On Subordinated Semigroups and Hardy Spaces Associated to Fractional Powers of Operators

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