A LIPSCHITZ ESTIMATE FOR MULTILINEAR OSCILLATORY SINGULAR INTEGRALS WITH ROUGH KERNELS

Abstract

Abstract:

In this paper, for the multilinear oscillatory singular integral operators
TA1,A2,···,Ar defined by
TA1,A2,···,Ar f(x) = p.v.
Z
Rn
eiP(x,y)
(x − y)
|x − y|n+M
r Y
s=1
Rms+1(As; x, y)f(y)dy, n > 2,
where P(x, y) is a nontrivial and real-valued polynomial defined on Rn × Rn,
(x) is
homogeneous of degree zero on Rn, As(x) has derivatives of order ms in ˙ s (0 < s < 1),
Rms+1(As; x, y) denotes the (ms+1)-st remainder of the Taylor series of As at x expended
about y (s = 1, 2, · · · , r), M =
P
r
s=1ms, the author proves that if 0 < =
P
r
s=1 s < 1,
and
2 Lq(Sn−1) for some q > 1/(1 − ), then for any p 2 (1,1), and some appropriate
0 < < 1, TA1,A2,···,Ar is bounded on Lp(Rn).

Key words: Multilinear operator, oscillatory singular integral, Lipschitz spaces, rough
kernel

CLC Number:

42B20

Wu Huoxiong. A LIPSCHITZ ESTIMATE FOR MULTILINEAR OSCILLATORY SINGULAR INTEGRALS WITH ROUGH KERNELS[J].Acta mathematica scientia,Series B, 2005, 25(4): 761-770.

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