$$\mathbf {L^p\rightarrow L^q}$$ bounds for spherical maximal operators

Anderson, Theresa C.; Hughes, Kevin; Roos, Joris; Seeger, Andreas

doi:10.1007/s00209-020-02546-0

Cited by 25 publications

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“…3 Dimensions of IFS attractors 6. 4 Ahlfors regularity and quasi-self-similarity 7 Self-similar sets 7.1 Self-similar sets and the Hutchinson-Moran formula 7.2…”

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Assouad Dimension and Fractal Geometry

Fraser

2020

111

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Dimension theory of orthogonal projections vi Contents 10.2.2 The Assouad dimension of orthogonal projections 10.2.3 An application to the box dimensions of projections 10.2.4 The lower dimension and projections 10.3 Slices and intersections 11 Two famous problems in geometric measure theory 11.1 Distance sets 11.2 Kakeya sets 12 Conformal dimension 12.1 Lowering the Assouad dimension by quasi-symmetry PART THREE APPLICATIONS 13 Applications in embedding theory 13.1 Assouad's embedding theorem 13.1.1 Doubling and uniformly perfect metric spaces 13.1.2 Assouad's embedding theorem 13.2 The spiral winding problem 13.3 Almost bi-Lipschitz embeddings 14 Applications in number theory 14.1 Arithmetic progressions 14.1.1 Discrete Kakeya sets 14.2 Diophantine approximation 14.3 Definability of the integers 15 Applications in probability theory 15.1 Uniform dimension results 15.2 Dimensions of random graphs 16 Applications in functional analysis 16.1 Hardy inequalities 16.2 L p → L q bounds for spherical maximal operators 16.3 Connection with L p -norms 17 Future directions 17.1 Finite stability of modified lower dimension 17.2 Dimensions of measures 17.3 Weak tangents 17.4 Further questions of measurability 17.5 IFS attractors Contents vii 17.6 Random sets 17.7 General behaviour of the Assouad spectrum 17.8 Projections 17.9 Distance sets 17.10 The Hölder mapping problem and dimension 17.11 Dimensions of graphs References List of notation Index PART ONE THEORY 1 Fractal geometry and dimension theoryIn this introductory chapter we briefly discuss the history and development of fractal geometry and dimension theory. We introduce and motivate some important concepts such as Hausdorff and box dimension. As part of this discussion we encounter covers and packings, which are central notions in dimension theory, and introduce the dimension theory of measures. The emergence of fractal geometryA fractal can be described as an object which exhibits interesting features on a large range of scales, see Figure 1.1. In pure mathematics, the Sierpiński triangle, the middle third Cantor set, the boundary of the Mandelbrot set, and the von Koch snowflake are archetypal examples and, in 'real life', examples include the surface of a lung, the horizon of a forest, and the distribution of stars in the galaxy. The fractal story began in the nineteenth century with the appearance of a multitude of strange examples exhibiting what we now understand as fractal phenomena. These included Weierstrass' example of a continuous nowhere differentiable function, Cantor's construction of an uncountable set with zero length, and Brown's observations on the path taken by a piece of pollen suspended in water (Brownian motion). During the first half of the twentieth century the mathematical foundations for fractal geometry were laid down by, for example, Besicovitch, Bouligand, Hausdorff, Julia, Marstrand and Sierpiński, and the theory was unified and popularised by the extensive writings of Mandelbrot in the 1970s, for example [197]. It was Mandelbrot who co...

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“…3 Dimensions of IFS attractors 6. 4 Ahlfors regularity and quasi-self-similarity 7 Self-similar sets 7.1 Self-similar sets and the Hutchinson-Moran formula 7.2…”

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“…3 The Assouad spectrum of self-similar sets 7. 4 Dimensions of self-similar measures 8 Self-affine sets 8.1 Self-affine sets and two strands of research 8. 2 Falconer's formula and the affinity dimension 8.…”

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Assouad Dimension and Fractal Geometry

Fraser

2020

111

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“…Proof of (2.29). For λ ≫ 1, let us take f (x) = ϕ λ(1 − |x|) for a smooth function ϕ such that ϕ ≥ 0, ϕ = 1 on [1,2], and supp ϕ ⊂ [1/2, 4]. Using polar coordinates and changing variables r → 1 − r, we have…”

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“…open. See also [1,29,2] for recent developments regarding the circular and spherical maximal functions.…”

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Circular average relative to fractal measures

Ham,

Ko,

Lee

2021

Preprint

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We prove new L p -L q estimates for averages over dilates of the circle with respect to α-dimensional fractal measure, which unify different types of maximal estimates for the circular average. Our results are consequences of L p -L q smoothing estimates for the wave operator relative to fractal measures. We also discuss similar results concerning the spherical averages. 2020 Mathematics Subject Classification. 42B25. Key words and phrases. circular average, general measures.

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$$L^{p}$$-Improving Bounds and Weighted Estimates for Maximal Functions Associated with Curvature

Wang

Zhai

2023

J Fourier Anal Appl

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$$\mathbf {L^p\rightarrow L^q}$$ bounds for spherical maximal operators

Cited by 25 publications

References 23 publications

Assouad Dimension and Fractal Geometry

Assouad Dimension and Fractal Geometry

Circular average relative to fractal measures

$$L^{p}$$-Improving Bounds and Weighted Estimates for Maximal Functions Associated with Curvature

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