Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Power weight inequalities for spherical maximal functions

Marco Fraccaroli, Joris Roos, Andreas Seeger

Abstract

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted $L^p(|x|^α)$ spaces. Aside from endpoint cases, a complete description of the allowable ranges of $p$, $α$ is given in terms of the Legendre--Assouad function of the dilation set. This settles, up to endpoints, an open problem of Duoandikoetxea and Seijo.

Power weight inequalities for spherical maximal functions

Abstract

This paper is about spherical maximal functions with general dilation sets acting on functions in weighted spaces. Aside from endpoint cases, a complete description of the allowable ranges of , is given in terms of the Legendre--Assouad function of the dilation set. This settles, up to endpoints, an open problem of Duoandikoetxea and Seijo.
Paper Structure (13 sections, 6 theorems, 93 equations, 1 figure)

This paper contains 13 sections, 6 theorems, 93 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Type set convexity
  4. An equivalent condition
  5. Scaling and a global to local reduction
  6. Examples of type sets
  7. Assouad regular sets
  8. Finite unions
  9. Lower bounds
  10. Proof of Lemma \ref{['lem:lowerboundgoal']} for small k
  11. Proof of Lemma \ref{['lem:lowerboundgoal']} for large k
  12. Upper bounds
  13. Proof of Proposition \ref{['prop:local']}

Key Result

Theorem 1.1

Let $d\ge 2$, ${{\mathcal{E}}}\subset (0,\infty)$ nonempty. Then

Figures (1)

  • Figure 1: Typical shape of the type set ${{\mathfrak{T}}}_{{\mathcal{E}}}$ (shaded). The containing trapezoid is defined by the necessary conditions $p\ge p_\beta$ and \ref{['eqn:oldneccond']}. Here $x_1=\tfrac{1}{p_1}$, $x_\beta=\tfrac{1}{p_\beta}$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1: DuoandikoetxeaSeijo
  • Remark 3.1
  • Lemma 4.1
  • Remark 4.2
  • Proposition 5.1
  • Remark 5.2
  • Lemma 5.3
  • proof
  • Lemma 5.4
  • ...and 1 more