Globally bisingular elliptic operators

Abstract

The main goal of this work is to extend the notion of bisingular pseudo-differential operators, already introduced on compact manifolds, to Shubin type operators on ℝn = ℝn1 ⊕ ℝn2, n1 + n2 = n. First, we prove global calculus (an analogue of the Γ calculus in the work of Shubin) for such operators, we introduce the notion of bisingular globally elliptic operators and we derive estimates for the action in anisotropic weighted Sobolev spaces, recently introduced by Gramchev, Pilipović, Rodino. Next, we investigate the complex powers of such operators and we demonstrate a Weyl type theorem for the spectral counting function of positive self-adjoint unbounded bisingular globally elliptic operators. The crucial ingredient for the proof is the use of the spectral zeta function. For particular classes of operators, defined as polynomials of P1×P2, P1×Iℝn2, Iℝn1 ×P2, Pj being globally elliptic in ℝnj, j = 1, 2, we are able to estimate and, in some cases, calculate explicitly the lower-order term in the asymptotic expansion of the spectral function.