arXiv subjects
Publications and source records attributed to Hasi Wulan.
Several upper bounds as well as one lower bound for the operator norm of the generalized Hilbert operator $\mathcal{H}_b$ acting on weighted Bergman spaces $A_α^p$ are established. Moreover, under some mild assumptions, we obtain the exact norm of $\mathcal{H}_b$ on $ A_α^p$.
Let $A_α^p$ be the weighted Bergman space on the unit disk, where $α>-1$. For $f(z)=\sum_{k=0}^{\infty}a_k z^k\in A_α^p$, consider the Hilbert matrix operator $\mathcal{H}f(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}\frac{a_k}{n+k+1}\right)z^n =\int_0^1\frac{f(t)}{1-tz}\,dt$. For even exponents $p=2m$, we prove that $\|\mathcal{H}\|_{A_α^{2m}\to A_α^{2m}}=B(a,1-a)$, where $a=(α+2)/(2m)$, whenever $0 B(a_0,1-a_0)$. The counterexample is based on the fixed function $f_0(z)=(1-z^2)^{-4/5}=\sum_{k=0}^{\infty}\frac{(4/5)_k}{k!}z^{2k}$. A rigorous interval estimate at $p=1100000$, together with monotonicity in $p$, yields the result on the entire half-line. In particular, the formula fails for every even exponent $p=2m$ with $m\geq 550000$.
Karapetrović conjectured that the norm of the Hilbert matrix operator on the Bergman space $A^p_α$ is equal to $π/\sin((2+α)π/p)$ when $-1<α \frac{1}{47}$ and $α\not=1$.
Suppose $(C_t)_{t\geq0}$ is the composition semigroup induced by a one-parameter semigroup $(φ_t)_{t\geq0}$ of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of $(C_t)_{t\geq0}$ acting on the weighted Bergman space induced by doubling weights, provided $(φ_t)_{t\geq0}$ is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of $(C_t)_{t\geq0}$ also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of $(C_t)_{t\geq0}$.
For $0<p<\infty$, we give a complete description of nonnegative radial weight functions $ω$ on the open unit disk $\mathbb{D}$ such that $$ \int_{\mathbb{D}} |f'(z)|^p (1-|z|^2)^{p-2}ω(z)dA(z)<\infty $$ if and only if $$ \int_{\mathbb{D}}\int_{\mathbb{D}}\frac{|f(z)-f(ζ)|^p}{|1-\overlineζz|^{4+τ+σ}}(1-|z|^2)^τ(1-|ζ|^2)^σω(ζ)dA(z)A(ζ)<\infty $$ for all analytic functions $f$ in $\mathbb{D}$, where $τ$ and $σ$ are some real numbers. As applications, we give some geometric descriptions of functions in Besove type spaces $B_p(ω)$ with doubling weights, and characterize the boundedness and compactness of Hankel type operators related to Besov type spaces with radial Békollé-Bonami weights. Some special cases of our results are new even for some standard weighted Besov spaces.
In this paper, we give a universal description of the boundedness and compactness of Toeplitz operator $\mathcal{T}_μ^ω$ between Bergman spaces $A_η^p$ and $A_\upsilon^q$ when $μ$ is a positive Borel measure, $1<p,q<\infty$ and $ω,η,\upsilon$ are regular weights. By using Khinchin's inequality and Kahane's inequality, we get a new characterization of the Carleson measure for Bergman spaces induced by regular weights.
In this paper, by describing characterizations of Carleson type measures on $[0,1)$, we determine the range of a Cesàro-like operator acting on $H^\infty$. A special case of our result gives an answer to a question posed by P. Galanopoulos, D. Girela and N. Merchán recently.
In this paper, the effect of absolute values on the behavior of functions $f$ in the spaces $\mathcal{Q}_K$ is investigated. It is clear that $f\in \mathcal{Q}_K(\partial {\mathbb{D}}) \Rightarrow |f|\in \mathcal{Q}_K(\partial {\mathbb{D}})$, but the converse is not always true. For $f$ in the Hardy space $H^2$, we give a condition involving the modulus of the function only, such that this condition together with $|f|\in \mathcal{Q}_K(\partial {\mathbb{D}})$ is equivalent to $f\in \mathcal{Q}_K$. As an application, a new criterion for inner-outer factorisation of $\mathcal{Q}_K$ spaces is given. These results are also new for $\mathcal{Q}_p$ spaces.
Using the Cauchy-Riemann operator, we characterize $Q_K$ spaces, Besov spaces and analytic Morrey spaces in terms of pseudoanalytic extensions of primitive functions. Our results are also true on some classical Banach spaces, such as the Bloch space, $BMOA$ and the Dirichlet space.
We obtain new characterizations for Bergman spaces with standard weights in terms of Lipschitz type conditions in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As a consequence, we prove optimal embedding theorems when an analytic function on the unit disk is symmetrically lifted to the bidisk.