3 problems

Let $A_t$ denote the spherical averaging operator, let $\partial_t^\gamma$ be the fractional time derivative, and let $L^p_s(\mathbb{R}^d)$ denote the inhomogeneous Sobolev space.…

Let $E\subset [1,2]$ satisfy $\dim_M E=\dim_A E=\beta$. For each scale $j$, let $\mathcal{Z}_j(E)$ index a minimal collection of $\delta$-separated points $t_\nu$ whose intervals…

For $t>0$, let $A_t$ be the spherical averaging operator on $\mathbb{R}^d$ defined by … where $\mathrm{d}\sigma$ is normalized surface measure on the unit sphere, and let $V_rA$ de…