Besov spaces generated by the Neumann Laplacian

Abstract

The purpose of this paper is to give a definition and prove the fundamental properties of Besov spaces generated by the Neumann Laplacian. As a by-product of these results, the fractional Leibniz rule in these Besov spaces is obtained.

Bilinear estimates in Besov spaces generated by the Dirichlet Laplacian

“…Hence, the bilinear estimate (2.2) in Theorem 2.1 holds in such a domain. In particular, when Ω is bounded, (2.1) holds for any t > 0, since the infimum of the spectrum is strictly positive (see, e.g., Taniguchi [17] and the references therein). Hence, the bilinear estimate (2.3) in Theorem 2.1 holds.…”

On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

“…we have the same estimate as in the case of the whole space R n . Indeed, in this case, the gradient estimate (4) holds for any t > 0 and 1 ≤ p ≤ ∞ (see, e.g., [17] and [30]).…”

Endpoint Strichartz estimates for the Schrödinger equation on an exterior domain

“…These Besov spaces enjoy the fundamental properties such as completeness, duality and embedding relations, etc. More precisely, we have the following: Proposition 1.3 (Sections 2 and 3 in [18], and also [16], [17], [29]). Let s, s 0 ∈ R and 1 ≤ p, q, q 0 , r ≤ ∞.…”