Some applications of the dual spaces of Hardy-amalgam spaces

Abstract

Thanks to the characterizations of the dual spaces of the Hardy-amalgam spaces $$\mathcal H^{(q,p)}$$ and $$\mathcal{H}_{\mathrm{loc}}^{(q,p)}$$ for $$0<q\leq1$$ and $$q\leq p<\infty$$ , obtained in one of our recent papers, we prove that the inclusion of $$\mathcal H^{(1,p)}$$ in $$(L^1,\ell^p)$$ for $$1\leq p<\infty$$ is strict, and more generally, that the one of $$\mathcal H^{(q,p)}$$ in $$\mathcal{H}_{\mathrm{loc}}^{(q,p)}$$ for $$0<q\leq1$$ and $${q\leq p<\infty}$$ , but also for $$0<p<q\leq1$$ , is strict. Moreover, as other applications, we obtain some results of boundedness of Calderón–Zygmund and convolution operators, generalizing those known in the setting of the spaces $$\mathcal H^1$$ and $$\mathrm{BMO}(\mathbb{R}^d)$$ , and more generally, in the setting of $$\mathcal H^q$$ and its dual space.