Spherical maximal operators with fractal sets of dilations on radial functions

Abstract

For a given set of dilations $E\subset [1,2]$, Lebesgue space mapping properties of the spherical maximal operator with dilations restricted to $E$ are studied when acting on radial functions. In higher dimensions, the type set only depends on the upper Minkowski dimension of $E$, and in this case complete endpoint results are obtained. In two dimensions we determine the closure of the $L^p\to L^q$ type set for every given set $E$ in terms of a dimensional spectrum closely related to the upper Assouad spectrum of $E$.

Figures (2)

• Figure 1: The triangle $\Delta_\beta$.
• Figure 2: If $d=2, \beta=0.5, \gamma=1$, then $\overline{\mathcal{T}^\mathrm{rad}_E}$ is contained in $\Delta_\beta$ and contains $\mathcal{Q}_{\beta,\gamma}^\mathrm{rad}$. The condition $2\gamma-\beta>1$ says that the point $P_3^\mathrm{rad}=P_{3,\beta}^\mathrm{rad}$ lies above the horizontal line $q=2(1+\gamma)$.

Theorems & Definitions (42)

• Theorem 1.1
• Remark
• Theorem 1.2
• Remark
• Corollary 1.3
• Theorem 1.4
• Remark
• Lemma 2.1
• proof
• proof : Proof of Corollary \ref{['cor:d=2']}, given Theorem \ref{['thm:maintwodim']}