On subordinated semigroups and Hardy spaces associated to fractional powers of operators
The Anh Bui, Michael G. Cowling, Xuan Thinh Duong
TL;DR
This work develops a unified analysis of subordinated semigroups $\{e^{-tL^{\alpha}}\}_{t>0}$ for a positive self-adjoint operator $L$ on a $\sigma$-finite metric measure space, leveraging subordination from the heat semigroup and Lévy densities. It derives sharp time-derivative and complex-parameter bounds for the Poisson semigroup ($\alpha=1/2$) and extends these to general $\alpha\in(0,1)$, establishing maximal-function estimates, sectorial extensions, and $L^p$-contraction properties. A key contribution is the weak-type $(1,1)$ bound for the subordinated maximal operator and the $L^p$-bounds for $p>1$, along with the Hardy-space framework: Hardy spaces $H^p_{L^\alpha}(X)$ coincide with $H^p_L(X)$ under Poisson-type upper bounds, and a Littlewood–Paley square function $G_{L^\alpha}$ provides a robust characterization. The results yield a stable, operator-theoretic Hardy space theory under fractional powers, with equivalent BMO spaces and implications for analysis on spaces with ADR geometry and Poisson kernel upper bounds.
Abstract
Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $σ$-finite metric measure space. When $α\in (0,1)$, the subordinated semigroup $\{\exp(-tL^α):t \in \mathbb{R}^+\}$ can be defined on $L^2(X)$ and extended to $L^p(X)$. We prove various results about the semigroup $\{\exp(-tL^α):t \in \mathbb{R}^+\}$, under different assumptions on $L$. These include the weak type $(1,1)$ boundedness of the maximal operator $f \mapsto \sup _{t\in \mathbb{R}^+}\exp(-tL^α)f$ and characterisations of Hardy spaces associated to the operator $L$ by the area integral and vertical square function.