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Vector-valued properties of the Weyl transform

Ritika Singhal, N. Shravan Kumar

Abstract

In this paper, we introduce and study the Weyl transform of functions which are integrable with respect to a vector measure on a phase space associated to a locally compact abelian group. We also study the Weyl transform of vector measures. Later, we also introduce and study the convolution of functions from $L^p$-spaces associated to a vector measure. We also study the Weyl transform of vector-valued functions and prove a vector-valued analogue of the Hausdorff-Young inequality.

Vector-valued properties of the Weyl transform

Abstract

In this paper, we introduce and study the Weyl transform of functions which are integrable with respect to a vector measure on a phase space associated to a locally compact abelian group. We also study the Weyl transform of vector measures. Later, we also introduce and study the convolution of functions from -spaces associated to a vector measure. We also study the Weyl transform of vector-valued functions and prove a vector-valued analogue of the Hausdorff-Young inequality.

Paper Structure

This paper contains 16 sections, 36 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weyl transform and twisted convolution
  4. Operator spaces
  5. Vector measures
  6. Tensor integrability
  7. Weyl transform of vector-valued functions
  8. Weyl type w.r.t a locally compact abelian group
  9. Weyl cotype w.r.t. a locally compact abelian group
  10. Weyl transform on $L^1(G\times\widehat{G},\nu),$ $L^1_w(G\times\widehat{G},\nu)$ and $M(G\times\widehat{G};X)$
  11. Weyl transform on $L^1(G\times\widehat{G},\nu)$
  12. Weyl transform on $M(G\times\widehat{G};X)$
  13. Twisted convolution
  14. Twisted convolution associated to a vector measure
  15. Vector-valued twisted convolution
  16. ...and 1 more sections

Key Result

Proposition 2.1

Suppose $1 \leq p \leq \infty, f \in L^{1}(G \times \widehat{G})$ and $g \in L^{p}(G \times \widehat{G})$.

Theorems & Definitions (72)

  • Proposition 2.1
  • Proposition 2.2
  • Theorem 2.3
  • Theorem 2.4: P2, Chapter 1
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • Theorem 2.7
  • Proposition 2.8
  • ...and 62 more