Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Journal of Operator Theory đề tài: Đó là số ít các nhà khai thác đa chiều tách rời. I: Các trường hợp bán không gian. | J. operator theory 11 1984 41-76 g Copyright by INCREST 1984- ON MULTIDIMENSIONAL SINGULAR INTEGRAL OPERATORS. I THE HALF-SPACE CASE ROLAND DUDUCHAVA INTRODUCTION Fredholm properties of a multidimensional singular integral operator equation A p x a0 p x i p y dy f x J x -Tl r re R R xR -1 R 0 oo is investigated when p fe Zp R 1 p oo in particular criterion for the existence of left and right regularizers of left and right inverses are obtained cf. Theorems . Those results will be used in the second part of the paper for the investigation of systems of multidimensional singular integral equations on a compact manifold with boundary in vector Sobolev-Slobodeckiĩ spaces. Most interesting in 1 which deals with auxiliary propositions is Theorem it shows that Calderon-Zygmund cf. 4 5 and other similar theorems on the boundedness of the operator in Lp R space is valid also for the space with weight p R X the weight is not pointwise here as in Stein s Theorem cf. 22 . Besides the interest of its own singular integral equations play an important role in mathematical physics mechanics and boundary value problems for the partial differential equations cf. 8 13 18 20 22 24 26 30 32 . It is impossible to observe here all results obtained earlier on this subject and we refer to books 8 13 18 20 22 32 . We mention only a few results closely related to our investigations. Classical theory of multidimensional singular integral equations on manifolds without boundary were developed by Tricomi Michlin Calderon Zygmund and others cf. 22 later Simonenko 31 treated the case of a half space R and. of a compact manifold M with boundary SM 0 but operators were investigated in the space Z R and L Af cf. also 13 24 26 27 . Simultaneously with 42 ROLAND DUDUCHAVA Simonenko Wisik and Eskin cf. 13 treated the case of Sobolev-SIobodeckiĩ spaces Hs2 M ÕM 0 and used obtained results for the investigation of boundary value problems for partial differential equations. There was .
