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

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.