A Boundedness Criterion for Generalized Calderon-Zygmund Operators

David, Guy; Journé, Jean-Lin

doi:10.2307/2006946

Cited by 405 publications

(314 citation statements)

References 9 publications

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“…We will be interested in the case when M is low-dimensional, for example a k dimensional manifold, or a k Ahlfors regular k-rectifiable set [34,35], with k ≪ D. More generally, M may be just approximated by a low-dimensional set, in a sense that our assumptions below will make precise. Let N be a random variable, for example N ∼ N (0, I D ), that will think of as as noise, and letX = X + σN .…”

Section: Problem Settingmentioning

confidence: 99%

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

Little

Maggioni

Rosasco

2017

Applied and Computational Harmonic Analysis

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Large data sets are often modeled as being noisy samples from probability distributions µ in R D , with D large. It has been noticed that oftentimes the support M of these probability distributions seems to be well-approximated by low-dimensional sets, perhaps even by manifolds. We shall consider sets that are locally well approximated by k-dimensional planes, with k ≪ D, with k-dimensional manifolds isometrically embedded in R D being a special case. Samples from µ are furthermore corrupted by D-dimensional noise. Certain tools from multiscale geometric measure theory and harmonic analysis seem well-suited to be adapted to the study of samples from such probability distributions, in order to yield quantitative geometric information about them. In this paper we introduce and study multiscale covariance matrices, i.e. covariances corresponding to the distribution restricted to a ball of radius r, with a fixed center and varying r, and under rather general geometric assumptions we study how their empirical, noisy counterparts behave. We prove that in the range of scales where these covariance matrices are most informative, the empirical, noisy covariances are close to their expected, noiseless counterparts. In fact, this is true as soon as the number of samples in the balls where the covariance matrices are computed is linear in the intrinsic dimension of M. As an application, we present an algorithm for estimating the intrinsic dimension of M.

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Section: Problem Settingmentioning

confidence: 99%

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

Little

Maggioni

Rosasco

2017

Applied and Computational Harmonic Analysis

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“…The conditions in (13) are motivated by one of the well-known versions of the original T 1 Theorem of David and Journé [18]. The necessity of the conditions follows from Theorem 5.…”

Section: The Multilinear T 1 Theoremmentioning

confidence: 99%

On multilinear singular integrals of Calderón-Zygmund type

Grafakos

Torres²

2002

Publicacions Matemàtiques

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Abstract. A variety of results regarding multilinear Calderón-Zygmund singular integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T 1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal operator associated with multilinear singular integrals is also introduced and employed to obtain weighted norm inequalities.

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“…They showed that these operators are bounded on L 2 (as well as many other spaces) and in so doing provided a unified approach to their study. Their work culminated in a celebrated theorem of G. David and J.-L. Journé [14].…”

Section: \K{x Y) -K(x' Y)\ + \K(y X') -K(y X)\ < \X' -X\ E \X -Y\mentioning

confidence: 99%

“…The classical Helson-Szegö theorem characterizes those weights w for which the Hilbert transform is bounded on L 2 . It turns out that weights for which inequalities such as (14) are valid have close connections with BMO functions: roughly speaking, the logarithms of such "good weights" belong to BMO. The study of such inequalities and their applications has been one of the very active areas of research during the last twenty years.…”

Section: The Tl Theorem a Calderón-zygmund Operator T Is Bounded On mentioning

confidence: 99%

Book Review: Real variable methods in harmonic analysis

Weiss¹

1989

Bull. Amer. Math. Soc.

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A Boundedness Criterion for Generalized Calderon-Zygmund Operators

Cited by 405 publications

References 9 publications

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

Multiscale geometric methods for data sets I: Multiscale SVD, noise and curvature

On multilinear singular integrals of Calderón-Zygmund type

Book Review: Real variable methods in harmonic analysis

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