L p L^{p} boundedness for a maximal singular integral operator

Abstract

Abstract Let Ω be homogeneous of degree zero and have vanishing moment of order one, let A be a function on ℝ d {\mathbb{R}^{d}} such that ∇ ⁡ A ∈ BMO ⁡ ( ℝ d ) {\nabla A\in\operatorname{BMO}(\mathbb{R}^{d})} , and let T Ω , A {T_{\Omega,A}} be the singular integral operator defined by T Ω , A ⁢ f ⁢ ( x ) = p . v . ∫ ℝ d Ω ⁢ ( x - y ) | x - y | d + 1 ⁢ ( A ⁢ ( x ) - A ⁢ ( y ) - ∇ ⁡ A ⁢ ( y ) ⁢ ( x - y ) ) ⁢ f ⁢ ( y ) ⁢ 𝑑 y . T_{\Omega,A}f(x)=\mathrm{p.v.}\int_{\mathbb{R}^{d}}\frac{\Omega(x-y)}{\lvert x% -y\rvert^{d+1}}(A(x)-A(y)-\nabla A(y)(x-y))f(y)\,dy. In this paper, the authors prove that if Ω ∈ L ⁢ ( log ⁡ L ) 2 ⁢ ( S d - 1 ) {\Omega\in L(\log L)^{2}(S^{d-1})} , then the maximal singular integral operator associated to T Ω , A {T_{\Omega,A}} is bounded on L p ⁢ ( ℝ d ) {L^{p}(\mathbb{R}^{d})} for all p ∈ ( 1 , ∞ ) {p\in(1,\infty)} .