3 problems

Let $S\subset\mathbb R^n$. A set of injectivity for the spherical means for $C_c(\mathbb R^2)$ is a set such that, for every $f$ in that class, $f\ast\mu_r(x)=0$ for all $r>0$ and…

Let $V_q(\mathcal{A}^{0})$ denote the $q$-variation operator associated with the classical spherical averages on $\mathbb{R}^{d}$. At the critical exponent $q=p/d$, with $p>2d$, th…

Range characterization conjecture. A function $\varphi_\theta(t)$ belongs to the range of the operator $f\mapsto(Mf)(\theta,t)$, with $f\in C^\infty(B)$, if and only if