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This contributed volume collects papers based on courses and talks given at the 2017 CIMPA school Harmonic Analysis, Geometric Measure Theory and Applications, which took place at the University of Buenos Aires in August 2017. These articles highlight recent breakthroughs in both harmonic analysis and geometric measure theory, particularly focusing on their impact on image and signal processing. The wide range of expertise present in these articles will help readers contextualize how these breakthroughs have been instrumental in resolving deep theoretical problems. Some topics covered include:Gabor framesFalconer distance problemHausdorff dimensionSparse inequalitiesFractional Brownian motionFourier analysis in geometric measure theoryThis volume is ideal for applied and pure mathematicians interested in the areas of image and signal processing. Electrical engineers and statisticians studying these fields will also find this to be a valuable resource.

Inhaltsverzeichnis

Chapter 1. CAZAC Sequences and Haagerup’s Characterization of Cyclic N-roots

Abstract
Constant amplitude zero autocorrelation (CAZAC) sequences play an important role in waveform design for radar and communication theory. They also have deep and intricate connections in several topics in mathematics, including Fourier analysis, Hadamard matrices, and cyclic N-roots. Our goals are to describe these mathematical connections, to provide a unified exposition of the theory of CAZAC sequences integrating several diverse ideas, to introduce new techniques for constructing CAZAC sequences alongside established methods, and to give an exposition of the fascinating unpublished theorem of Uffe Haagerup (1949–2015), which proves that the number of CAZAC generating cyclic N-roots is finite. The role of the uncertainty principle in the proof is essential.
John J. Benedetto, Katherine Cordwell, Mark Magsino

Chapter 2. Hardy Spaces with Variable Exponents

Abstract
In this paper, we make a survey on some recent developments of the theory of Hardy spaces with variable exponents in different settings.
Víctor Almeida, Jorge J. Betancor, Estefanía Dalmasso, Lourdes Rodríguez-Mesa

Chapter 3. Regularity of Maximal Operators: Recent Progress and Some Open Problems

Abstract
This is an expository paper on the regularity theory of maximal operators, when these act on Sobolev and BV functions, with a special focus on some of the current open problems in the topic. Overall, a list of fifteen research problems is presented. It summarizes the contents of a talk delivered by the author in the CIMPA 2017 Research School—Harmonic Analysis, Geometric Measure Theory, and Applications, in Buenos Aires, Argentina.
Emanuel Carneiro

Chapter 4. Gabor Frames: Characterizations and Coarse Structure

Abstract
This chapter offers a systematic and streamlined exposition of the most important characterizations of Gabor frames over a lattice.
Karlheinz Gröchenig, Sarah Koppensteiner

Chapter 5. On the Approximate Unit Distance Problem

Abstract
The Erdös unit distance conjecture in the plane says that the number of pairs of points from a point set of size n separated by a fixed (Euclidean) distance is $$\le C_{\varepsilon } n^{1+\varepsilon }$$ for any $$\varepsilon >0$$. The best known bound is $$Cn^{\frac{4}{3}}$$. We show that if the set under consideration is homogeneous, or, more generally, s-adaptable in the sense of [12], and the fixed distance is much smaller than the diameter of the set, then the exponent $$\frac{4}{3}$$ is significantly improved, even if we consider a small range of distances instead of a fixed value. Corresponding results are also established in higher dimensions. The results are obtained by solving the corresponding continuous problem and using a continuous-to-discrete conversion mechanism. The degree of sharpness of results is tested using the known results on the distribution of lattice points in dilates of convex domains.
Alex Iosevich

Chapter 6. Hausdorff Dimension, Projections, Intersections, and Besicovitch Sets

Abstract
This is a survey on recent developments on the Hausdorff dimension of projections and intersections for general subsets of Euclidean spaces, with an emphasis on estimates of the Hausdorff dimension of exceptional sets and on restricted projection families. We shall also discuss relations between projections and Hausdorff dimension of Besicovitch sets.
Pertti Mattila

Chapter 7. Dyadic Harmonic Analysis and Weighted Inequalities: The Sparse Revolution

Abstract
We will introduce the basics of dyadic harmonic analysis and how it can be used to obtain weighted estimates for classical Calderón–Zygmund singular integral operators and their commutators. Harmonic analysts have used dyadic models for many years as a first step toward the understanding of more complex continuous operators. In 2000, Stefanie Petermichl discovered a representation formula for the venerable Hilbert transform as an average (over grids) of dyadic shift operators, allowing her to reduce arguments to finding estimates for these simpler dyadic models. For the next decade, the technique used to get sharp weighted inequalities was the Bellman function method introduced by Nazarov, Treil, and Volberg, paired with sharp extrapolation by Dragičević et al. Other methods where introduced by Hytönen, Lerner, Cruz-Uribe, Martell, Pérez, Lacey, Reguera, Sawyer, and Uriarte-Tuero, involving stopping time and median oscillation arguments, precursors of the very successful domination by positive sparse operators methodology. The culmination of this work was Tuomas Hytönen’s 2012 proof of the $$A_2$$ conjecture based on a representation formula for any Calderón–Zygmund operator as an average of appropriate dyadic operators. Since then domination by sparse dyadic operators has taken central stage and has found applications well beyond Hytönen’s $$A_p$$ theorem. We will survey this remarkable progression and more in these lecture notes.
María Cristina Pereyra

Chapter 8. Sharp Quantitative Weighted Estimates and a New Proof of the Harboure–Macías–Segovia’s Extrapolation Theorem

Abstract
In this paper, we are concerned with quantitative weighted $$\mathop {\mathrm {BMO}}$$-type estimates. We provide a new quantitative proof for a result due to Harboure, Macías and Segovia (Amer J Math 110 (1988), 383–397, [15]) that also allows to slightly weaken the hypothesis. We also obtain some sharp weighted $$L_c^\infty -\mathop {\mathrm {BMO}}$$-type estimates for Calderón–Zygmund operators.
Alberto Criado, Carlos Pérez, Israel P. Rivera-Ríos

Chapter 9. Dimensions of Self-similar Measures and Applications: A Survey

Abstract
We present a self-contained proof of a formula for the $$L^q$$ dimensions of self-similar measures on the real line under exponential separation (up to the proof of an inverse theorem for the $$L^q$$ norm of convolutions). This is a special case of a more general result of the author from Shmerkin (Ann Math, 2019), and one of the goals of this survey is to present the ideas in a simpler, but important, setting. We also review some applications of the main result to the study of Bernoulli convolutions and intersections of self-similar Cantor sets.
Pablo Shmerkin

Chapter 10. Sample Paths Properties of the Set-Indexed Fractional Brownian Motion

Abstract
For $$0 < H \le 1/2$$, let $$\mathbf {B}^H = \{ \mathbf {B}^H(t);\; t\in \mathbb R^N_+ \}$$ be the Gaussian random field obtained from the set-indexed fractional Brownian motion restricted to the rectangles of $$\mathbb R^N_+$$. We prove that $$\mathbf {B}^H$$ is tangent to a multiparameter fBm which is isotropic in the $$l^1$$-norm and we determine the Hausdorff dimension of the inverse image of $$\mathbf {B}^H$$ and its hitting probabilities. By applying the Lamperti transform and a Fourier analytic method, we show that $$\mathbf {B}^H$$ has the property of strong local nondeterminism (SLND) for $$N=2$$. By applying SLND, we obtain the exact uniform and local moduli of continuity and Chung’s law of iterated logarithm for $$\mathbf {B}^H = \{ \mathbf {B}^H(t);\; t\in \mathbb R^2_+ \}$$. These results show that, away from the axes of $$\mathbb R^2_+$$, the local behavior of $$\mathbf {B}^H$$ is similar to the ordinary fractional Brownian motion of index H.
Erick Herbin, Yimin Xiao

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