Related papers: On a planar Pierce--Yung operator

We examine the averaging operator $T$ corresponding to the manifold in $\mathbb{R}^{2d}$ of pairs of points $(u,v)$ satisfying $|u| = |v| = |u - v| = 1$, so that $\{0,u,v\}$ is the set of vertices of an equilateral triangle. We establish…

This paper investigates the $L^p$ boundedness of wave operators for the Laplace operator with finite rank perturbations \begin{equation*} H=-\Delta+\sum\limits_{i=1}^N\langle\cdot\,, \varphi_i\rangle \varphi_i \qquad \mbox{on}\,\,\, \R^d.…

We obtain $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded…

We find necessary and sufficient conditions on weights $u_1, u_2, v_1, v_2$, i.e. measurable, positive, and finite, a.e. on $(a,b)$, for which there exists a positive constant $C$ such that for given $0 < p_1,q_1,p_2,q_2 <\infty$ the…

Let $T\colon L^p({\mathcal M})\to L^p({\mathcal N})$ be a bounded operator between two noncommutative $L^p$-spaces, $1\leq p<\infty$. We say that $T$ is $\ell^1$-bounded (resp. $\ell^1$-contractive) if $T\otimes I_{\ell^1}$ extends to a…

It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb{R})$ for all $1<p<\infty$. In this note it is shown that $$ \| S_{\mathcal{I}_{E_2}} \|_{L^p (\mathbb{R})…

The paper provides a complement to the classical results on Fourier multipliers on $L^p$ spaces. In particular, we prove that if $q\in (1,2)$ and a function $m:\mathbb{R} \rightarrow \mathbb{C}$ is of bounded $q$-variation uniformly on the…

$T$ is a Ritt operator in $L^p$ if $\sup_n n\|T^n-T^{n+1}\|<\infty$. From \cite{LeMX-Vq}, if $T$ is a positive contraction and a Ritt operator in $L^p$, $1<p<\infty$, the square function $\left( \sum_n n^{2m+1} |T^n(I-T)^{m+1}f|^2…

Let $\A$ be a $C^*-$algebra with unit element $1$ and unitary group $U.$ Let $a=(a_1, ..., a_k)$ and $b=(b_1, ..., b_k)$ two $k-$tuples of elements in $\A.$ The elementary operator associated to $a$ and $b$ is defined by $ R_{a, b} (x)=…

In this dissertation we explore the $[L^{\mathrm{p}},\ L^{q}]$-boundedness of certain integral operators on weighted spaces on cones in ${\mathbb R}^{n}.$ These integral operators are of the type $\displaystyle \int_{V}k(x,\ y)f(y)dy$…

We solve the equation $f(x,y)^3 + g(x,y)^3 = x^3 + y^3$ for homogeneous $f, g \in \mathbb C(x,y)$, completing an investigation begun by Vi\`ete in 1591. The usual addition law for elliptic curves and composition give rise to two binary…

We consider the higher order Schr\"odinger operator $H=(-\Delta)^m+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2m$, $m\in \mathbb N$, $m>1$. When $n$ is odd, we prove that the wave operators extend to bounded operators on…

Let $\mathcal{H}$ be an infinite dimensional Hilbert space and $\mathcal{B}(\mathcal{H})$ be the C*-algebra of all bounded linear operators on $\mathcal{H}$, equipped with the operator-norm. By improving the Brown-Pearcy construction,…

Let $1\le p\le \infty$. In this paper, we consider solving a nonlinear functional equation $$f(x)=y,$$ where $x, y$ belong to $\ell^p$ and $f$ has continuous bounded gradient in an inverse-closed subalgebra of ${\mathcal B}(\ell^2)$, the…

In this paper we study minimal realizations in $L^p(\mathbb{R}^N)$ of the second order elliptic operator \begin{equation*} { A_{b,c}} := (1+|x|^\alpha)\Delta + b|x|^{\alpha-2}x\cdot\nabla - c |x|^{\alpha-2} - |x|^{\beta} , \quad x \in…

We introduce an algebra $\mathcal W_t$ of linear operators that act continuously on each of the Fock spaces $F_t^p$, $1 \leq p \leq \infty$, and contains all Toeplitz operators with bounded symbols. We show that compactness, the spectrum,…

We study mapping properties of the centered Hardy--Littlewood maximal operator $\mathcal{M}$ acting on Lorentz spaces. Given $p \in (1,\infty)$ and a metric measure space $\mathfrak{X}$ we let $\Omega^p_{\rm HL}(\mathfrak{X}) \subset…

We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, the integral operator \begin{eqnarray*} S_{\varphi}F(z)=\int_{\mathbb{C}^n} F(w) e^{z…

We consider divergence form operators with complex coefficients on an open subset of Euclidean space. Boundary conditions in the corresponding parabolic problem are dynamical, that is, the time derivative appears on the boundary. As a…

We prove that if $X,Y$ are Banach spaces, $\Omega$ a compact Hausdorff space and $U\hbox{\rm :} C(\Omega,X)\to Y$ is a bounded linear operator, and if $U$ is a Dunford--Pettis operator the range of the representing measure $G(\Sigma)…