ekoll\'e-Bonami weights. Some special cases of our results are new even for some standard weighted Besov spaces.Comment: V2, made some corrections on V1 and added some result

Boundary multipliers of a family of Möbius invariant spaces

For $1<p<\infty$ and $0<s<1$, we consider the function spaces $\mathcal{Q}_s^p(\mathbb{T})$ that appear naturally as the space of boundary values of a certain family of analytic Möbius invariant function spaces on the the unit disk. In this paper, we give a complete description of the pointwise multipliers going from $Q_s^{p_1}(\mathbb{T})$ to $Q_r^{p_2}(\mathbb{T})$ for all ranges of $1<p_1, p_2<\infty$ and $0<s,r<1$. The spectra of such multiplication operators is also obtained

Hankel matrices acting on the Dirichlet space

The characterization of the boundedness of operators induced by Hankel matrices on analytic function spaces can be traced back to the work of Z. Nehari and H. Widom on the Hardy space, and has been extensively studied on many other analytic function spaces recently. However, this question remains open in the context of the Dirichlet space [20]. By Carleson measures, the Widom type condition and the reproducing kernel thesis, this paper provides a comprehensive solution to this question. As a beneficial product, characterizations of the boundedness and compactness of operators induced by Ces\`aro type matrices on the Dirichlet space are given. In addition, we also show that a random Dirichlet function almost surely induces a compact Hankel type operator on the Dirichlet space

𝒬

We introduce a new space, K(ℝn) space, of several real variables with nondecreasing functions K. By giving basic properties of the weighted function K, by establishing a Stegenga-type estimate, and by introducing the K-Carleson measure on ℝ+n+1, we obtain various characterizations of K(ℝn) space