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

2014; Springer Science+Business Media; Volume: 109; Issue: 2 Linguagem: Inglês

10.1007/s13398-014-0209-z

1579-1505

Joaquín Motos, María Jesús Planells, C.F. Talavera,

Advanced Mathematical Physics Problems

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.