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

Abstract

Let T be a smooth homogeneous Calderón-Zygmund singular integral operator in R n .In this paper we study the problem of controlling the maximal singular integral T f by the singular integral T f .The most basic form of control one may consider is the estimate of the L 2 (R n ) norm of T f by a constant times the L 2 (R n ) norm of T f .We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality, where C is a constant and M is the Hardy-Littlewood maximal operator.We prove that the L 2 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 L 2 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 ∞ (S n-1 ) and with zero integral on the unit sphere S n-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.

Locations

• arXiv (Cornell University) - View - PDF
• Annals of Mathematics - View - PDF