Sharp Resolvent Estimates Outside of the Uniform Boundedness Range.

In this paper we are concerned with resolvent estimates for the Laplacian Δ in Euclidean spaces. Uniform resolvent estimates for Δ were shown by Kenig et al. (Duke Math J 55(2):329–347, 1987) who established rather a complete description of the Lebesgue spaces allowing such estimates. However, the problem of obtaining sharp L p – L q bounds depending on z has not been considered in a general framework which admits all possible p, q. In this paper, we present a complete picture of sharp L p – L q resolvent estimates, which may depend on z. We also obtain the sharp resolvent estimates for the fractional Laplacians and a new result for the Bochner–Riesz operators of negative index. [ABSTRACT FROM AUTHOR]