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

Let L=-δ+μ be the generalized Schrödinger operator on Rⁿ, 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^-tL,0≤K_t(x,y)≤Ch_t(x-y)e^{-εdμ(x,y,t)}, where h_t(x)=(4πt)^-n/2e-|x|²/(4t), and d_μ(x, y, t) is some parabolic type distance function associated with μ. As a consequence,0≤K_t(x,y)≤Ch_t(x-y)exp (-c0(1+m(x,μ)max {|x-y|,√t})1/k₀+1),where m(x, μ) is some auxiliary function associated with μ. We then study a Hardy space H_L¹ by means of a maximal function associated with the heat semigroup e^-tL generated by -L to obtain its characterizations via atomic decomposition and Riesz transforms. Also the dual space BMO_L of H_L¹ is studied in this paper.