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Characterization of non-linear Besov spaces

Chong Liu, David J. Prömel, Josef Teichmann

Abstract

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how non-linear Besov spaces embed into non-linear $p$-variation spaces and vice versa. We emphasize that we neither assume the UMD property of the involved spaces nor their separability.

Characterization of non-linear Besov spaces

Abstract

The canonical generalizations of two classical norms on Besov spaces are shown to be equivalent even in the case of non-linear Besov spaces, that is, function spaces consisting of functions taking values in a metric space and equipped with some Besov-type topology. The proofs are based on atomic decomposition techniques and metric embeddings. Additionally, we provide embedding results showing how non-linear Besov spaces embed into non-linear -variation spaces and vice versa. We emphasize that we neither assume the UMD property of the involved spaces nor their separability.

Paper Structure

This paper contains 9 sections, 9 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Main results and function spaces
  3. Embeddings of metric spaces into Banach spaces
  4. Notation
  5. Three equivalent norms on vector-valued Besov spaces
  6. Atomic decompositions and harmonic representations
  7. Besov spaces on intervals
  8. Three equivalent norms
  9. Embedding results between Besov and p-variation spaces

Key Result

Theorem \oldthetheorem

Suppose that $(E,d)$ is a metric space. Let $s \in (0,1)$ and $p,q\in [1,\infty]$ be such that $s >1/p$. Then, one has and $\|\cdot\|_{s,p,q}$ and $\|\cdot\|_{s,p,q;(1)}$ are equivalent, i.e., there exist constants $C_1,C_2 > 0$ only depending on $s$, $p$ and $q$ such that for all $f \in C([0,1];E)$.

Theorems & Definitions (30)

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  • ...and 20 more