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

Let L be a positive self-adjoint operator on L²(X) L 2 ( X ) , where X is a σ σ -finite metric measure space. When α ∈ (0,1) α ∈ ( 0 , 1 ) , the subordinated semigroup exp (-tLα):t ∈ R⁺\ { exp ( - t L α ) : t ∈ R + } can be defined on L²(X) L 2 ( X ) and extended to Lᵖ(X) L p ( X ) . We prove various results about the semigroup exp (-tLα):t ∈ 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 ↦ t∈ R⁺exp (-tLα)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.