This paper is a summary of the research on the mapping properties of sin-gular integrals with rough kernels and the corresponding maximal operators.More precisely,the author presents some recent progress,interesting p...This paper is a summary of the research on the mapping properties of sin-gular integrals with rough kernels and the corresponding maximal operators.More precisely,the author presents some recent progress,interesting problems and typi-cal methods used in the theory concerning the boundedness and continuity for the rough singular integral operators and maximal singular integral operators along cer-tain submanifolds such as polynomial mappings,polynomial curves,homogeneous mappings,surfaces of revolution and real-analytic submanifolds on the Lebesgue spaces,Triebel-Lizorkin spaces,Besov spaces and mixed radial-angular spaces.展开更多
The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) ...The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) is a real-valued polynomial on R n× R n , Ω is homogeneous of degree zero, R m j (A j;x,y) denotes the m j -th order Taylor series remainder of A j at x expanded about y , M=∑kj=1 m j . It is shown that if Ω belongs to the space L log +L(S n-1 ) and has vanishing moment up to order M , then‖T A 1,A 2,…,A k f‖ q C ∏kj=1∑|α|=mj‖D αA j‖ r j ‖f‖ p, provided that 1<p,q<∞ , 1<r j ∞ (j=1,2,...,k) and 1/q=1/p+∑kj=1 1/r j . The corresponding maximal operator is also considered.展开更多
基金survey report given in the Workshop on Operators with Rough Kernels in China(Jinhua,July 15-18,2022)supported partly by the National Natural Sci-ence Foundation of China(Grant No.11701333)。
文摘This paper is a summary of the research on the mapping properties of sin-gular integrals with rough kernels and the corresponding maximal operators.More precisely,the author presents some recent progress,interesting problems and typi-cal methods used in the theory concerning the boundedness and continuity for the rough singular integral operators and maximal singular integral operators along cer-tain submanifolds such as polynomial mappings,polynomial curves,homogeneous mappings,surfaces of revolution and real-analytic submanifolds on the Lebesgue spaces,Triebel-Lizorkin spaces,Besov spaces and mixed radial-angular spaces.
文摘The authors consider the multilinear oscillatory singular integral operator defined by T A 1,A 2,…,A k f(x)=∫ R n e iP(x,y) ∏k j=1 R m j (A j;x,y)Ω(x-y)|x-y| n+M f(y)dy,where P(x,y) is a real-valued polynomial on R n× R n , Ω is homogeneous of degree zero, R m j (A j;x,y) denotes the m j -th order Taylor series remainder of A j at x expanded about y , M=∑kj=1 m j . It is shown that if Ω belongs to the space L log +L(S n-1 ) and has vanishing moment up to order M , then‖T A 1,A 2,…,A k f‖ q C ∏kj=1∑|α|=mj‖D αA j‖ r j ‖f‖ p, provided that 1<p,q<∞ , 1<r j ∞ (j=1,2,...,k) and 1/q=1/p+∑kj=1 1/r j . The corresponding maximal operator is also considered.
