Boundedness Of A Singular Integral With Variable Nuclear Operator In Some Space

Since Reisz transform is very usfull in partial differential equation , in harmonic analysis, the researches about Reisz transform attract the interest of all and many beautiful results arc obtained. Fractional integral operators with homogeneous or rough kernels are very active subject just surrounding the Reisz transform. On the other hand, Calderon and Zygmund extended the Reisz transform to the convolution operator -the classic Calderon-Zygmund operator, more generally, to the integral operators with variable kernels. The integral operators with variable kernels are not real convolution operator, but part of they are convolution, so we can call them "incomplete convolution". Integral operators of this type are the second generation of Calderon-Zygmund operators. Which we discussed in this thesis are the singular integral operators and fractional integral with variable kernels.The boundedness of the fractional integral with homogenous kernels on most of function spaces is solved, but the boundedness of the integral operators with variable kernels still has many aspects to be improved. During recent years, the improvement of the decomposition theory of function spaces, especially the molecular characterization of the Herz type Hardy spaces, helps us to study the boundedness of the integral operators with variable kernels more easily. Applying the given result and the decomposition theory of Hardy spaces and the Herz type Hardy spaces, the new results of the boundedness of the singular and fractional operators with variable kernels on these spaces are obtained. The main results are as following:(i) For some 0 < p≤1, when the kernelsΩ(x, z) satisfy some Diniconditionsand vanishing moment condition, the singular integral operators T_Ωwith variable kernels are bounded from Hardy spaces H^p(Rⁿ) to themselves;(ii) Under similar conditions, the fractional integral operators T_Ω,μ with variable kernels are bounded from Hardy spaces H^p(Rⁿ) to Hardy spaces H^q(Rⁿ) (where q satisfies:1/q=1/p-μ/n); (iii) When the kernels satisfy some given conditions, the singular integral operators T_Ωwith variable kernels are bounded from Herz type Hardy spaces HK_q^α,p(Rⁿ) to themselves;(iv) Under similar conditions, the fractional integral operators T_Ω,μ with variable kernels are bounded from HK_q1^α,p1(Rⁿ) to HK_q2^α,p2(Rⁿ) (where q₂ satisfies1/q2=1/q1-μ/n).