A new perspective on the Riesz potential

This paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only \mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains {I_{\alpha}L^{p,\kappa}} under the conditions {1<p<\infty}, {1\leq q<\infty}, q(\kappa/p-\alpha)\leq\kappa\leq n, {0<\alpha<\min\{n,1+\kappa/p\}}, but also {\mathfrak{L}^{q,\lambda}} exists as an associate space under the condition {-q<\lambda<n}, where {I_{\alpha}L^{p,\kappa}} and {\mathfrak{L}^{q,\lambda}} are the Morrey–Sobolev and Campanato spaces on {\mathbb{R}^{n}} respectively. On the other hand, a nonnegative Radon measure μ is completely characterized to produce a continuous map {I_{\alpha}:L_{p,1}\to L^{q}_{\mu}} under the condition {1<p<\min\{q,{n}/{\alpha}\}} or {1<q\leq p<\min\{{q(n-\alpha p)}/({n-\alpha q(q-1)^{-1}}),{n}/{\alpha}\}}, where {L_{p,1}} and {L^{q}_{\mu}} are the {(p,1)}-Lorentz and {(q,\mu)}-Lebesgue spaces on {\mathbb{R}^{n}} respectively.