A formula for the resolvent of (-Δ)^{𝑚}+𝑀^{2𝑚}_{𝑞} with applications to trace class

Abstract

ABSTRACT. We derive a formula for the resolvent of the elliptic operator H = (-A)m + Mqm on Z-2(RN) in terms of bounded integral operators Sx and T\ whose kernels we know explicitly. We use this formula to specify the domain of the operator A\ = (H + \I)MP on ¿2(RJV), and to estimate the Hilbert-Schmidt norm of its inverse A^1, for A > 0. Finally we exploit the last two results to prove a trace class criterion for an integral operator K on L2(RN)-0. Introduction.It is quite common that the resolvent of an elliptic partial differential operator can be represented as an integral operator. Our main result is a simple formula for the resolvent (H + A/)-1, A > 0, of the selfadjoint elliptic operator H = (-A)m+M2m on L2(RN), where N > 1 and m > 1 are integers, A is the Laplacian, and Mq denotes the operator of pointwise multiplication by a positive continuous function q on R^. We state this formula as a part of our Theorem 1.3. It conveys sufficient information about the kernel of the integral operator (H + XI)-1, so that we can decide, after a short computation, whether the inverse A^"1 of the (nonselfadjoint) closed operator A\ = (H + XI)MP on L2(RN) is Hilbert-Schmidt or not. Here p is another positive continuous function on RN. In this manner we obtain a Hilbert-Schmidt criterion for the operator A^1 which we state as Theorem 1.4. To complete our study of the operator A\ we identify its domain in Theorem 1.5. As an application of Theorems 1.4 and 1.5 we formulate a trace class criterion for an integral operator K on L2(RN) which we state as Theorem 1.6. In this criterion we formulate sufficient conditions on the kernel k(x,y), x,y E RN, of the integral operator K which imply that K is of trace class. These conditions require that the kernel k(x,y) have both sufficient smoothness and decay at infinity with respect to the x-variable. As a direct consequence of our Theorem 1.6, we state Corollaries 1.7 and 1.8. The latter one shows overlapping between our results and those of Kamp, Lorentz, and Rejto [9]. Finally we illustrate the optimality of our trace class criterion with Example 1.9.As for the organization and methods of this paper, we state our main results as Theorems 1.3 through 1.6 in §1.In §2 we prove Proposition 1.2. To prove the boundedness on L2(RN) of the integral operators Sx and Tx from Definition 1.1, we make use of a singular integral method which involves basic facts about the Hardy-Littlewood maximal function.