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Reduced inequalities for vector-valued functions

Tuomas P. Hytönen

Abstract

Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product $\vec f(x)\cdot\vec g(y)$ instead of the product of lengths $|\vec f(x)| |\vec g(y)|$ that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of $L^p$ norms, also with $p<1$, that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato-Ponce inequality (or fractional Leibnitz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method.

Reduced inequalities for vector-valued functions

Abstract

Building on the notion of convex body domination introduced by Nazarov, Petermichl, Treil, and Volberg, we provide a general principle of bootstrapping bilinear estimates for scalar-valued functions into vector-valued versions with a reduced right-hand side involving iterated norms of a pointwise dot product instead of the product of lengths that would result from a naïve extension of the scalar inequality. On the way, we study connections between convex body domination and tensor norms. In order to cover the full regime of norms, also with , that naturally arise in bilinear harmonic analysis, we develop a framework in general quasi-normed spaces. A key application is a vector-valued Kato-Ponce inequality (or fractional Leibnitz rule) with a reduced right-hand side, which we obtain as a soft corollary of the known scalar-valued version and our general bootstrapping method.
Paper Structure (5 sections, 12 theorems, 127 equations)

This paper contains 5 sections, 12 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Tensor products of quasi-normed spaces
  3. Convex bodies and reducing operators
  4. Bilinear operators with a simple bound
  5. Bilinear operators with a two-term bound

Key Result

Theorem 1.10

Let $p_k,q_k\in(1,\infty]$ and $r\in(\frac{1}{2},\infty]$ satisfy $\frac{1}{r}=\frac{1}{p_k}+\frac{1}{q_k}$ for $k=0,1$, and let $s>d(\frac{1}{r}-1)_+$ or $s\in 2\mathbb{N}$. Then for both $D\in\{(-\triangle)^{s/2},(I-\triangle)^{s/2}\}$ and all Schwartz test functions $f,g\in\mathscr S(\mathbb{R}^d)$, where the implied constant may depend on all other parameters, but not on the functions $f$ and

Theorems & Definitions (27)

  • Example 1.7
  • Theorem 1.10: GraOh:14, Theorem 1
  • Theorem 1.14
  • Corollary 1.17
  • proof
  • Example 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 17 more