NDLI: Duals of variable exponent Hörmander spaces ( $$0< p^- \le p^+ \le 1$$ ) and some applications

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Duals of variable exponent Hörmander spaces ( $$0< p^- \le p^+ \le 1$$ ) and some applications

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Duals of variable exponent Hörmander spaces ( $$0< p^- \le p^+ \le 1$$ ) and some applications

Content Provider	Springer Nature Link
Author	Motos, Joaquín Planells, María Jesús Talavera, César F.
Copyright Year	2014
Abstract	In this paper we characterize the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ of the variable exponent Hörmander space $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ when the exponent $$p(\cdot )$$ satisfies the conditions $$0 < p^- \le p^+ \le 1$$ , the Hardy-Littlewood maximal operator $$M$$ is bounded on $$L_{p(\cdot )/p_0}$$ for some $$0 < p_0 < p^-$$ and $$\Omega $$ is an open set in $$\mathbb {R}^n$$ . It is shown that the dual $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )'$$ is isomorphic to the Hörmander space $$\mathcal {B}^{\mathrm {loc}}_\infty (\Omega )$$ (this is the $$p^+ \le 1$$ counterpart of the isomorphism $$\bigl (\mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq \mathcal {B}^{\mathrm {loc}}_{\widetilde{p'(\cdot )}} (\Omega )$$ , $$1 < p^- \le p^+ < \infty $$ , recently proved by the authors) and hence the representation theorem $$\bigl ( \mathcal {B}^c_{p(\cdot )} (\Omega ) \bigr )' \simeq l^{\mathbb {N}}_\infty $$ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of $$\mathcal {B}^c_{p(\cdot )} (\Omega )$$ and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for $$p(\cdot ) \equiv p$$ , $$0 < p < 1$$ , are also given (e.g. $$\mathcal {B}^c_p (\Omega )$$ does not contain any infinite-dimensional $$q$$ -Banach subspace with $$p < q \le 1$$ or the quasi-Banach space $$\mathcal {B}_p \cap \mathcal {E}'(Q)$$ contains a copy of $$l_p$$ when $$Q$$ is a cube in $$\mathbb {R}^n$$ ). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent Hörmander spaces is proposed.
Starting Page	657
Ending Page	668
Page Count	12
File Format	PDF
ISSN	15787303
Journal	Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas
Volume Number	109
Issue Number	2
e-ISSN	15791505
Language	English
Publisher	Springer Milan
Publisher Date	2014-12-12
Publisher Place	Milan
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Variable exponent Hardy-Littlewood maximal operator Banach envelope $$L_{p(\cdot )}$$ -spaces of entire analytic functions Hörmander spaces Topological linear spaces of test functions, distributions and ultradistributions Spaces of differentiable or holomorphic functions on infinite-dimensional spaces Not locally convex spaces Maximal functions, Littlewood-Paley theory Mathematics Applications of Mathematics Theoretical, Mathematical and Computational Physics
Content Type	Text
Resource Type	Article
Subject	Geometry and Topology Applied Mathematics Algebra and Number Theory Analysis Computational Mathematics

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