Jiecheng Chen

Singular integral operators on function spaces

We study the singular integral operator fx,t(y � ) = f( x− ty � ), defined on all test functions f ,w hereb is a bounded function, α 0, Ω is suitable distribution on the unit sphere S n−1 satisfying some cancellation conditions. We prove certain boundedness properties of TΩ,α on the Triebel–Lizorkin spaces and on the Besov spaces. We also use our results to study the Littlewood–Paley functions. These results improve and extend some well-known results.

Certain Operators with Rough Singular Kernels

We study the singular integral operator TΩ,α f (x) = p.v. ∫ Rn b(|y|)Ω(y )|y| f (x − y) dy, defined on all test functions f ,where b is a bounded function, α ≥ 0, Ω(y ) is an integrable function on the unit sphere Sn−1 satisfying certain cancellation conditions. We prove that, for 1 < p < ∞, TΩ,α extends to a bounded operator from the Sobolev space L p α to the Lebesgue space L p with Ω being a distribution in the Hardy space Hq(Sn−1) where q = n−1 n−1+α . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for TΩ,α on the Hardy spaces, as well as the boundedness for the truncated maximal operator T Ω,m. Received by the editors July 18, 2001; revised June 14, 2002. Supported by 973project (G1999075105), Major project of NNSFC, NSFZJ and NECC. AMS subject classification: 42B20, 42B25, 42B15. c ©Canadian Mathematical Society 2003. 504

^{} bounds for oscillatory hyper-Hilbert transform along curves

We study the oscillatory hyper-Hilbert transform (1) H n,α,β f(x) = ∫ 1 0 f(x-Γ(t))e ιt- t -1-α dt along the curve F(t) = (t p1 , t p2 , ···, t pn ), where p 1 , p 2 , ···, p n,α,β are some real positive numbers. We prove that if β > (n+ 1)α, then H n,α,β is bounded on L p whenever p ∈ (2β 2β-(n+1)α, (2β (n+1)α). Furthermore, we also prove that H n,α,β is bounded on L 2 when β = (n + 1)α. Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an L p boundedness result for some strongly parabolic singular integrals with rough kernels.

BOUNDEDNESS OF MAXIMAL SINGULAR INTEGRALS

The authors study the singular integrals under the Hormander condition and the measure not satisfying the doubling condition. At first, if the corresponding singular integral is bounded from L2 to itself, it is proved that the maximal singular integral is bounded from L∞ to RBMO except that it is infinite μ-a.e. on Rd. A sufficient condition and a necessary condition such that the maximal singular integral is bounded from L2 to itself are also obtained. There is a small gap between the two conditions.