Estimates for the maximal singular integral in terms of the singular integral: The case of even kernels

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Let T be a smooth homogeneous Calderón-Zygmund singular integral operator in R{double-struck}n. In this paper we study the problem of controlling the max-imal singular integral T*f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L2(R{double-struck})n norm of T*f by a constant times the L2(R{double-struck})n) 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 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 trans-form associated with a homogeneous harmonic polynomial P which divides each Pj in the ring of polynomials in n variables with real coefficients.
Original languageEnglish
Pages (from-to)1429-1483
JournalAnnals of Mathematics
Publication statusPublished - 1 Nov 2011


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