On sharp estimates for Schrödinger groups of fractional powers of nonnegative self-adjoint operators

Abstract

Let L be a non negative, selfadjoint operator on L2(X), where X is a metric space endowed with a doubling measure. Consider the Schrödinger group for fractional powers of L. If the heat flow e−tL satisfies suitable conditions of Davies–Gaffney type, we obtain the following estimate in Hardy spaces associated to L:‖(I+L)−β/2eiτLγ/2f‖HLp(X)≤C(1+|τ|)nsp‖f‖HLp(X) where p∈(0,1], γ∈(0,1], β/γ=n|12−1p|=nsp and τ∈R.If in addition e−tL satisfies a localized Lp0→L2 polynomial estimate for some p0∈[1,2), we obtain‖(I+L)−β/2eiτLγ/2f‖p0,∞≤C(1+|τ|)nsp0‖f‖p0,∀τ∈R, provided 0<γ≠1, β/γ=n|12−1p|=nsp and τ∈R. By interpolation, the second estimate implies also, for all p∈(p0,p0′), the strong (p,p) type estimate‖(I+L)−β/2eiτLγ/2f‖p≤C(1+|τ|)nsp‖f‖p. The applications of our theory span a diverse spectrum, ranging from the Schrödinger operator with an inverse square potential to the Dirichlet Laplacian on open domains. It showcases the effectiveness of our theory across various settings.