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

• 成果类型: Article
• 署名作者: Mateu, Joan; Orobitg, Joan; Verdera, Joan
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2011.174.3.2
• 发表日期: 2011
• 页码: 1429-1483
• 关键词: lipschitz approximation; elliptic-equations

摘要

Let T be a smooth homogeneous Calderon-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 T* f(x) <= C M(T f)(x), 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 Calderon-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 Omega(x)/vertical bar x vertical bar(n) of T, where Omega is an even homogeneous function of degree 0, of class C-infinity(Sn-1) and with zero integral on the unit sphere Sn-1. Let Omega = Sigma P-j be the expansion of Q in spherical harmonics P-j of degree j. Let A stand for the algebra generated by the identity and the smooth homogeneous Calderon-Zygmund operators. Then our characterizing condition states that T is of the form RoU, 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 P-j in the ring of polynomials in n variables with real coefficients.