Boundedness of spectral multipliers for Schrödinger operators on open sets

Abstract

Let H_V be a self-adjoint extension of the Schrödinger operator -\Delta+V(x) with the Dirichlet boundary condition on an arbitrary open set \Omega of \mathbb R^d , where d \ge 1 and the negative part of potential V belongs to the Kato class on \Omega . The purpose of this paper is to prove L^p - L^q -estimates and gradient estimates for an operator \varphi(H_V) , where \varphi is an arbitrary rapidly decreasing function on \mathbb{R} , and \varphi(H_V) is defined via the spectral theorem.

On fractional Leibniz rule for Dirichlet Laplacian in exterior domain

“…we have (8) for small values of t and 1 < p < ∞. The estimate ( 8) for small values of t and for the endpoint case p = 1 can be deduced from the results in [22], [29].…”

“…is studied for several situations. The case 1 ≤ p ≤ 2 is studied in [22], [29], where the estimate (4) is proved for any t > 0 in an arbitrary open set. On the other hand, the situation in the case p > 2 is more complicated.…”

Besov spaces generated by the Neumann Laplacian

“…Estimates in amalgam spaces. Following [12] (see also Jensen and Nakamura [13] and the references therein), let us define the amalgam spaces as follows:…”

“…The proof of Lemma B.1 is similar to that of Lemmas 6.3 and 7.1 in [12]. Here, we use the fact that C ∞ 0 (R n )| Ω is dense in H 1 (Ω), which is the main difference from the previous paper [12]. Indeed, instead of this fact, in Dirichlet Laplacian case we used the density of C ∞ 0 (Ω) in H 1 0 (Ω).…”

“…There are a lot of literatures on characterization of Besov spaces (see, e.g., Triebel [18][19][20]). We are concerned with Besov spaces characterized by differential operators via the spectral approach (see [1][2][3]6,7,[10][11][12]14] and the references therein). The purpose of this paper is to give a definition of Besov spaces generated by the Neumann Laplacian on a domain, and prove their fundamental properties; completeness and embedding relations etc.…”