Some Sobolev embeddings of fractional type and applications

Abstract

Using the Riemann-Liouville derivatives as a basis, we let us introduce in depth fractional Sobolev spaces, characterizing their distinctive nature. We also define derivatives weak fractional values and demonstrate their agreement with the derivatives of RiemannLiouville. Subsequently, we established the equivalence between certain norms within these spaces, thus deducing their exhaustiveness, reflexivity, and separability. In an unconventional way, we highlight certain Sobolev embeddings which are not generally classical, thus enriching our understanding of these spaces. Finally, we apply these notions to a specified boundary problem.