Related papers: On several problems in p-Bergman theory

Let $M$ be a closed complex submanifold in ${\mathbb C}^N$ with the complete K\"ahler metric induced by the Euclidean metric. Several finiteness theorems on the $L^p$ Bergman space of holomorphic sections of a given Hermitian line bundle…

We investigate the heat equation corresponding to the Bessel operators on a symmetric cone $\Omega=G/K$. These operators form a one-parameter family of elliptic self-adjoint second order differential operators and occur in the Lie algebra…

Using the machinery of unitary spherical harmonics due to Koornwinder, Folland and other authors, we~obtain expansions for the Szeg\"o and the weighted Bergman kernels of $M$-harmonic functions, i.e.~functions annihilated by the invariant…

We introduce a nonlinear potential theory problem for the Laplacian, the solution of which characterizes the Berezin density $B(z,\cdot)$ for the polynomial Bergman space, where the point $z\in\mathbb{C}$ is fixed. When $z=\infty$, the…

In this paper we study the compactness of operators on the Bergman space of the unit ball and on very generally weighted Bargmann-Fock spaces in terms of the behavior of their Berezin transforms and the norms of the operators acting on…

In this paper we consider the reproducing kernel thesis for boundedness and compactness for operators on $\ell^2$--valued Bergman-type spaces. This paper generalizes many well--known results about classical function spaces to their…

In this paper, we introduce a class of singular integral operators which generalize Calder\'on-Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a…

In this paper we investigate some properties of the harmonic Bergman spaces $\mathcal A^p(\sigma)$ on a $q$-homogeneous tree, where $q\geq 2$, $1\leq p<\infty$, and $\sigma$ is a finite measure on the tree with radial decreasing density,…

This is a continuation of recent work on the general definition of pseudo-differential operators of type $1,1$, in H\"ormander's sense. Continuity in $L_p$-Sobolev spaces and H\"older--Zygmund spaces, and more generally in Besov and…

It was recently proved by Fischer, Keller, and Pogorzelski in [Integr. Equ. Oper. Theory, 95(24), 2023] that the classical discrete $p$-Hardy inequality admits an improvement, and the optimal $p$-Hardy weight $\omega_{p}$ was determined…

Given a sequence of positive Hermitian holomorphic line bundles $(L_p,h_p)$ on a K\"ahler manifold $X$, we establish the asymptotic expansion of the Bergman kernel of the space of global holomorphic sections of $L_p$, under a natural…

In this work, we extend the theory of B\'ekoll\`e-Bonami $B_p$ weights. Here we replace the constant $p$ by a non-negative measurable function $p(\cdot),$ which is log-H\"older continuous function with lower bound $1$. We show that the…

We consider mainly the Hilbert space of bianalytic functions on a given domain in the plane, square integrable with respect to a weight. We show how to obtain the asymptotic expansion of the corresponding bianalytic Bergman kernel for power…

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. The alpha-harmonic functions and their first-order partial derivative functions on unit disk are estimated…

We consider the weighted $A^p(\omega)$ and $B_p(\omega)$ spaces of holomorphic functions on the polydisk (in the case of $p>1$). We prove some theorems about the boundedness of Toeplitz operators on weighted Besov spaces $B_p(\omega)$ and…

Let $\Omega\subset \mathbb{C}$ be an arbitrary domain in the one-dimensional complex plane equipped with a positive Radon measure $\mu$. For any $1\le p< \infty$, it is shown that the weighted Bergman space $A^p(\Omega, \mu)$ of holomorphic…

We prove that for $1<p\le q<\infty$, $qp\geq {p'}^2$ or $p'q'\geq q^2$, $\frac{1}{p}+\frac{1}{p'}=\frac{1}{q}+\frac{1}{q'}=1$, $$\|\omega P_\alpha(f)\|_{L^p(\mathcal{H},y^{\alpha+(2+\alpha)(\frac{q}{p}-1)}dxdy)}\le…

We introduce a wider class of bounded Hartogs domains, which contains some generalizations of the classical Hartogs triangle. A sharp criteria for the $L^p-L^q$ boundedness of the Toeplitz operator with symbol $K^{-t}$ is obtained on these…

We show that Petermichl's dyadic operator $\mathcal{P}$ (S. Petermichl (2000), Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol) is a Calder\'{o}n-Zygmund type operator on an adequate metric normal space of…

We introduce $p$-adic Kummer spaces of continuous functions on $\mathbb{Z}_p$ that satisfy certain Kummer type congruences. We will classify these spaces and show their properties, for instance, ring properties and certain decompositions.…