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Paraproducts on local dyadic fractional Sobolev spaces

Valentia Fragkiadaki, Mishko Mitkovski, Cody B. Stockdale

Abstract

We characterize the boundedness and compactness of dyadic paraproducts on local dyadic fractional Sobolev spaces. Our conditions are stated in terms of new dyadic fractional BMO and CMO conditions involving the dyadic fractional Sobolev capacity, and our proofs use a new dyadic fractional version of the Carleson embedding theorem.

Paraproducts on local dyadic fractional Sobolev spaces

Abstract

We characterize the boundedness and compactness of dyadic paraproducts on local dyadic fractional Sobolev spaces. Our conditions are stated in terms of new dyadic fractional BMO and CMO conditions involving the dyadic fractional Sobolev capacity, and our proofs use a new dyadic fractional version of the Carleson embedding theorem.

Paper Structure

This paper contains 7 sections, 8 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Local dyadic fractional derivatives, integrals, and Sobolev spaces
  4. Dyadic fractional Sobolev capacity, , and -Carleson sequences
  5. Main results
  6. Dyadic fractional Carleson embedding theorem
  7. Proof of Theorem \ref{['Thm: ParaproductInequality']}

Key Result

Theorem 1.1

Let $s \in (0,1)$ and $b \in L^1$. Then $\Pi_b$ is bounded on $H^s$ if and only if $b \in \text{BMO}^s$, and, in this case, Moreover, $\Pi_b$ is compact on $H^s$ if and only if $b \in \text{CMO}^s$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 9 more