Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels | Annals of Mathematics

Abstract Let T be a smooth homogeneous Calderón-Zygmund singular integral operator in ℝn. In this paper we study the problem of controlling the maximal singular integral T⋆f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L2(ℝn) norm of T⋆f by a constant times the L2(ℝn) norm of Tf. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T⋆f(x)≤CM(Tf)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L2 estimate of T⋆ by T is equivalent, for even smooth homogeneous Calderón-Zygmund operators, to the pointwise inequality between T⋆ and M(T). Our main result characterizes the L2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel Ω(x)|x|n of T, where Ω is an even homogeneous function of degree 0, of class C∞(Sn−1) and with zero integral on the unit sphere Sn−1. Let Ω=∑Pj be the expansion of Ω in spherical harmonics Pj of degree j. Let A stand for the algebra generated by the identity and the smooth homogeneous Calderón-Zygmund operators. Then our characterizing condition states that T is of the form R∘U, where U is an invertible operator in A and R is a higher order Riesz transform associated with a homogeneous harmonic polynomial P which divides each Pj in the ring of polynomials in n~variables with real coefficients.

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数学年刊(Annals of Mathematics)

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