Sharp Resolvent Estimates Outside of the Uniform Boundedness Range

Abstract

In this paper we are concerned with resolvent estimates for the Laplacian ∆ in Euclidean spaces. Uniform resolvent estimates for ∆ were shown by Kenig, Ruiz and Sogge [32] 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.2010 Mathematics Subject Classification. 35B45, 42B15.

Time-Harmonic Solutions for Maxwell’s Equations in Anisotropic Media and Bochner–Riesz Estimates with Negative Index for Non-elliptic Surfaces

“…We also show that the strong bounds are sharp for α ≥ 1 2 . In the elliptic case the currently best results were shown by Kwon-Lee [33,Section 2.6]. This also shows that our strong bounds are not sharp for α < 1 2 .…”

“…The body of literature concerned with Bochner-Riesz estimates with negative index is huge, see, e.g., [5,10,24,33,41]. In Sect.…”

A limiting absorption principle for Helmholtz systems and time-harmonic isotropic Maxwell's equations

“…For any fixed ζ ∈ C \ R ≥0 , however, the estimate (11) actually holds for a larger range of exponents, which is due to the improved properties of the Fourier symbol 1/(|ξ| 2 − ζ) and related Bessel potential estimates. We refer to [24] for more details about sharp L p − L q resolvent estimates of the form (11). Theorem 6 extends in an obvious way to the system case that we shall need in the following.…”

An annulus multiplier and applications to the limiting absorption principle for Helmholtz equations with a step potential

“…Here, the last equality comes from the fact that the third number inside the bracket of the second last line lies between the fourth and the fifth number. Combining this with (32) gives the desired bound.…”

“…Another reference for this result and for the optimality of the asserted range can be found in [32,Theorem 2.14]. We will also need several Fourier restriction theorems for the Fourier transforms F n , F n−1 restricted to spheres in R n−1 respectively R n .…”