A class of generalized integral operators.

Abstract

In this paper , we introduce a class of generalized integral operators that includes Fourier integral operators . We establish some conditions on these operators such that they do not have bounded extension on L2(Rn): This permit us in particular to construct a class of Fourier integral operators with bounded symbols in S0 1;1(Rn Rn) and in T0< <1S0 ;1(Rn Rn) which cannot be extended to bounded operators in L2(Rn): 1 . Introduction The integral operators of type A'(x) = Z eiS(x; )a(x; )F'( )d (1:1) appear naturally for solving the hyperbolic partial di erential equations and ex - pressing the C1􀀀 solution of the associate Cauchy problem ' s ( see e . g . [ 1 0 , 1 1 ] ) . If we write formally the expression of the Fourier transform F'( ) in ( 1 . 1 ) , we obtain the following Fourier integral operators , so - called canonical transformations , A'(x) = Z ei(S(x; )􀀀y )a(x; y; )'(y)dyd (1:2) in which appear two C1􀀀 functions , the phase function (x; y; ) = S(x; ) 􀀀 y and the amplitude a called the symbol of the operator A: In the particular case where S(x; ) = x ; one recovers the notion of pseudodi erential operators ( see e . g [ 6 , 1 5 ] ) . Since 1 970 , many of Mathematicians have been interested to these type of op - erators : Duistermaat [ 3 ] , H o rmander [ 6 , 7 ] Kumano - Go [ 8 ] , and Fuj iwara [ 2 ] . We mention also the works of Hasanov [ 4 ] , and the recent results of Messirdi Senous - saoui [ 1 2 ] and Aiboudi - Messirdi - Senoussaoui [ 1 ] . In this paper we study a general class of integral operators including the class of Fourier integral operators , specially we are interested in their continuity on L2(Rn): The continuity of the operator A on L2(Rn) is guaranteed if the weight of the symbol a is bounded , if this weight tends to zero then A is compact on L2(Rn)( see eg . [ 1 2 ] ) .