[Paper Review] Maximal function characterizations for Hardy spaces associated to nonnegative self-adjoint operators on spaces of homogeneous type

Liang Song, Lixin Yan|arXiv (Cornell University)|May 25, 2016

Advanced Harmonic Analysis Research3 references3 citations

TL;DR

This paper establishes the equivalence of maximal and atomic Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type, under Gaussian heat kernel bounds and without requiring regularity of the kernel in spatial variables. It proves that $ H^p_{L, ext{max}}(X) \simeq H^p_{L,\text{at},q,M}(X) $ for $ 0 < p \leq 1 $, $ q > p $, and $ M > \frac{n}{2}(\frac{1}{p} - 1) $, using a Calderón-type decomposition and maximal function estimates.

ABSTRACT

Let $X$ be a metric measure space with a doubling measure and $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. In this article we continue a study in \cite{SY} to give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(X)$ in terms of the nontangential maximal function associated with the heat semigroup of $L$, and hence we establish characterizations of Hardy spaces associated to an operator $L$, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of $ H^p_{L, max}(X)$ in terms of the radial maximal function.

Motivation & Objective

• To establish the equivalence between maximal and atomic Hardy spaces associated with nonnegative self-adjoint operators on spaces of homogeneous type.
• To remove the need for Hölder regularity assumptions on the heat kernel's spatial variables, which were required in prior works.
• To prove that the maximal function characterization of $ H^p_L(X) $ is equivalent to its atomic decomposition under Gaussian upper bounds.
• To extend the equivalence to include radial maximal functions, completing the characterization of operator-based Hardy spaces.
• To generalize previous results from Euclidean space to the broader setting of spaces of homogeneous type with doubling measures.

Proposed method

• Adapts a Calderón-type decomposition technique to construct atomic decompositions directly from the maximal function norm.
• Uses the nontangential maximal function $ f^*_L(x) = \sup_{d(x,y)<t} |e^{-t^2L}f(y)| $ as the central tool for defining the maximal Hardy space $ H^p_{L,\text{max}}(X) $.
• Applies the Gaussian upper bound on the heat kernel $ p_t(x,y) \leq \frac{C}{V(x,t)} \exp\left(-\frac{d(x,y)^2}{ct}\right) $ to control operator norms and integral estimates.
• Employs dyadic decomposition of frequency and time scales via a partition of unity and a function $ \Psi $ with compactly supported Fourier transform.
• Estimates integral operators involving $ \Psi(t\sqrt{L}) $ and $ t^2Le^{-t^2L} $ using the Calderón reproducing formula and maximal function techniques.
• Establishes $ L^p $ bounds for the maximal function via weak-type estimates and the John-Nirenberg inequality, leveraging the doubling measure structure.

Experimental results

Research questions

• RQ1 Can the maximal function characterization of Hardy spaces associated with nonnegative self-adjoint operators be shown to be equivalent to the atomic decomposition without assuming spatial regularity of the heat kernel?
• RQ2 Does the equivalence $ H^p_{L,\text{max}}(X) \simeq H^p_{L,\text{at},q,M}(X) $ hold on spaces of homogeneous type under only Gaussian heat kernel bounds?
• RQ3 Is the radial maximal function equivalent to the nontangential maximal function in the operator-based Hardy space setting?
• RQ4 Can the Calderón-type decomposition be adapted to spaces of homogeneous type with doubling measures and no smoothness assumptions on the kernel?
• RQ5 What is the sharp range of parameters $ p, q, M $ for which the atomic and maximal Hardy spaces coincide?

Key findings

• The maximal Hardy space $ H^p_{L,\text{max}}(X) $ is equivalent to the atomic Hardy space $ H^p_{L,\text{at},q,M}(X) $ for all $ 0 < p \leq 1 $, $ q > p $, and integers $ M > \frac{n}{2}(\frac{1}{p} - 1) $, under the Gaussian upper bound assumption.
• The equivalence holds without requiring Hölder continuity of the heat kernel in the spatial variables, extending prior results that required such regularity.
• The radial maximal function $ f^*_{L,\text{rad}}(x) = \sup_{t>0} |e^{-t^2L}f(x)| $ is equivalent to the nontangential maximal function, hence $ H^p_{L,\text{rad}}(X) \simeq H^p_{L,\text{max}}(X) $.
• The area function characterization $ H^p_{L,S}(X) $ is equivalent to the maximal and atomic spaces, completing a full characterization chain: $ H^p_{L,\text{at},q,M} \simeq H^p_{L,S} \simeq H^p_{L,\text{max}} \simeq H^p_{L,\text{rad}} $.
• The proof relies on a modified Calderón reproducing formula and $ L^p $ estimates for integral operators involving $ \Psi(t\sqrt{L}) $, with uniform bounds in time and space.
• The constant in the equivalence depends only on the doubling constant and the Gaussian parameters, not on the regularity of the kernel.