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 $\mathbb{R}^n$. In this paper we study the problem of controlling the maximal singular integral $T^{\star}f$ by the singular integral $Tf$. The most basic form of control one may consider is the estimate of the $L^2(\mathbb{R}^n)$ norm of $T^{\star}f$ by a constant times the $L^2(\mathbb{R}^n)$ norm of $Tf$. We show that if $T$ is an even higher order Riesz transform, then one has the stronger pointwise inequality $T^{\star}f(x) \leq C \, M(Tf)(x)$, where $C$ is a constant and $M$ is the Hardy-Littlewood maximal operator. We prove that the $L^2$ estimate of $T^{\star}$ by $T$ is equivalent, for even smooth homogeneous Calderón-Zygmund operators, to the pointwise inequality between $T^{\star}$ 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 $\frac{\Omega(x)}{|x|^n}$ of $T$, where $\Omega$ is an even homogeneous function of degree $0$, of class $C^\infty(S^{n-1})$ and with zero integral on the unit sphere $S^{n-1}$. Let $\Omega= \sum P_j$ be the expansion of $\Omega$ in spherical harmonics $P_j$ 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 $R\circ U$, 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 $P_j$ in the ring of polynomials in $n$~variables with real coefficients.