A theorem about three quadratic forms

TL;DR

This paper proves a self-improvement property for quadratic forms on vector spaces and explores its implications for dimension-free L^p estimates of singular integral operators like Riesz transforms.

Contribution

It introduces a novel self-improvement theorem for quadratic forms and applies it to obtain new dimension-free estimates for Riesz transforms.

Findings

Self-improvement property for quadratic forms established

Dimension-free L^p estimates for Riesz transforms derived

Implications for analysis on arbitrary vector spaces

Abstract

We prove a self-improvement property regarding quadratic forms on arbitrary vector spaces. We discuss several consequences of this result, in particular those concerning dimension-free L^p estimates of certain singular integral operators (Riesz transforms).