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

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/2e^iτLγ/2f‖_HL^p(X)≤C(1+|τ|)^ns_p‖f‖_HL^p(X) where p∈(0,1], γ∈(0,1], [Formula presented] and τ∈R. If in addition e^−tL satisfies a localized L^p₀→L² polynomial estimate for some p₀∈[1,2), we obtain ‖(I+L)^−β/2e^iτLγ/2f‖_p0,∞≤C(1+|τ|)^ns_p0‖f‖_p0,∀τ∈R, provided 0<γ≠1, [Formula presented] and τ∈R. By interpolation, the second estimate implies also, for all p∈(p₀,p₀^′), the strong (p,p) type estimate ‖(I+L)^−β/2e^iτLγ/2f‖_p≤C(1+|τ|)^ns_p‖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.