On spaces of Riesz potentials

In connection with problems which arise in the theory of integral equations of the first kind with a potential-type kernel we investigate the space of Riesz potentials $I^\alpha(L_p)=\{f=K^\alpha\varphi;\varphi\in L_p(R^n),1<p<n/\alpha\}$, where $K^\alpha$ is the Riesz integration operator ($\widehat{K^\alpha\varphi}(x)=|(x)|^{-\alpha}\widehat\varphi(x)$). We give a description of the space $I^\alpha(L_p)$ in terms of differences of singular integrals, establish a theorem on denseness of $C^\infty_0(R^n)$ in $I^\alpha(L_p)$, and indicate a “weight” invariant description of $I^\alpha(L_p)$.
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