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Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces

Divyang G. Bhimani, Rupak K. Dalai

Abstract

We characterize weighted modulation spaces (data space) for which the heat semigroup $e^{-tL}f$ converges pointwise to the initial data $f$ as time $t$ tends to zero. Here $L$ stands for the standard Laplacian $-Δ$ or Hermite operator $H=-Δ+|x|^2$ on the Euclidean space. This is the first result on pointwise convergence with data in a weighted modulation spaces (which do not coincide with weighted Lebesgue spaces). We also prove that the Hardy-Littlewood maximal operator operates on certain modulation spaces. This may be of independent interest.

Pointwise convergence to initial data of heat and Hermite-heat equations in Modulation Spaces

Abstract

We characterize weighted modulation spaces (data space) for which the heat semigroup converges pointwise to the initial data as time tends to zero. Here stands for the standard Laplacian or Hermite operator on the Euclidean space. This is the first result on pointwise convergence with data in a weighted modulation spaces (which do not coincide with weighted Lebesgue spaces). We also prove that the Hardy-Littlewood maximal operator operates on certain modulation spaces. This may be of independent interest.

Paper Structure

This paper contains 11 sections, 15 theorems, 94 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. The weight-class in the framework of modulation spaces
  4. Main results
  5. Maximal operator
  6. Preliminaries
  7. Maximal operator on Lebesgue spaces
  8. Short-time Fourier transform (STFT)
  9. Hardy-Littlewood maximal operator on modulation spaces
  10. Heat equation with Laplacian
  11. Heat equation with Hermite operator

Key Result

Theorem 1.1

Let $v$ be a weight in $\mathbb{R}^n, 1 \leq p<\infty$, and heat kernel $\left\{h_t\right\}_t$ be as in heatKernel. Define for some $R,~ 0<R<\infty$. The following statements are equivalent:

Theorems & Definitions (39)

  • Theorem 1.1: vivianiPAMS
  • Remark 1.2
  • Remark 1.3
  • Theorem 2.1
  • Remark 2.2: examples
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 29 more