Multiplication operators on amalgam of $l^{q), θ}$ and $L^{p}$
Monika Singh, Jitendra Kumar
TL;DR
This work develops the theory of grand amalgam spaces by introducing the grand amalgam Lebesgue function space $l^{q),\theta}(L^p)$, proving its Banach space structure and relevant inclusion properties with respect to the parameters $p,q$ and $\theta$. It further defines a small Lebesgue sequence space $l^{q)',\theta}$ and derives a Hölder-type inequality within the framework, leveraging the auxiliary space to couple function-space and sequence-space norms. A central contribution is the boundedness analysis of multiplication operators on $l^{q),\theta}(L^p)$, establishing that for compactly supported multipliers $g$, boundedness on $l^{q),\theta}(L^p)$ is equivalent to $g \in L^\infty$ with operator norm $\|M_g\| = \|g\|_\infty$, and showing related mapping properties into $L^1$ under $g$-weights. The results extend classical amalgam-space theory to a grand amalgam setting, offering a rigorous functional-analytic framework for product-type operators with potential applications in harmonic analysis and PDE contexts.
Abstract
We define the grand amalgam Lebesgue function space $l^{q), θ}(L^p),$ and study the fundamental structural properties of the space, including completeness. Then we define the small Lebesgue sequence space and study its function space properties. Furthermore, we prove a version of the Hölders inequality on the frame work of these spaces. Finally, we study the multiplication operator in the setting of these spaces.