We call an $L^p$-multiplier m tame if for each complex homomorphism χ acting on the space of $L^p$ multipliers there is some ${\gamma }_0\in \Gamma$ and |a| ≤ 1 such that $\chi \left(\gamma m\right)=am\left({\gamma }_0\gamma \right)$ for all γ ∈ Γ. Examples of tame multipliers include tame measures and one-sided Riesz products. Tame multipliers show an interesting similarity to measures. Indeed we show that the only tame idempotent multipliers are measures. We obtain quantitative estimates on the size of $L^p$-improving tame multipliers which are similar to those obtained for measures, but...

The boundedness properties of commutators for operators are of central importance in Mathematical Analysis, and some of these commutators arise in a natural way from interpolation theory. Our aim is to present a general abstract method to prove the boundedness of the commutator $[T,\Omega ]$ for linear operators $T$ and certain unbounded operators $\Omega$ that appear in interpolation theory, previously known and a priori unrelated for both real and complex interpolation methods, and also to show how the abstract result...

We establish necessary and sufficient conditions on the real- or complex-valued potential $Q$ defined on ${ℝ}^n$ for the relativistic Schrödinger operator $\sqrt{-\Delta }+Q$ to be bounded as an operator from the Sobolev space $W_2^{1/2}\left({ℝ}^n\right)$ to its dual $W_2^{-1/2}\left({ℝ}^n\right)$.

In this article we prove for $1<p<\infty$ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega$. More precisely, $\Omega$ is considered to be given as the Cartesian product of a cube and a bounded domain $V$ having $C^1$-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in $\Omega$ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as...

Let 0 < p ≤ 1 < q < ∞ and α = n(1/p - 1/q). We introduce some new Hardy spaces $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ which are the local versions of $H^p\left({ℝ}^n\right)$ spaces at the origin. Characterizations of these spaces in terms of atomic and molecular decompositions are established, together with their φ-transform characterizations in M. Frazier and B. Jawerth’s sense. We also prove an interpolation theorem for operators on $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$ and discuss the $HK{̇}_q^{\alpha ,p}\left({ℝ}^n\right)$-boundedness of Calderón-Zygmund operators. Similar results can also be obtained for the non-homogeneous...

This article is concerned with the question of whether Marcinkiewicz multipliers on ${ℝ}^{2n}$ give rise to bilinear multipliers on ℝⁿ × ℝⁿ. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.

Figà-Talamanca characterized the space of Fourier multipliers as the dual space of a certain Banach space. In this paper, we characterize the space of maximal Fourier multipliers as a dual space.

Let ${\varphi }_1,\cdots ,{\varphi }_n$ be real homogeneous functions in $C^{\infty }({ℝ}^n-\left\{0\right\})$ of degree $k\ge 2$, let $\varphi \left(x\right)=({\varphi }_1\left(x\right),\cdots ,{\varphi }_n\left(x\right))$ and let $\mu$ be the Borel measure on ${ℝ}^{2n}$ given by $$\mu \left(E\right)={\int }_{{ℝ}^n}{\chi }_E(x,\varphi \left(x\right)){\left|x\right|}^{\gamma -n}\mathrm{d}x$$ where $\mathrm{d}x$ denotes the Lebesgue measure on ${ℝ}^n$ and $\gamma >0$. Let $T_{\mu }$ be the convolution operator $T_{\mu }f\left(x\right)=(\mu *f)\left(x\right)$ and let $$E_{\mu }=\{(1/p,1/q)\parallel T_{\mu }{\parallel }_{p,q}<\infty ,1\le p,q\le \infty \}.$$ Assume that, for $x\ne 0$, the following two conditions hold: $det\left({\mathrm{d}}^2\varphi \left(x\right)h\right)$ vanishes only at $h=0$ and $det\left(\mathrm{d}\varphi \right(x\left)\right)\ne 0$. In this paper we show that if $\gamma >n(k+1)/3$ then $E_{\mu }$ is the empty set and if $\gamma \le n(k+1)/3$ then $E_{\mu }$ is the closed segment with endpoints $D=\left(1-\frac{\gamma }{n(k+1)},1-\frac{2\gamma }{n(k+1)}\right)$ and $D^{\text{'}}=\left(\frac{2\gamma }{n(1+k)},\frac{\gamma }{n(1+k)}\right)$. Also, we give some examples.

The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^p\left({ℝ}^n\right)$ and $H^p{(}^n)$, 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty }\left({ℝ}^n\right)$ function m is a maximal multiplier on $H^p\left({ℝ}^n\right)$ if and only if it is a maximal multiplier on $H^p{(}^n)$. As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered.

We study convolution operators bounded on the non-normable Lorentz spaces $L^{1,q}$ of the real line and the torus. Here 0 < q < 1. On the real line, such an operator is given by convolution with a discrete measure, but on the torus a convolutor can also be an integrable function. We then give some necessary and some sufficient conditions for a measure or a function to be a convolutor on $L^{1,q}$. In particular, when the positions of the atoms of a discrete measure are linearly independent over the rationals,...

We use scaling properties of convex surfaces of finite line type to derive new estimates for two problems arising in harmonic analysis. For Riesz means associated to such surfaces we obtain sharp Lp estimates for p &gt; 4, generalizing the Carleson-Sjölin theorem. Moreover we obtain estimates for the remainder term in the lattice point problem associated to convex bodies; these estimates are sharp in some instances involving sufficiently flat boundaries.