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Orlicz Space Interpolation and Its Applications to Operator Convolution

Wolfram Bauer, Robert Fulsche, Joachim Toft

Abstract

We prove a strong-type interpolation result for noncommutative Orlicz spaces over semifinite von Neumann algebras. Based on this result, we obtain Young-type convolution estimates for the Weyl pseudodifferential symbols of operators in appropriate Orlicz-Schatten spaces. Equivalently, we prove convolution estimates of Young type for Werner's function-operator convolutions in quantum harmonic analysis.

Orlicz Space Interpolation and Its Applications to Operator Convolution

Abstract

We prove a strong-type interpolation result for noncommutative Orlicz spaces over semifinite von Neumann algebras. Based on this result, we obtain Young-type convolution estimates for the Weyl pseudodifferential symbols of operators in appropriate Orlicz-Schatten spaces. Equivalently, we prove convolution estimates of Young type for Werner's function-operator convolutions in quantum harmonic analysis.
Paper Structure (6 sections, 20 theorems, 132 equations)

This paper contains 6 sections, 20 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Interpolation in noncommutative Orlicz spaces
  3. Convolution estimates
  4. Dilated convolutions
  5. Passage to Quantum Harmonic Analysis
  6. Discussion

Key Result

Theorem 1

Assume that $\Phi$ is a quasi-Young function satisfying $0 < p_0 < q_{\Phi} \leq p_{\Phi} < p_1 \leq \infty$ and $T: L^0(\mathcal{M}) \rightarrow L^0(\mathcal{N})$ be a quasilinear operator which is of weak type $(p_i,p_i)$ for $i=0,1$ if $p_1< \infty$ and of strong type $(p_1,p_1)$ if $p_1= \inft

Theorems & Definitions (33)

  • Theorem
  • Theorem
  • Theorem
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Remark 2.6
  • proof
  • ...and 23 more