Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.4610

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

Joan Mateu·Joan Orobitg·Joan Verdera
View original source ↗Read source document ↗Export metadata ↓

Abstract

The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm of T*f by a constant times the L2 norm of Tf. The second is the pointwise estimate of T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewood maximal operator. Notice that this is an improved variant of Cotlar's inequality, because the term Mf(x) is missing on the right hand side. Our main result states that, for even operators, both are equivalent to a purely algebraic condition formulated in terms of the expansion of the kernel in spherical harmonics. The condition holds by higher order Riesz transforms, which then satisfy an improved version of Cotlar's inequality

Explore related subjects

math.CAmath.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joan Mateu, Joan Orobitg, Joan Verdera. 2010-12-20. Estimates for the maximal singular integral in terms of the singular integral:the case of even kernels. https://arxiv.org/abs/0707.4610

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Bounded lattice tiles that pack with another lattice

Suppose $L, M$ are full-rank lattices in Euclidean space, such that ${\mathrm{vol}} L < {\mathrm{vol}} M$. Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set $F$ (we can even take $F$ to be a finite union of polytopes) such that $F+L$ is a tiling and $F+M$ is a packing. In the related problem of finding a common fundamental domain for two lattices $L$ and $M$ we show that no such common fundamental domain exists if the volumes are different even, if we do not require measurability of this domain.

math.CA↗
arXiv2026

Parallelepipeds of maximal facet area and total edge length in ellipsoids, through prescribed boundary points

Let \[ \mathcal{E}_A=\{x\in\mathbb{R}^n:x^{\top}A^{-1}x\le 1\},\qquad n\ge2, \] where $A$ is real symmetric positive definite. We study full-dimensional parallelepipeds whose $2^n$ vertices lie on $\partial\mathcal{E}_A$. First we show that such parallelepipeds are necessarily centred at the origin and are precisely the images, under $A^{1/2}$, of orthotopes inscribed in the Euclidean unit sphere. This reduces the extremal questions to finite-dimensional linear algebra. For the total length $L$ of the one-skeleton we prove \[ L_{\max}(\mathcal{E}_A)=2^n\sqrt{\operatorname{tr} A}. \] Moreover, the prescribed-vertex problem for $L$ has the same answer in every dimension: for every $x_0\in\partial\mathcal{E}_A$ there is an inscribed parallelepiped with vertex $x_0$ and total edge length $2^n\sqrt{\operatorname{tr} A}$. The proof uses the Schur--Horn theorem applied to the trace-zero matrix $A-\operatorname{tr}(A)y_0y_0^{\top}$, where $y_0=A^{-1/2}x_0$. For the total $(n-1)$-dimensional measure $S$ of the facets we prove \[ S_{\max}(\mathcal{E}_A)=2^n n^{-(n-2)/2}\sqrt{\det A}\,\sqrt{\operatorname{tr}(A^{-1})}. \] For $n\ge3$ the maximisers are more rigid: on the sphere they are orthotopes with all edge lengths equal and with a Schur--Horn equal diagonal condition for $A^{-1}$. The prescribed-vertex facet-area problem is therefore equivalent to a restricted Schur--Horn problem with a prescribed barycentric basis. In dimension two this recovers the Connes--Zagier property for ellipses. In dimension three, however, the direct higher-dimensional analogue fails for triaxial ellipsoids at principal-axis vertices; an exact obstruction is given.

math.CA↗
arXiv2026

The manufacture of examples: the $(-1)$-classical orthogonal polynomials

The big $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials have appeared in the mathematical physics literature through differential- or difference-reflection eigenvalue problems. We show that these three families are governed by a single alternating mechanism, formulated at the level of orthogonality functionals, transpose operators, and structural identities. Starting from an ordinary orthogonal polynomial sequence in the quadratic variable, the resulting algorithm constructs the direct alternating family and transfers its moments, squared norms, and recurrence coefficients. One regular Geronimus inversion then produces the big $(-1)$-Jacobi and Bannai-Ito families, whereas the $(-1)$-Meixner-Pollaczek family is obtained directly.

math.CA↗