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Gagliardo-Nirenberg inequality via a new pointwise estimate

Karol Lesnik, Tomas Roskovec, Filip Soudsky

Abstract

We prove a new type of pointwise estimate of the Kalamajska-Mazya-Shaposhnikova type, where sparse averaging operators replace the maximal operator. It allows us to extend the Gagliardo-Nirenberg interpolation inequality to all rearrangement invariant Banach function spaces without any assumptions on their upper Boyd index, i.e. omitting problems caused by unboundedness of maximal operator on spaces close to $L^1$. In particular, we remove unnecessary assumptions from the Gagliardo-Nirenberg inequality in the setting of Orlicz and Lorentz spaces. The applied method is new in this context and may be seen as a kind of sparse domination technique fitted to the context of rearrangement invariant Banach function spaces.

Gagliardo-Nirenberg inequality via a new pointwise estimate

Abstract

We prove a new type of pointwise estimate of the Kalamajska-Mazya-Shaposhnikova type, where sparse averaging operators replace the maximal operator. It allows us to extend the Gagliardo-Nirenberg interpolation inequality to all rearrangement invariant Banach function spaces without any assumptions on their upper Boyd index, i.e. omitting problems caused by unboundedness of maximal operator on spaces close to . In particular, we remove unnecessary assumptions from the Gagliardo-Nirenberg inequality in the setting of Orlicz and Lorentz spaces. The applied method is new in this context and may be seen as a kind of sparse domination technique fitted to the context of rearrangement invariant Banach function spaces.
Paper Structure (5 sections, 10 theorems, 113 equations)

This paper contains 5 sections, 10 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Sparse domination
  3. Gagliardo--Nirenberg inequality
  4. Applications to Lorentz and Orlicz spaces
  5. Summary and comments

Key Result

Theorem 1.1

Let $X, Y$ be r.i. Banach function spaces over $\mathop{\mathrm{\mathbb{R}}}\nolimits^n$ and let $Z:=X^{j/k}Y^{1-j/k}$ be the Calderón--Lozanovskii space, where $1\leq j< k$. Then, there exists a positive constant $C_a$ (independent of dimension) such that for each $1\leq i\leq n$ inequality holds for each $u\in W^{k,1}_{\mathop{\mathrm{loc}}\limits}(\mathop{\mathrm{\mathbb{R}}}\nolimits^n)$. In

Theorems & Definitions (29)

  • Theorem 1.1: Gagliardo-Nirenberg interpolation inequality for r.i. Banach function spaces
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['crux']}
  • proof : Proof of Corollary \ref{['corcrux']}
  • Lemma 2.3
  • proof
  • Definition 3.1: Calderón--Lozanovskii construction
  • ...and 19 more