Boundedness Of Some Singular Integral Operators And The Compactness Of Commutator

This Ph.D. Thesis focus on the boundedness of some singular integral op-erators in harmonic analysis and the compactness of commutator. It is divided into six chapters.In Chapter 1, we will introduce the background and the main results ob-tained in this thesis.In Chapter 2, we use interpolation and iterative methods to study the frac-tional integral operator FΩ,α with variable kernel. We obtain the sharp size con-dition on Ω to ensure the (Lq(Rn), Lp(Rn)) boundedness of FΩ,α for 0<α< n, 1<p<∞. We also obtain some corresponding estimates of the rough bilinear fractional integral.In Chapter 3, as we all know, when we use the method of rotations for the fractional integral operator with variable kernel, the directional fractional integral plays an important role. We will use the spherical harmonic development and the interpolation of mixed norm to study the boundedness of the directional fractional integral. As a consequence of this result, we obtain a corollary for FΩ,α.In Chapter 4, we continuous to consider the singular integral with variable kernel. By Fourier transform estimates and some approximation, we prove that if b∈CMO(Rn), the commutator TΩ,α,b which is generated by TΩ,α with b is compact from L2n/n+2a(Rn) to L2(Rn).In Chapter 5, we study the oscillatory hyper-Hilbert transform Hnn,α,β along the curve Γ(t)= (tp1, tp2,..., tpn). Firstly, using the integral by parts and interpo-lation, we obtain that Hn,α,β is bounded from from Lγ2(Rn) to L2(Rn). Further-more, on the basis of the first step, we prove that Hn,α,β is bounded from Lγp(Rn) to LP(Rn).In Chapter 6, we assume (X, d,μ) be a geometrically doubling metric space and the measure μ satisfies the upper doubling condition. By invoking a Cotlar type inequality, we show that the maximal bilinear Calderon-Zygmund operators of type cu(t) is bounded from Lp1(μ) x Lp2(μ) into Lp(μ) for any pi ∈ (1,∞] and bounded from LP1(μ) × Lp2(μ) into Lp,∞)(μ) for p1=1 or P2= 1, where p∈[1/2,∞),1/p1+1/p2=1/p.Moreover, if w=(w1,w2) belongs to the weight class Ap�'ρ(μ), using the John-stromberg maximal operator and the John-stromberg sharp maximal operator, the authors obtain a weighted weak type esti-mate Lp1(w1)×LP2(w2)-> Lp,∞(vw) for the maximal bilinear Calderon-Zygmund operators of type ω(t). By weakening the assumption of ω∈ Dini(1/2) into ω∈Dini(1), the results obtained in this paper are substantial improvements and extensions of some known results, even on Euclidean spaces Rn.