L～P Estimates For Rough Singular Integrals Associated To Some Hypersurfaces In The Mixed Homogeneity Space

In the mixed homogeneity space, we consider a class of rough integral operatorsT that are associated to a Calderón-Zygmund tpye kernel K and a hypersurface givenby the graph {(y,φ(ψ(y))): y∈Rⁿ} in a mixed homogeneity space (Rⁿ,ρ). Hereφ(t)is an increasing convex C² function on [0,∞), andψis a smooth convex function onRⁿ, which is homogeneous of degree one with respect to A_t and of a-convex type. Theoperator T is defined by Tf(x,t)=p.v. integral from n=Rⁿ f(x-y,t-φ(ψ(y))b(ψ(y))K(y)dywhere b is a bounded function on [0,∞).Also, K|_S^n-1 satisfies a cancelation condition and some other hypotheses but mayfail to be smooth. We obtain L^p estimates for these operators assuming that the maximalfunction related to the functionφ(t) is bounded on L^p.This paper mainly proves the following two theorems:Theorem 1: LetΩ∈L^q(S^n-1) for someq＞1. Suppose thatφsatisfiesφ: [0,∞)→R and thatψis a a-convex type for some a＞0, which is a homogeneous functionof degree one with respect to A_t, such thatψ(A_tx)=rψ(x)。Then T is a boundedoperator in L^P(Rⁿ⁺¹) for all 1＜p＜∞, provided that the maximal operator V_φ: V_φg(t)=sup_k∈Z|integral from n=2^k to 2^k+1 f(t-φ(r))dr/r|is bounded on L^p(Rⁿ) for all 1＜p＜∞。Theorem 2: LetΩ∈LLog⁺L(S^n-1). Suppose thatφsatisfiesφ: [0,∞)→R andψis a homogeneous function of degree one with respect to A_t, such thatψ(A_tx)=tψ(x), which is a a-convex type for some a＞0. Then T is a bounded operator on L^P(Rⁿ⁺¹)for all 1＜p＜∞, provided that the maximal operatorV_φ^(m)g(t)=sup_k∈z|integral from n=2^mk to 2^m(k+1) g(t-φ(r))dr/r]çš„V_φ^(m) satisfies‖V_φ^(m)g‖_L^p(R)≤C_pm‖g‖_L^p(R)。...