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| Author | : Guy David |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 370 |
| Release | : 1993 |
| Genre | : Mathematics |
| ISBN | : 0821815377 |
* The only available reference on uniform rectifiabilityThe text covers the understanding of uniform rectifiability of a given set in terms of the approximate behaviour of the set at most locations and scales.
| Author | : Guy David |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 146 |
| Release | : 2000 |
| Genre | : Mathematics |
| ISBN | : 0821820486 |
This book is intended for graduate students and research mathematicians interested in calculus of variations and optimal control; optimization.
| Author | : Pertti Mattila |
| Publisher | : Cambridge University Press |
| Total Pages | : 181 |
| Release | : 2023-01-12 |
| Genre | : Mathematics |
| ISBN | : 1009288083 |
A broad survey of the theory of rectifiability and its deep connections to numerous different areas of mathematics.
| Author | : Xavier Tolsa |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 142 |
| Release | : 2017-01-18 |
| Genre | : Mathematics |
| ISBN | : 1470422522 |
This monograph is devoted to the proof of two related results. The first one asserts that if is a Radon measure in satisfyingfor -a.e. , then is rectifiable. Since the converse implication is already known to hold, this yields the following characterization of rectifiable sets: a set with finite -dimensional Hausdorff measure is rectifiable if and only ifH^1x2EThe second result of the monograph deals with the relationship between the above square function in the complex plane and the Cauchy transform . Assuming that has linear growth, it is proved that is bounded in if and only iffor every square .
| Author | : Guy David |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 114 |
| Release | : 2012 |
| Genre | : Mathematics |
| ISBN | : 0821853104 |
The authors extend the proof of Reifenberg's Topological Disk Theorem to allow the case of sets with holes, and give sufficient conditions on a set $E$ for the existence of a bi-Lipschitz parameterization of $E$ by a $d$-dimensional plane or smooth manifold. Such a condition is expressed in terms of square summability for the P. Jones numbers $\beta_1(x,r)$. In particular, it applies in the locally Ahlfors-regular case to provide very big pieces of bi-Lipschitz images of $\mathbb R^d$.
| Author | : Hervé M. Pajot |
| Publisher | : Springer |
| Total Pages | : 133 |
| Release | : 2002-01-01 |
| Genre | : Mathematics |
| ISBN | : 3540360743 |
Based on a graduate course given by the author at Yale University this book deals with complex analysis (analytic capacity), geometric measure theory (rectifiable and uniformly rectifiable sets) and harmonic analysis (boundedness of singular integral operators on Ahlfors-regular sets). In particular, these notes contain a description of Peter Jones' geometric traveling salesman theorem, the proof of the equivalence between uniform rectifiability and boundedness of the Cauchy operator on Ahlfors-regular sets, the complete proofs of the Denjoy conjecture and the Vitushkin conjecture (for the latter, only the Ahlfors-regular case) and a discussion of X. Tolsa's solution of the Painlevé problem.
| Author | : Carlos E. Kenig |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 345 |
| Release | : 2020-12-14 |
| Genre | : Education |
| ISBN | : 1470461277 |
The origins of the harmonic analysis go back to an ingenious idea of Fourier that any reasonable function can be represented as an infinite linear combination of sines and cosines. Today's harmonic analysis incorporates the elements of geometric measure theory, number theory, probability, and has countless applications from data analysis to image recognition and from the study of sound and vibrations to the cutting edge of contemporary physics. The present volume is based on lectures presented at the summer school on Harmonic Analysis. These notes give fresh, concise, and high-level introductions to recent developments in the field, often with new arguments not found elsewhere. The volume will be of use both to graduate students seeking to enter the field and to senior researchers wishing to keep up with current developments.
| Author | : Benjamin Jaye |
| Publisher | : American Mathematical Soc. |
| Total Pages | : 97 |
| Release | : 2020-09-28 |
| Genre | : Mathematics |
| ISBN | : 1470442132 |
Fix $dgeq 2$, and $sin (d-1,d)$. The authors characterize the non-negative locally finite non-atomic Borel measures $mu $ in $mathbb R^d$ for which the associated $s$-Riesz transform is bounded in $L^2(mu )$ in terms of the Wolff energy. This extends the range of $s$ in which the Mateu-Prat-Verdera characterization of measures with bounded $s$-Riesz transform is known. As an application, the authors give a metric characterization of the removable sets for locally Lipschitz continuous solutions of the fractional Laplacian operator $(-Delta )^alpha /2$, $alpha in (1,2)$, in terms of a well-known capacity from non-linear potential theory. This result contrasts sharply with removability results for Lipschitz harmonic functions.
| Author | : Dorina Mitrea |
| Publisher | : Springer Nature |
| Total Pages | : 940 |
| Release | : 2022-11-04 |
| Genre | : Mathematics |
| ISBN | : 3031059506 |
This monograph presents a comprehensive, self-contained, and novel approach to the Divergence Theorem through five progressive volumes. Its ultimate aim is to develop tools in Real and Harmonic Analysis, of geometric measure theoretic flavor, capable of treating a broad spectrum of boundary value problems formulated in rather general geometric and analytic settings. The text is intended for researchers, graduate students, and industry professionals interested in applications of harmonic analysis and geometric measure theory to complex analysis, scattering, and partial differential equations. Volume I establishes a sharp version of the Divergence Theorem (aka Fundamental Theorem of Calculus) which allows for an inclusive class of vector fields whose boundary trace is only assumed to exist in a nontangential pointwise sense.
| Author | : Guy David |
| Publisher | : Springer Science & Business Media |
| Total Pages | : 592 |
| Release | : 2006-03-10 |
| Genre | : Mathematics |
| ISBN | : 3764373024 |
The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. It is largely self-contained, and should be accessible to graduate students in analysis. The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.
