A Comparative Study of Classical Lebesgue Spaces and Variable Exponent Lebesgue Spaces with Their Applications

Abstract

Lebesgue spaces are fundamental in functional analysis and the theory of partial differential equa tions. They extend the classical notion of integrability based on Riemann integration by relying on measure theory. The spaces L p (Ω) classify functions according to the integrability of |f| p and form Banach spaces, while L 2 (Ω) is a Hilbert space. However, constant-exponent L p spaces are not suitable for modeling heterogeneous phenomena where properties vary locally. This limitation motivates the introduction of variable exponent spaces L p(x) (Ω), where the exponent depends on the spatial position. These spaces are widely used in applications such as fluid mechanics and image processing.