Regularity Of Thermal Semigroups On Heisenberg Groups And Their Applications In Function Space

Let L=-△G+V be a Schrodinger operator on Heisenberg groups,where AG is the sub-Laplacian and V belongs to the reverse H(?)lder class.Due to its applications in some mathematical fields such as harmonic analysis and partial differential equations,the study of both function spaces and the boundedness of singular integral operators associated with Schr(?)dinger operators has become an active area of research in the last few years.This paper is mainly divided into two parts.Firstly,By the aid of subordinate formula,we investigate the regularity properties of the time-fractional derivatives of semigroups {e-tL}t>0 and {e-t√L}t>0,respectively.As an application,using fractional square functions,we characterize the Hardy-Sobolev type space HL1,α(G)associated with L.Moreover,the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.Secondly,As the first part of generalization,we study the regularity properties of fractional heat semigroup {e-tLβ}r>0,0<β<1.Moreover,we introduce a new Campanato type space CLγ(G)of vanishing mean oscillation associated with L.By Carleson measures related to fractional heat semigroups,we establish an equivalent characterization of CLγ(G).As an application,we prove that the dual of CLγ(G)is BLp(G),where BLp(G)is the completeness of HLp(G).