Exponential-square integrability, weighted inequalities for the square functions associated to operators and applications

TL;DR

This paper proves exponential-square integrability for functions with bounded square functions related to certain operators on metric spaces, leading to new weighted inequalities and eigenvalue estimates, even extending classical results to more general settings.

Contribution

It establishes a novel exponential-square integrability result for functions associated with operators satisfying Gaussian bounds, without the preservation condition, and applies it to various inequalities and eigenvalue problems.

Findings

Proves exponential-square integrability for functions with bounded square functions.

Derives weighted norm inequalities for these functions.

Provides eigenvalue estimates for Schrödinger operators.

Abstract

Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and H\"older's continuity in $x$ but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (i) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (ii) weighted norm inequalities for the square functions; and (iii) eigenvalue estimates for Schr\"odinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.