Radon measures and Lipschitz graphs

AbstractI prove that closed n-regular sets $$E \subset {\mathbb {R}}^{d}$$ E ⊂ R d with plenty of big projections have big pieces of Lipschitz graphs. In particular, these sets are uniformly n-rectifiable. This answers a question of David and Semmes from 1993.

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Lusin type theorems for Radon measures

Rendiconti del Seminario Matematico della Università di Padova ◽

10.4171/rsmup/138-9 ◽

2017 ◽

Vol 138 ◽

pp. 193-207 ◽

Cited By ~ 1

Author(s):

Andrea Marchese

Keyword(s):

Radon Measures

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On weak compactness in the space of Radon measures

Journal of Functional Analysis ◽

10.1016/0022-1236(70)90030-3 ◽

1970 ◽

Vol 5 (2) ◽

pp. 259-298 ◽

Cited By ~ 7

Author(s):

Samuel Kaplan

Keyword(s):

Weak Compactness ◽

Radon Measures

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Set-Theoretical Aspects and Radon Measures

Radon Integrals - Progress in Mathematics ◽

10.1007/978-1-4612-0377-3_4 ◽

1995 ◽

pp. 177-288

Author(s):

Bernd Anger ◽

Claude Portenier

Keyword(s):

Radon Measures

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Two examples related to conical energies

Annales Fennici Mathematici ◽

10.54330/afm.113378 ◽

2022 ◽

Vol 47 (1) ◽

pp. 261-281

Author(s):

Damian Dąbrowski

Keyword(s):

Singular Integral ◽

Recent Article ◽

Integral Operators ◽

Singular Integral Operators ◽

The Other ◽

Independent Interest ◽

Sufficient Condition ◽

Lipschitz Graphs ◽

Big Pieces

In a recent article (2021) we introduced and studied conical energies. We used them to prove three results: a characterization of rectifiable measures, a characterization of sets with big pieces of Lipschitz graphs, and a sufficient condition for boundedness of nice singular integral operators. In this note we give two examples related to sharpness of these results. One of them is due to Joyce and Mörters (2000), the other is new and could be of independent interest as an example of a relatively ugly set containing big pieces of Lipschitz graphs.

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