Heat kernels, upper bounds and Hardy spaces associated to the generalized Schrödinger operators

Abstract Let L = − Δ + μ be the generalized Schrodinger operator on R n , n ≥ 3 , where μ ≢ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Based on Shen's work for the fundamental solution of L in [23] , we establish the following upper bound for semigroup kernels K t ( x , y ) , associated to e − t L , 0 ≤ K t ( x , y ) ≤ C h t ( x − y ) e − e d μ ( x , y , t ) , where h t ( x ) = ( 4 π t ) − n / 2 e − | x | 2 / ( 4 t ) , and d μ ( x , y , t ) is some parabolic type distance function associated with μ . As a consequence, 0 ≤ K t ( x , y ) ≤ C h t ( x − y ) exp ⁡ ( − c 0 ( 1 + m ( x , μ ) max ⁡ { | x − y | , t } ) 1 k 0 + 1 ) , where m ( x , μ ) is some auxiliary function associated with μ . We then study a Hardy space H L 1 by means of a maximal function associated with the heat semigroup e − t L generated by − L to obtain its characterizations via atomic decomposition and Riesz transforms. Also the dual space BMO L of H L 1 is studied in this paper.