The completeness role of the function ϕ in generating the Riesz potential operator

Vinsensia, Desi
Utami, Yulia
Addini, Puteri Fadjar

International Journal of Basic and Applied Science
Vol. 13 No. 4 (2025): Computer Science, Engineering, Basic and Applied mathematics Science

The Riesz potential operator is a central tool in harmonic analysis and the theory of partial differential equations, commonly defined via convolution with a singular Kernel. In many modern frameworks, function space are generated by a mappings involving such operators. In this paper, we explore the dual role of the generating function- in: (i). Defining the Riesz function space and (ii). Ensuring its completeness. We introduce a Riesz function space whose norm is induced growth function- (a Young function). We establish, through several examples and proofs, that under suitable conditions (specifically, the condition on ), the space is complete. Furthemore, we illustrate discrete analogues and applications to Orlicz space, thereby underscoring the fundamental importance of in both the construction and Banach space structure of these function spaces.