On a characterisation of the space of Riesz potential of functions in Banach spaces with some à priori properties

We consider the problem of describing the space $I^\alpha(X)$ of functions representable by the Riesz potential ${I}^\alpha \varphi$ with density $\varphi$ in the given space $X.$ It is assumed that $X\subset \Phi'$, where $\Phi'$ is the space of distributions over the Lizorkin test function space $\Phi$, invariant with respect to Riesz integration, and the range $I^\alpha(X)$ is understood in the sense of distributions. In this general setting, we study the question under what assumptions on the space $X$ the inclusion of the element $f$ in to the range $I^\alpha (X) $ is equivalent to the convergence of the truncated hypersingular integrals $\mathbb D_\varepsilon^\alpha f$ in the space $X.$ For this purpose, this question is first investigated in the context of the topology of the space $ \Phi. $ Namely, for any linear subset $X$ in $\Phi'$ it is shown that the inclusion of $f$ into the range $I^\alpha (X)$ is equivalent to the convergence of truncated hypersingular integrals on the set $X$ in the topology of the space $\Phi'$. If $X$ is a Banach space, the passage from the inclusion into the range to the convergence of truncated hypersingular integrals in the norm is proved up to an additive polynomial term under the assumption that some special convolution is an identity approximation in the space $X$. It is known that the latter holds for many Banach function spaces and is valid for function spaces $X$ where the maximal operator is bounded. The inverse passage is proved for the Banach function space $X$ enjoying the property that the associated space $X'$ includes the Lizorkin test function space.