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Matrix weighted modulation spaces

Morten Nielsen

Abstract

Given a matrix-weight $W$ in the Muckenhoupt class $\mathbf{A}_p(\mathbb{R}^n)$, $1\leq p<\infty$, we introduce corresponding vector-valued continuous and discrete $α$-modulation spaces $M^{s,α}_{p,q}(W)$ and $m^{s,α}_{p,q}(W)$ and prove their equivalence through the use of adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and an application to the study of pseudo-differential operators on vector valued spaces is given.

Matrix weighted modulation spaces

Abstract

Given a matrix-weight in the Muckenhoupt class , , we introduce corresponding vector-valued continuous and discrete -modulation spaces and and prove their equivalence through the use of adapted tight frames. Compatible notions of molecules and almost diagonal matrices are also introduced, and an application to the study of pseudo-differential operators on vector valued spaces is given.
Paper Structure (12 sections, 16 theorems, 134 equations)

This paper contains 12 sections, 16 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Vector-valued smoothness spaces
  3. The family of $\alpha$-coverings of the frequency domain
  4. Matrix-weighted $L^p$-spaces and Muckenhoupt weights
  5. Vector valued modulation spaces
  6. Discrete vector valued modulation spaces and norm characterzations
  7. Stable expansions and almost diagonal matrices
  8. Reducing operators and the connection to scalar spaces
  9. Almost diagonal matrices
  10. An application to Fourier multipliers
  11. Fourier Multipliers
  12. Completeness of the $\alpha$-modulation spaces

Key Result

Lemma 2.7

The family of functions $\{\varphi_k\}_k$ defined in Eq. eq:b satisfies Definition de:bapu.

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Remark 2.6
  • Lemma 2.7
  • proof
  • Remark 2.8
  • Definition 2.9
  • ...and 41 more