On integral representations of functions determined by one class of pseudodifferential operators
We construct an integral representation for pseudodifferential operators \[ p(D) u= \int_{E_n} e^{ix\xi} p(i\xi) \widehat u(\xi) d\xi, \] with \(p(i\xi)\) a continuous function in \(E_n\) satisfying some conditions. Earlier, S. V. Uspenskiĭ (1972) obtained such a representation for hypoelliptic differential operators and pseudodifferential operators with symbol homogeneous with respect to some vector \(\alpha= (\alpha_1, \alpha_2,\dots, \alpha_n)\). The kernels of the integral representations constructed in (loc. cit.) are regularizations with respect to a parameter of fundamental solutions to the corresponding equations. This approach proved convenient for studying the differential properties and the behavior at infinity of solutions to quasielliptic equations as well as for studying boundary value problems for such equations.
integral representation for pseudodifferential operators
scientific article; zbMATH DE number 923570