K\"othe-Herz Spaces: The Amalgam-Type Spaces of Infinite Direct Sums

In this paper, we introduce a class of function spaces called K\"othe-Herz spaces $E(\mathcal{X})$. These spaces are similar to amalgam spaces and are characterized by a local component given by a countable family $\mathcal{X}=\left( X_{\alpha }\right) _{\alpha \in I}$ of quasi-normed function spaces, and a global component $E$, which is a quasi-normed sequence space. We investigate various geometric and topological properties inherited by $E(\mathcal{X})$ from its components, such as their completeness, duality, order continuity, ideal and Fatou properties, in an abstract setting. In addition, we provide a Banach function space characterization for $E(\mathcal{X})$, which allows us to understand its structure and behavior more deeply. Furthermore, by appropriate amalgamation of Lorentz spaces (Orlicz spaces) and Lebesgue sequence spaces, we define Lorentz-Herz spaces (Orlicz-Herz spaces) as a particular case of $E(\mathcal{X})$, which are still generalizations of the classical Herz spaces. In this context (especially Lorentz-Herz spaces), we establish previously studied properties, demonstrate interpolation results, and prove the boundedness of important sublinear integral operators with kernels that satisfy a size condition.