Mixed-norm Amalgam Spaces

Abstract

We introduce the mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}})({\mathbb{R}}^n)$ and $(L^{\vec{p}},L^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)$, and show their some basic properties. In addition, we find the predual $\mathcal{H}({\vec{p}}^{\prime },\vec{s}{}^{\prime },{\alpha }^{\prime })$ of mixed-norm amalgam spaces $(L^{\vec{p}},{\ell }^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)$ by the dual spaces $(L^{{\vec{p}}^{\prime }},{\ell }^{\vec{s}{}^{\prime }})({\mathbb{R}}^n)$ of $(L^{\vec{p}},{\ell }^{\vec{s}})({\mathbb{R}}^n)$, where $(L^{\vec{p}},L^{\vec{s}})({\mathbb{R}}^n)=(L^{\vec{p}},{\ell }^{\vec{s}})({\mathbb{R}}^n)$ and $(L^{\vec{p}},L^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)=(L^{\vec{p}},{\ell }^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)$. Then, we study the strong-type estimates for fractional integral operators $I_{\gamma }$ on mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)$. And, the strong-type estimates of linear commutators $[b,I_{\gamma }]$ generated by $b\in BMO({\mathbb{R}}^n)$ and $I_{\gamma }$ on mixed-norm amalgam spaces $(L^{\vec{p}},L^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)$ are established as well. Furthermore, based on the dual theorem, the characterization of $BMO({\mathbb{R}}^n)$ by the boundedness of $[b,I_{\gamma }]$ from $(L^{\vec{p}},L^{\vec{s}}{)}^{\alpha }({\mathbb{R}}^n)$ to $(L^{\vec{q}},L^{\vec{s}}{)}^{\beta }({\mathbb{R}}^n)$ is given, which is a new result even for the classical amalgam spaces.

References in corpus (2)

• Integrable fractional mean functions on spaces of homogeneous type
• Necessary and sufficient conditions for boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces