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. II: Trường hợp của đa tạp compact. | J. OPERATOR THEORY 11 1984 199-214 Copyright by INCREST 1984 ON MULTIDIMENSIONAL SINGULAR INTEGRAL OPERATORS. II THE CASE OF COMPACT MANIFOLDS ROLAND DUDUCHAVA INTRODUCTION Singular integral operators equations A p x a x p x i x pịy dy x X e M J k - y n u are investigated where M is a compact manifold with the boundary ÕM 0 f x e 77sp A Af p e 1 p co co 5 co proofs here are based on 3 where the first part of the present investigations the half-space case M R was published. As it was already mentioned in 3 the operators were investigated by Simonenko 6 the case p 2 s 0 and by Wisik and Eskin 7 the case p 2 co 5 co . The main tool of investigation here is the local principle cf. which is a slight modification of the local principle from 5 cf. extended with some notions from 6 We set forth here the numeration of sections thus the references to 1 2 will mean the reference to 3 all notations from 3 are used without the further explanations. 3. PRELIMINARIES 1 . On the isomorphism of soBOLEV-SLOBODECKn SPACES. Consider the operators A p - g cp As p p e Co R co s co 200 ROLAND DUDUCHAVA where gi t ici T iè i F 1 i ễi o G R gỉU 1 - ỉ these operators have continuous extensions As A Hrp R - Hir sip Ra - oo r oo and these extensions are isomorphisms 4s arranges even the isometricai isomorphism ll plirp iMM r-S p obviously ASA S I A Afs I. The operators 4 P1 A l1 where p1 and 1 are restricting and extending operators also arrange isomorphisms cf. 2 4 ÌỊ R H f-S R rfAf - ÃL 7 7 Rn - H - ịR -OO s r oo . The following two lemmas are easy to prove cf. for example 2 4 . Lemma . The operator w ae P Hsp N Rn is isomorphic wo b-swoBs B s Ai4 4 Sj 4 to itself wẳ G Lf Lp Rnf operating in the space Lp Rn therefore G G S Hsp N Rn if and only if aif G Mp N Rn 1 p oo oo .V oo . Lemma . The operator Wl G J fHsf N Rn tfĩsp N Rn is isomorphic wi Zlz lt to the operator jyi G J5 Z R s O Ya O 5 V Ki - f ị - 1 hence is valid if and only if a f G Aip A R . We .
