Related papers: On several problems in p-Bergman theory

It's well known that the functional Hilbert space over the unit ball in $B_{d} \in C^d$, with kernel function $K(z,w)=\frac{1}{1-z_{1}w_{1}-... -z_{d}w_{d}}$, admits a natural $A(B_{d})$-module structure. We show the rank of a nonzero…

Based on Harnack's inequality and convex analysis we show that each plurisubharmonic function is locally BUO (bounded upper oscillation) with respect to polydiscs of finite type but not for arbitrary polydiscs. We also show that each…

We show that if $\Omega$ is a vanishing chord-arc domain and $L$ is a divergence-form elliptic operator with H\"older-continuous coefficient matrix, then $\log k_L \in VMO$, where $k_L$ is the elliptic kernel for $L$ in the domain $\Omega$.…

In this note, we introduce a novel norm, termed the $t-$Berezin norm, on the algebra of all bounded linear operators defined on a reproducing kernel Hilbert space $\mathcal{H}$ as $$\|A\|_{t-ber} = \sup_{ \lambda, \mu \in \Omega} \left\{…

We study interior $L^p$-regularity theory, also known as Calderon-Zygmund theory, of the equation \[ \int_{\mathbb{R}^n} \int_{\mathbb{R}^n} \frac{K(x,y)\ (u(x)-u(y))\, (\varphi(x)-\varphi(y))}{|x-y|^{n+2s}}\, dx\, dy = \langle f, \varphi…

In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_\omega$, whose growth conditions are given in terms of a translation invariant measure $\omega$ on the closed half-plane…

We study linear extremal problems in the Bergman space $A^p$ of the unit disc, where $1 < p < \infty$. Given a functional on the dual space of $A^p$ with representing kernel $k \in A^q$, where $1/p + 1/q = 1$, we show that if $q \le q_1 <…

In this paper, we present an explicit description for the boundary behavior of the Bergman kernel function, the Bergman metric, and the associated curvatures along certain sequences converging to an $h$-extendible boundary point.

Off-diagonal upper bounds are established away from the diagonal for the Bergman kernels associated to high powers of holomorphic line bundles over compact complex manifolds, asymptotically as the power tends to infinity. The line bundle is…

The Bergman theory of domains $\{ |{z_{1} |^{\gamma}} < |{z_{2}} | < 1 \}$ in $\mathbb{C}^2$ is studied for certain values of $\gamma$, including all positive integers. For such $\gamma$, we obtain a closed form expression for the Bergman…

The aim of the present paper is three folds. Firstly, we complete the study of the weighted hyperholomorphic Bergman space of the second kind on the ball of radius $R$ centred at the origin. The explicit expression of its Bergman kernel is…

Let $K$ be a standard H\"older continuous Calder\'on--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed…

We show that if $H$ is a group of polynomial growth whose growth rate is at least quadratic then the $L_p$ compression of the wreath product $\Z\bwr H$ equals $\max{\frac{1}{p},{1/2}}$. We also show that the $L_p$ compression of $\Z\bwr \Z$…

We give a purely complex geometric proof of the existence of the Bergman kernel expansion. Our method provides a sharper estimate, and in the case that the metrics are real analytic, we prove that the remainder decays faster than any…

We consider the Szeg\H{o} reproducing kernel associated with the space of $H$-harmonic functions on the unit ball in n-dimensional space, i.e. functions that are characterized by being annihilated by the hyperbolic Laplacian. This paper…

We establish local asymptotic estimates of partial Bergman kernels on closed, $S^1$-symmetric K\"{a}hler manifolds. The main result concerns the scaling asymptotics of partial Bergman kernels at generic off-diagonal points in which they are…

We study connections between orthogonal polynomials, reproducing kernel functions, and polynomials $p$ minimizing Dirichlet-type norms $\|pf-1\|_{\alpha}$ for a given function $f$. For $\alpha\in [0,1]$ (which includes the Hardy and…

For~weights $\rho$ which are either radial on the unit ball or depend only on the vertical coordinate on the upper half-space, we describe the asymptotic behaviour of the corresponding weighted harmonic Bergman kernels with respect to…

Bounded Bergman projections $P_\omega:L^p_\omega(v)\to L^p_\omega(v)$, induced by reproducing kernels admitting the representation $$ \frac{1}{(1-\overline{z}\zeta)^\gamma}\int_0^1\frac{d\nu(r)}{1-r\overline{z}\zeta}, $$ and the…

In this paper we prove an analogue of the discrete spherical maximal theorem of Magyar, Stein, and Wainger, an analogue which concerns maximal functions associated to homogenous algebraic surfaces. Let $\mathfrak{p}$ be a homogenous…