Weak type (p,p) bounds for Schrödinger groups via generalized Gaussian estimates

Abstract

Let L be a non-negative self-adjoint operator acting on L2(X), where X is a space of homogeneous type with a dimension n. Suppose that the heat operator e−tL satisfies the generalized Gaussian (p0,p0′)-estimates of order m for some 1≤p0<2. It is known that the operator (I+L)−seitL is bounded on Lp(X) for s≥n|1/2−1/p| and p∈(p0,p0′) (see for example, [5,7,9,10,13,26]). In this paper we study the endpoint case p=p0 and show that for s0=n|12−1p0|, the operator (I+L)−s0eitL is of weak type (p0,p0), that is, there is a constant C>0, independent of t and f so thatμ({x:|(I+L)−s0eitLf(x)|>α})≤C(1+|t|)n(1−p02)(‖f‖p0α)p0,t∈R for α>0 when μ(X)=∞, and α>(‖f‖p0/μ(X))p0 when μ(X)<∞. Our results can be applied to Schrödinger operators with rough potentials and higher order elliptic operators with bounded measurable coefficients although in general, their semigroups fail to satisfy Gaussian upper bounds.