On the capacity of sets of divergence associated with the spherical partial integral operator

Abstract:

In this article, we study the pointwise convergence of the spherical partial integral operator $S_Rf(x)=\int _{B(0,R)} \hat {f} (y) e^{2\pi ix\cdot y}dy$ when it is applied to functions with a certain amount of smoothness. In particular, for $f\in \mathcal {L}_{\alpha }^p(\mathbb {R}^n)$, $\tfrac {n-1}{2} <\alpha \leq \tfrac {n}{p}$, $2\leq p<\tfrac {2n}{n-1}$, we prove that $S_Rf(x)\to G_{\alpha } *g(x)$ $C_{\alpha ,p}$-quasieverywhere on $\mathbb {R}^n$, where $g\in L^p({\mathbb {R}}^n )$ is such that $f=G_{\alpha }*g$ almost everywhere. A weaker version of this result in the range $0<\alpha \leq \tfrac {n-1}{2}$ as well as some related localisation principles are also obtained. For $1\leq p<2-\tfrac {1}{n}$ and $0\leq \alpha <\tfrac {(2-p)n-1}{2p}$, we construct a function $f\in \mathcal {L}_\alpha ^p(\mathbb {R}^n)$ such that $S_Rf(x)$ diverges everywhere.