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Register a new userComposition operators on Hardy-Sobolev spaces with bounded reproducing kernels
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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 $H^{2}_{beta }$.
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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 $B$.
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.
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 $J_b$ are equivalent on $H^p$ for any $1 le p < infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $ell^2$ when $p e 2.$
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 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 kernel is a Green kernel of differential operator $L:=mathbf{P}^{ast T}mathbf{P}$ with homogeneous or nonhomogeneous boundary conditions given by $mathbf{B}$, where we ensure that the distributional adjoint operator $mathbf{P}^{ast}$ of $mathbf{P}$ is well-defined in the distributional sense. We represent the inner product of the reproducing-kernel Hilbert space in terms of the operators $mathbf{P}$ and $mathbf{B}$. In addition, we find relationships for the eigenfunctions and eigenvalues of the reproducing kernel and the operators with homogeneous or nonhomogeneous boundary conditions. These eigenfunctions and eigenvalues are used to compute a series expansion of the reproducing kernel and an orthonormal basis of the reproducing-kernel Hilbert space. Our theoretical results provide perhaps a more intuitive way of understanding what kind of functions are well approximated by the reproducing kernel-based interpolant to a given multivariate data sample.
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