Related papers: $L^p$-theory for Cauchy-transform on the unit disk

We study the behaviour of singular integral operators $T_{k_t}$ of convolution type on $\mathbb{C}$ associated with the parametric kernels $$ k_t(z):=\frac{(\Re z)^{3}}{|z|^{4}}+t\cdot \frac{\Re z}{|z|^{2}}, \quad t\in \mathbb{R},\qquad…

Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"older continuous condition and $L$ a one to one operator of type $\omega$ in $L^2({\mathbb R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a…

We present several Liouville type results for the $p$-Laplacian in $\R^N$. Suppose that $h$ is a nonnegative regular function such that $$ h(x) = a|x|^\gamma\ {\rm for}\ |x|\ {\rm large},\ a>0\ {\rm and}\ \gamma> -p. $$ We obtain the…

We prove that, for r>2, the r-variation and oscillation for the smooth truncations of the Cauchy transform on Lipschitz graphs are bounded in L^p for 1<p finite. The analogous result holds for the n-dimensional Riesz transform on…

A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.

Yanni Chen extended the classical Beurling-Helson-Lowdenslager Theorem for Hardy spaces on the unit circle $\mathbb{T}$ defined in terms of continuous gauge norms on $L^{\infty}$ that dominate $\Vert\cdot\Vert_{1}$. We extend Chen's result…

Recently, Charpentier showed that there exist holomorphic functions $f$ in the unit disk such that, for any proper compact subset $K$ of the unit circle, any continuous function $\phi$ on $K$ and any compact subset $L$ of the unit disk,…

We obtain Fourier inequalities in the weighted $L_p$ spaces for any $1<p<\infty$ involving the Hardy-Ces\`aro and Hardy-Bellman operators. We extend these results to product Hardy spaces for $p\le 1$. Moreover, boundedness of the…

Given $\alpha > -1$, consider the second order differential operator in $(0,\infty)$, $$L_\alpha f \equiv (x^2 \frac{d^2}{dx^2} + (2\alpha+3)x \frac{d}{dx} + x^2 + (\alpha+1)^2)(f), $$ which appears in the theory of Bessel functions. The…

Using a transference result, several inequalities of approximation by entire functions of exponential type in $\mathcal{C}(\mathbf{R})$, the class of bounded uniformly continuous functions defined on $\mathbf{R}:=\left( -\infty ,+\infty…

Parabolic integro-differential nondegenerate Cauchy problem is considered in the scale of L_{p} spaces of functions whose regularity is defined by a Levy measure with O-regulary varying radial profile. Existence and uniqueness of a solution…

This paper is concerned with the $L^p$ integrability of $N$-harmonic functions with respect to the standard weights $(1-|x|^2)^{\alpha}$ on the unit ball $\mathbb{B}$ of $\mathbb{R}^n$, $n\geq 2$. More precisely, our goal is to determine…

We consider the Cauchy problem for one-dimensional p-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangean coordinate. Our concern is an asymptotic behavior of…

For $d = 2, 3, \ldots$ and $p \in [1, \infty),$ we define a class of representations $\rho$ of the Leavitt algebra $L_d$ on spaces of the form $L^p (X, \mu),$ which we call the spatial representations. We prove that for fixed $d$ and $p,$…

For the non-local space-time reaction-diffusion equation involving fractional $p$-Laplacian \begin{equation*} \begin{cases} \frac{\partial^{\alpha }u}{\partial t^{\alpha }}+(-\Delta)_{p}^{s} u=\mu u^{2}(1-kJ*u)-\gamma…

By a theorem of Bayart, $\varphi$ generates a bounded composition operator on the Hardy space $\Hp$of Dirichlet series ($1\le p<\infty$) only if $\varphi(s)=c_0 s+\psi(s)$, where $c_0$ is a nonnegative integer and $\psi$ a Dirichlet series…

In this paper, we characterize invertible Toeplitz products on a number of Banach spaces of analytic functions, including weighted Bergman space $L^p_a (\mathbb{B}_n, dv_\gamma)$, the Hardy space $H^p(\partial \mathbb{D})$, and the weighted…

Dual pairs of interior and exterior Hardy spaces associated to a simple closed Lipschitz planar curve are considered, leading to a M\"obius invariant function bounding the norm of the Cauchy transform $\bf{C}$ from below. This function is…

In this paper, we show the equivalence between the boundedness of the Riesz transform $d\Delta^{-1/2}$ on $L^p$, $p\in (2,p_0)$, and the equality $H^p=L^p$, $p\in(2,p_0)$, in the class of manifold whose measure is doubling and for which the…

We prove that for any $2<p<\infty$ and for every $n$-dimensional subspace $X$ of $L_p$, represented on $\mathbb R^n$, whose unit ball $B_X$ is in Lewis' position one has the following two-level Gaussian concentration inequality: \[ \mathbb…