A New Framework for Generalised Hajasz-Besov Spaces on Metric Measure Spaces

What is it about?

This article introduces a new class of function spaces called generalised Hajasz-Besov spaces, defined on metric measure spaces with the doubling and reverse doubling properties (RD-spaces). The work extends classical results for Besov spaces by combining two important ideas: using slowly varying functions to modify smoothness and replacing the basis space with rearrangement invariant spaces. The paper proves several embedding theorems, Sobolev-type embeddings, and results on essential continuity and Morrey-type embeddings for these new spaces, showing their relevance in both abstract and Euclidean settings.

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Photo by Andrew Childress on Unsplash

Photo by Andrew Childress on Unsplash

Why is it important?

Generalised Hajasz-Besov spaces provide a unified and more flexible framework for studying function spaces on metric measure spaces. This allows for finer control over the smoothness and integrability properties of functions, which is crucial for applications in partial differential equations, variational problems, and harmonic analysis. The embedding and continuity results obtained in the paper are essential for understanding the structure and regularity of solutions in these generalized contexts, making the work significant for both theoretical and applied mathematics.

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