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 𝔏q,qp-1(n-αp)∩𝔏p,κ-αp\mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains IαLp,κ{I_{\alpha}L^{p,\kappa}} under the conditions 1<p<∞{1<p<\infty}...
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| Format: | Article |
| Language: | English |
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De Gruyter
2017-08-01
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| Series: | Advances in Nonlinear Analysis |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/anona-2015-0183 |
| Summary: | This paper offers a new perspective to look at the Riesz potential. On the one hand, it is shown that not only 𝔏q,qp-1(n-αp)∩𝔏p,κ-αp\mathfrak{L}^{q,qp^{-1}(n-\alpha p)}\cap\mathfrak{L}^{p,\kappa-\alpha p} contains IαLp,κ{I_{\alpha}L^{p,\kappa}} under the conditions 1<p<∞{1<p<\infty}, 1≤q<∞{1\leq q<\infty}, q(κ/p-α)≤κ≤nq(\kappa/p-\alpha)\leq\kappa\leq n, 0<α<min{n,1+κ/p}{0<\alpha<\min\{n,1+\kappa/p\}}, but also 𝔏q,λ{\mathfrak{L}^{q,\lambda}} exists as an associate space under the condition -q<λ<n{-q<\lambda<n}, where IαLp,κ{I_{\alpha}L^{p,\kappa}} and 𝔏q,λ{\mathfrak{L}^{q,\lambda}} are the Morrey–Sobolev and Campanato spaces on ℝn{\mathbb{R}^{n}} respectively. On the other hand, a nonnegative Radon measure μ is completely characterized to produce a continuous map Iα:Lp,1→Lμq{I_{\alpha}:L_{p,1}\to L^{q}_{\mu}} under the condition 1<p<min{q,n/α}{1<p<\min\{q,{n}/{\alpha}\}} or 1<q≤p<min{q(n-αp)/(n-αq(q-1)-1),n/α}{1<q\leq p<\min\{{q(n-\alpha p)}/({n-\alpha q(q-1)^{-1}}),{n}/{\alpha}\}}, where Lp,1{L_{p,1}} and Lμq{L^{q}_{\mu}} are the (p,1){(p,1)}-Lorentz and (q,μ){(q,\mu)}-Lebesgue spaces on ℝn{\mathbb{R}^{n}} respectively. |
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| ISSN: | 2191-9496 2191-950X |