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For any real $beta$ let $H^2_beta$ be the Hardy-Sobolev space on the unit disk $D$. $H^2_beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $beta>1/2$. In this paper, we study composition operators $C_varphi$ on $H^2_beta$ for $1/2<beta<1$. Our main result is that, for a non-constant analytic function $varphi:DtoD$, the operator $C_{varphi }$ has dense range in $H_{beta }^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $varphi(D)$. It follows that if the range of $C_{varphi }$ is dense in $H_{beta }^{2}$, then $varphi $ is a weak-star generator of $H^{infty}$. Note that this conclusion is false for the classical Dirichlet space $mathfrak{D}$. We also characterize Fredholm composition operators on $H^{2}_{beta }$.
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,p
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 t
We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $ell^p$-singularity of
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
We introduce a vector differential operator $mathbf{P}$ and a vector boundary operator $mathbf{B}$ to derive a reproducing kernel along with its associated Hilbert space which is shown to be embedded in a classical Sobolev space. This reproducing ker