On subordinated semigroups and Hardy spaces associated to fractional powers of operators

Let $L$ be a positive self-adjoint operator on $L^2(X)$, where $X$ is a $\sigma$-finite metric measure space. When $\alpha \in (0,1)$, the subordinated semigroup $\{\exp (-t{L}^{\alpha }):t\in {\mathbb{R}}^{+}\}$ can be defined on $L^2(X)$ and extended to $L^p(X)$. We prove various results about the semigroup $\{\exp (-t{L}^{\alpha }):t\in {\mathbb{R}}^{+}\}$, under different assumptions on $L$. These include the weak type $(1,1)$ boundedness of the maximal operator $f\mapsto {\sup }_{t\in {\mathbb{R}}^{+}}\exp (-t{L}^{\alpha })f$ and characterisations of Hardy spaces associated to the operator $L$ by the area integral and vertical square function.