Generalized modular and orlicz-paranormed sequence spaces and their matrix domains

Abstract

We develop a unified modular framework for Orlicz-type sequence spaces generated by either a single Orlicz function or a coordinatewise family of Orlicz functions. Under a uniform ∆2 growth hypothesis we prove that the associated Luxemburg functional induces a complete BK-topology. We then introduce Maddox-type variable-exponent Orlicz spaces ℓψ(p) and show that they inherit the same BK-structure. Finally, we study Orlicz-paranormed matrix domains ℓψ(A) under lower triangular matrices with nonzero diagonal, establishing completeness, inclusion criteria driven by global dominance of Orlicz functions and by weight monotonicity, explicit descriptions of the Kothe-Toeplitz α-, β-, and γ-duals in terms of the conjugate Orlicz function, and concrete Schauder bases transported via A−1.