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

Let L=−Δ+μL=−Δ+μ be the generalized Schrödinger operator on RnRn, n≥3n≥3, where μ≢0μ≢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 LL in [23], we establish the following upper bound for semigroup kernels Kt(x,y), associated to e−tLe−tL,0≤Kt(x,y)≤Cht(x−y)e−εdμ(x,y,t), where ht(x)=(4πt)−n/2e−|x|2/(4t)ht(x)=(4πt)−n/2e−|x|2/(4t), and dμ(x,y,t)dμ(x,y,t) is some parabolic type distance function associated with μ. As a consequence,0≤Kt(x,y)≤Cht(x−y)exp⁡(−c0(1+m(x,μ)max⁡{|x−y|,t})1k0+1),where m(x,μ)m(x,μ) is some auxiliary function associated with μ  . We then study a Hardy space HL1 by means of a maximal function associated with the heat semigroup e−tLe−tL generated by −L−L to obtain its characterizations via atomic decomposition and Riesz transforms. Also the dual space BMOLBMOL of HL1 is studied in this paper.