Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces
We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in...
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| Format: | Article |
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Wiley
2016-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2016/1393496 |
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| author | Joaquín Motos María Jesús Planells César F. Talavera |
| author_facet | Joaquín Motos María Jesús Planells César F. Talavera |
| author_sort | Joaquín Motos |
| collection | DOAJ |
| description | We show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed. |
| format | Article |
| id | doaj-art-937e0141bfb34a0ba5b17b10b70e418f |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-937e0141bfb34a0ba5b17b10b70e418f2025-08-20T03:37:19ZengWileyAbstract and Applied Analysis1085-33751687-04092016-01-01201610.1155/2016/13934961393496Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander SpacesJoaquín Motos0María Jesús Planells1César F. Talavera2Departamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, SpainDepartamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, SpainDepartamento de Matemática Aplicada, Universidad Politécnica de Valencia, Camino de Vera, s/n, 46022 Valencia, SpainWe show that the dual Bp·locΩ′ of the variable exponent Hörmander space Bp(·)loc(Ω) is isomorphic to the Hörmander space B∞c(Ω) (when the exponent p(·) satisfies the conditions 0<p-≤p+≤1, the Hardy-Littlewood maximal operator M is bounded on Lp(·)/p0 for some 0<p0<p- and Ω is an open set in Rn) and that the Fréchet envelope of Bp(·)loc(Ω) is the space B1loc(Ω). Our proofs rely heavily on the properties of the Banach envelopes of the p0-Banach local spaces of Bp(·)loc(Ω) and on the inequalities established in the extrapolation theorems in variable Lebesgue spaces of entire analytic functions obtained in a previous article. Other results for p(·)≡p, 0<p<1, are also given (e.g., all quasi-Banach subspace of Bploc(Ω) is isomorphic to a subspace of lp, or l∞ is not isomorphic to a complemented subspace of the Shapiro space hp-). Finally, some questions are proposed.http://dx.doi.org/10.1155/2016/1393496 |
| spellingShingle | Joaquín Motos María Jesús Planells César F. Talavera Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces Abstract and Applied Analysis |
| title | Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces |
| title_full | Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces |
| title_fullStr | Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces |
| title_full_unstemmed | Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces |
| title_short | Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces |
| title_sort | frechet envelopes of nonlocally convex variable exponent hormander spaces |
| url | http://dx.doi.org/10.1155/2016/1393496 |
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