Résumé:
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 ] ) .
