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Multilinear oscillatory integrals with Calderón-Zygmund kernel. (English) Zbl 0954.42008

The author establishes an L p -boundedness criterion for a class of multilinear oscillatory singular integrals with standard Calderón-Zygmund kernels. Let

R m (A;x,y)=A(x)- |α|<m D α A(y) α!(x-y) α ,
T A 1 ,A 2 f(x)=p.v. n e iP(x,y) K(x,y) |x-y| M-1 j=1 2 R m j (A j ;x,y)f(y)dy,

where M=m 1 +m 2 ; P(x,y) is a real-valued polynomial in x and y; A 1 (x) is a function that satisfies D α A 1 BMO( n ) for all multi-indices α with |α|=m 1 -1; A 2 has derivatives of order m 2 in L s ( n ), 1<s, and K(x,y) is a standard Calderón-Zygmund kernel. The truncated operator S A 1 ,A 2 f(x) is defined by

S A 1 ,A 2 f(x)=p.v. |x-y|<1 K(x,y)|x-y| -M+1 j=1 2 R m j (A j ;x,y)f(y)dy·

The author proved that if P(x,y) satisfies certain conditions, then, for 1/r=1/p+1/s, 1<r, p<,

T A 1 ,A 2 f L r C 1 |α|=m 1 -1 D α A 1 BMO |β|=m 2 D β A 2 L s f L p

if and only if

S A 1 ,A 2 f L r C 2 |α|=m 1 -1 D α A 1 BMO |β|=m 2 D β A 2 L s f L p ,

where C 1 , C 2 are constants depending only on n, p, and deg(P).

This result is also true if r=p, A 2 has derivatives of order m 2 in BMO( n ) and |β|=m 2 D β A 2 L s is replaced by |β|=m 2 D β A 2 BMO .

MSC:
42B20Singular and oscillatory integrals, several variables
42B30H p -spaces (Fourier analysis)
42B35Function spaces arising in harmonic analysis
References:
[1]Ricci, F., Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals I: oscillatory integral,J. of Funct. Anal., 1987, 73: 179. · Zbl 0622.42010 · doi:10.1016/0022-1236(87)90064-4
[2]Lu, S., Zhang, Y., Criterion onL P-boundedness for a class of oscillatory singular integral with mugh kernel,Rev. Mat. Iberoameriana, 1992, 8: 201.
[3]Chen, W., Hu, G., Lu, S., A criterion of (L p,L r ) boundedness for a of multilinear oscillatory singular integrals,Nagoya Math. J., 1998, 149: 33.
[4]Cohen J, Gosselin, J., A BMO estimate for multilinear singular integrals,Illinois J. of Math., 1985, 30: 445.
[5]Chen, W., Hu, G., Lu, S., On a multilinear oscillatory singular integral operatory I,Chin. Ann. of Math., 1997, 18B (2): 181.