A generalization of Orlicz–Sobolev spaces on metric measure spaces via Banach function spaces

Abstract

To cite this article: Marcelina Mocanu (2010) A generalization of Orlicz-Sobolev spaces on metric measure spaces via Banach function spaces, Complex Variables andWe show that several basic definitions and results regarding modulus, capacity and Orlicz-Sobolev spaces on metric measure spaces can be generalized to the case where the role of the Orlicz space is played by an abstract Banach function space B. This new general setting could bring a new perspective in the study of Sobolev-type spaces on metric measure spaces, due to the great generality of Banach function spaces. We prove several properties of the newly introduced Sobolev-type space N 1,B (X ), including its completeness and a Mazur-type theorem.

Newtonian Spaces Based on Quasi-Banach Function Lattices

“…Harjulehto, Hästö, and Pere further developed the theory in [13], where they discussed the Newtonian spaces based on Orlicz-Musielak variable exponent spaces, where the exponent function was essentially bounded. Mocanu [23] worked with Banach function spaces as defined in Bennett and Sharpley [3, Definition I.1.3]. The paper [23] however suffers from improper work with equivalence classes, which eventually leads to invalidity of some of the claims therein.…”

“…Mocanu [23] worked with Banach function spaces as defined in Bennett and Sharpley [3, Definition I.1.3]. The paper [23] however suffers from improper work with equivalence classes, which eventually leads to invalidity of some of the claims therein. Some of the results there also rely on uniform convexity of the function space which considerably lessens the generality.…”

“…Note that we follow the notation of Björn and Björn [5], where N 1 X denotes the space of functions defined everywhere while N 1 X denotes the space of equivalence classes. Some authors, e.g., Shanmugalingam [26], Tuominen [27] and Mocanu [23], use the corresponding symbols the other way around.…”

A Poincaré Inequality for Orlicz–Sobolev Functions with Zero Boundary Values on Metric Spaces

“…B-capacity is an outer measure and represents the correct gauge for distinguishing between two functions in N 1,B (X ) [14].…”

Lebesgue Points for Orlicz-Sobolev Functions on Metric Measure Spaces

“…These spaces are essential in order to extend quasiconformal theory and nonlinear potential theory to the metric setting. A recent direction of research in analysis on metric spaces deals with Orlicz-Sobolev spaces [1], [19] and some of their generalizations [15].…”