Cones, rectifiability, and singular integral operators
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TL;DR: In this paper, the authors define conical energies, which quantify the portion of the Lipschitz graph lying in the cone with vertex vertex vertices, direction vertices and aperture vertices.
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Abstract: Let $\mu$ be a Radon measure on $\mathbb{R}^d$. We define and study conical energies $\mathcal{E}_{\mu,p}(x,V,\alpha)$, which quantify the portion of $\mu$ lying in the cone with vertex $x\in\mathbb{R}^d$, direction $V\in G(d,d-n)$, and aperture $\alpha\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $\mu$ has polynomial growth, we give a sufficient condition for $L^2(\mu)$-boundedness of singular integral operators with smooth odd kernels of convolution type.
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Citations
Plenty of big projections imply big pieces of Lipschitz graphs
TL;DR: In this paper, it was shown that a closed regular set with Lipschitz graphs has big pieces of regular sets, i.e., a regular set that has a large number of Lipschi-graphs.
Plenty of big projections imply big pieces of Lipschitz graphs
TL;DR: In this article, it was shown that closed n-regular sets with plenty of big projections have big pieces of Lipschitz graphs and that these sets are uniformly n-rectifiable.
Two examples related to conical energies
TL;DR: In this article , a characterization of rectifiable measures and a sufficient condition for boundedness of nice singular integral operators are given. But these results are based on conical energies, which are not necessarily rectifiable.
Quantitative Besicovitch projection theorem for irregular sets of directions
D. Dkabrowski
- 30 Nov 2022
TL;DR: In this article , it was shown that if a planar set R 2 is AD-regular and there exists a set of direction G ⊂ S 1 with H 1 (G ) (cid:38) 1 such that for every θ ∈ G we have k π θ H 1 | E k L ∞ (cID:46) 1, then a big piece of R 2 can be covered by a Lipschitz graph Γ with Lip(Γ) (Cid: 46) 1.
Radon measures and Lipschitz graphs
TL;DR: In this paper, the authors studied the problem of detecting when a general Radon measure charges a Lipschitz graph by analyzing the behavior of coarse doubling ratios on dyadic cubes that intersect conical annuli.
References
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Classical Fourier Analysis
Loukas Grafakos
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TL;DR: In this paper, the Fourier Transform and Distributions of convolutional neural networks have been studied in the context of Trigonometric Identities and Inequalities and Mean Value Theorem in Variables.
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Modern Fourier analysis
Loukas Grafakos
- 01 Jan 2009
TL;DR: In this paper, the Carleson-Hunt Theorem is used to describe the smoothness and function spaces of non-convolutional non-convolutional types.
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Analysis of and on uniformly rectifiable sets
Guy David,Stephen Semmes +1 more
- 01 Jan 1993
TL;DR: The notion of uniform rectifiability of sets (in a Euclidean space), which emerged only recently, can be viewed in several different ways as mentioned in this paper, as a quantitative and scale-invariant substitute for the classical notion of Rectifiability; as the answer (sometimes only conjecturally) to certain geometric questions in complex and harmonic analysis; as a condition which ensures the parametrizability of a given set, with estimates, but with some holes and self-intersections allowed, as an achievable baseline for information about the structure of a set.
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