Hardy spaces for semigroups with Gaussian bounds

TL;DR

This paper studies Hardy spaces associated with semigroups of operators that satisfy Gaussian bounds, establishing atomic decompositions with weighted cancellation conditions.

Contribution

It introduces a new atomic decomposition for Hardy spaces linked to semigroups with Gaussian bounds, incorporating a weighted cancellation condition.

Findings

Established atomic decomposition for H^1_L

Identified a weight function x(x) for atoms

Proved boundedness of the semigroup operators

Abstract

Let T_t=e^{-tL} be a semigroup of self-adjoint linear operators acting on L^2(X,mu), where (X,d mu) is a space of homogeneous type. We assume that T_t has an integral kernel T_t(x,y) which satisfies the upper and lower Gaussian bounds: \frac{C_1}{mu(B(x,\sqrt{t}))} \exp(-c_1d(x,y)^2/t)\leq T_t(x,y) \leq \frac{C_2}{\mu(B(x,\sqrt{t}))} \exp(-c_2 d(x,y)^2/t). By definition, f belongs to H^1_L if \| f\|_{H^1_L}=\|\sup_{t>0}|T_t f(x)|\|_{L^1(X,\mu)} <\infty. We prove that there is a function \omega(x), 0<c \leq \omega(x) \leq C, such that H^1_L admits an atomic decomposition with atoms satisfying: supp a \subset B, \|a\|_{L^\infty} \leq mu(B)^{-1}, and the weighted cancellation condition \int a(x)\omega(x) dmu(x)=0.