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# L～P Estimates for Rough Singular Integrals Associated to Some Hypersurfaces in the Mixed Homogeneity Space

Author: TaoYing
Tutor: WangSiLei；ChenJieCheng
School: Zhejiang University
Course: Basic mathematics
Keywords: Mixed homogeneous space Rough Singular Integral Operators Hypersurface Boundedness
CLC: O186.5
Type: Master's thesis
Year: 2007
Quote: 0

### Abstract

 In the the mixed homogeneous space , considering the rough nuclear family Calderán-Zygmund kernel K singular integral operator T, and the family of operators Son and { (y , φ (ψ (y )) ): y ∈ R n } , where φ (t) is [ 0 , ∞ ) on increasing convex C 2 < / sup > function , ψ R n on smooth convex function A t is a homogeneous and a- convex in the the mixed homogeneous space ( R n < / sup> , ρ ) . Operator T is defined as Tf (x, t) = p. v. integral from n = R n f (xy, t-φ (ψ (y)) b (ψ (y)) K (y) dy where b is [ 0, ∞ ) on the boundary function. K | S n - 1 also meet certain vanishing conditions and some assumptions , but not necessarily smooth, we get the same family of operators L p bounded φ (t) the maximal operator . This paper prove the following two theorems : Theorem 1 : Let Ω ∈ L q ( S n - 1 < / sup >) , q > 1 , assuming mixed homogeneous space φ satisfies φ : [0 , ∞) → R , ψ is a a- convex and a t is a homogeneous , ie ψ ( the a t x ) = tψ (x). Then T L P ( R n < / sup> ) bounded for all 1 < p < ∞ , and as long as the form the V φ g ( t) = sup k ∈ Z | integral from n = 2 k to 2 k 1 g (t-φ (r)) dr / r | V φ L p (R n ) bounded , 1 n - 1 < / sup >) , assuming φ satisfy φ : [0 , ∞) → R , ψ is a a- convex , and About A t is a homogeneous , i.e. ψ (A t x ) = tψ ( x ) , then T is L P ( R n 1 ) bounded operator for all 1 < p < ∞ shaped like a V φ m < / sup > g (t ) = sup k ∈ z | integral from n = 2 mk to 2 ( m (k 1) g (t-φ (r)) dr ╱ r | of the V < of sub> φ m satisfies ‖ V the φ m g ‖ L p (R) ≤ C p m ‖ g ‖ L p (R).

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