Some applications of the dual spaces of Hardy-amalgam spaces

TL;DR

This paper explores the dual spaces of Hardy-amalgam spaces, demonstrating strict inclusions and applying these results to establish boundedness of key operators like Calderón-Zygmund and convolution operators.

Contribution

It generalizes the dual space characterizations of Hardy-amalgam spaces and applies these to prove operator boundedness, extending classical results.

Findings

Inclusion of $\\mathcal{H}^{(1,p)}$ in $(L^1,\ell^p)$ is strict.

Inclusion of $\mathcal{H}^{(q,p)}$ in $\mathcal{H}_{\mathrm{loc}}^{(q,p)}$ is strict.

Boundedness results for Calderón-Zygmund and convolution operators.

Abstract

In this paper, thanks to the generalizations 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 our earlier paper, 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, the one of $\mathcal H^{(q,p)}$ in $\mathcal{H}_{\mathrm{loc}}^{(q,p)}$ for $0<q\leq1$ and $q\leq p<\infty$. Moreover, as other applications, we obtain results of boundedness of Calder\'on-Zygmund and convolution operators, generalizing those known in the context of the spaces $\mathcal H^1$ and $BMO(\mathbb{R}^d)$.