# 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.

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## Authors

Joan Mateu

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Catalonia

Joan Orobitg

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Catalonia

Joan Verdera

Departament de Matemàtiques
Universitat Autònoma de Barcelona
08193 Bellaterra (Barcelona)
Catalonia