Index computation for a bisingular operator

Weyl asymptotics of bisingular operators and Dirichlet divisor problem

“…In (27), (a m 1 ,· ) l and (a ·,m 2 ) l are the symbols of the complex powers of the operators a m 1 ,· (x 1 , x 2 , ξ 1 , D 2 ) and a ·,m 2 (x 1 , x 2 , D 1 , ξ 2 ). In order to obtain the terms in (26), (27), (28), we notice that the constant τ in (22) is arbitrary and the Laurent coefficients clearly do not change if we change the partition of R n 1 × R n 2 , therefore we can let τ tend to infinity. In this way both the fourth and fifth integral in (22) vanish, due to the continuity of the integral w.r.t.…”

“…For the index of bisingular operators Ubertino Battisti, Università degli Studi di Torino, via Carlo Alberto 10, 10123 Torino Tel. : +39 011 6702877 Fax: +39 011 6702878 E-mail: ubertino.battisti@unito.it see also the work of V. S. Pilidi [27] and of R. V. Dudučava [5,6]. In [23], R. Melrose and F. Rochon introduced pseudodifferential operators of product type, a class of operators close to bisingular operators.…”

On a class of pseudodifferential operators on the product of compact Lie groups

“…In this section, we define what we shall call class of bisingular symbols , since, as pointed out by Rodino in [12], it contains symbols of operators of bisingular type (see [9],[10], and [15]). Notation In what follows, we call $$x=(x_1,x_2)$$ an element of $$G=G_1\times G_2$$ and $$\xi :=\xi _1\otimes \xi _2$$ an element of $$\widehat{G}$$, where $$\xi _j\in \widehat{G}_j$$.…”

“…The study of these operators goes back to 1971, when Pilidi in [9] reduced the boundary value problem for functions of two complex variables in bicylinders to the analysis of a bisingular equation on the two distinguished boundaries. In [10] the same author also developed a product calculus to deal with these objects and considered the corresponding index problem. Afterward, a priori estimates and Fredholm properties for bisingular operators were studied by Rabinovič in [11], while in 1975 Rodino in [12] introduced the so‐called calculus of bisingular pseudodifferential operators.…”

A Two Dimensional Adler-Manin Trace and Bi-singular Operators

“…The literature in this connection is wide, involving problems of operator theory, harmonic analysis and several complex variables, see for example [34,32,19], coming from the school of A. Zygmund, and [30,26,27,20], from the school of F. D. Gahov.…”