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On a class of Mikhlin multipliers which do not preserve $L^1$-, $L^\infty$-regularity and continuity

Pavel Dimovski, Stevan Pilipovic, Bojan Prangoski

Abstract

We show that every Fourier multiplier with real-valued and positively homogeneous symbol of order 0, supported in a cone whose dual cone has a nonempty interior and such that the average of the positive part is sufficiently larger than the average of the negative part does not preserve the $L^1$- nor the $L^\infty$ regularity and neither the continuity.We also construct wave front sets which measure the microlocal regularity with respect to a large class of Banach spaces. As a consequence of the first part, we argue that one can never construct wave front sets that behave in a natural way and measure the microlocal $L^1$- nor $L^\infty$-regularity and neither the continuity

On a class of Mikhlin multipliers which do not preserve $L^1$-, $L^\infty$-regularity and continuity

Abstract

We show that every Fourier multiplier with real-valued and positively homogeneous symbol of order 0, supported in a cone whose dual cone has a nonempty interior and such that the average of the positive part is sufficiently larger than the average of the negative part does not preserve the - nor the regularity and neither the continuity.We also construct wave front sets which measure the microlocal regularity with respect to a large class of Banach spaces. As a consequence of the first part, we argue that one can never construct wave front sets that behave in a natural way and measure the microlocal - nor -regularity and neither the continuity

Paper Structure

This paper contains 3 sections, 7 theorems, 34 equations.

Table of Contents

  1. Introduction and preliminaries
  2. The main result
  3. Wave front sets for a class of Banach spaces

Key Result

Lemma 2.1

Let $V\subseteq \mathbb R^n$ be a closed cone such that $V\backslash\{0\}\neq \emptyset$ and $V':=\operatorname{int}V^*\neq \emptyset$. Let $\varphi_1\in L^{\infty}(\mathbb{S}^{n-1})$ be a real-valued function satisfying $\operatorname{supp}\varphi_1\subseteq \mathbb{S}^{n-1}\cap V$ and set $\varphi If $\operatorname{supp}\varphi_{1,-}\neq \emptyset$ then $\kappa_0\in(0,1]$, otherwise $\kappa_0=\i

Theorems & Definitions (23)

  • Lemma 2.1
  • Remark 2.2
  • proof : Proof of Lemma \ref{['lem-for-con-exampl']}
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • ...and 13 more