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Dyadic fractional Sobolev spaces: Embeddings and algebra property

Patricia Alonso Ruiz, Valentia Fragkiadaki

Abstract

This paper studies a dyadic version of fractional Sobolev spaces in $\mathbb{R}^n$ for $n\geq 1$. It provides new proofs of the corresponding fractional Sobolev embedding as well as the algebra property of the spaces, which rely solely on dyadic techniques and in particular bypass the Fourier transform. Specific counterexamples are constructed to verify the failure of the algebra property in low-regularity ranges.

Dyadic fractional Sobolev spaces: Embeddings and algebra property

Abstract

This paper studies a dyadic version of fractional Sobolev spaces in for . It provides new proofs of the corresponding fractional Sobolev embedding as well as the algebra property of the spaces, which rely solely on dyadic techniques and in particular bypass the Fourier transform. Specific counterexamples are constructed to verify the failure of the algebra property in low-regularity ranges.

Paper Structure

This paper contains 13 sections, 18 theorems, 79 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and background
  3. The dyadic setting
  4. Useful equalities
  5. Fractional Sobolev spaces
  6. Fractional embeddings
  7. Case [small]
  8. Case [critical]
  9. Case [large]
  10. Algebra property
  11. Case [small]
  12. Case [critical]
  13. High regularity: [large]

Key Result

Theorem 1

Let $s>0$ and $1<q<\infty$. Then, it holds that $H^{s}_\mathcal{D}(\mathbb{R}^n)\subseteq L^q(\mathbb{R}^n)$ when

Figures (1)

  • Figure 1: Dyadic cubes involved in the counterexample \ref{['E:counterex_low_reg']} ($n=2$).

Theorems & Definitions (36)

  • Theorem
  • Theorem
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Lemma 3
  • Lemma 4
  • proof
  • ...and 26 more