Refined localization spaces, Kondratiev spaces with fractional smoothness and extension operators

TL;DR

This paper introduces Kondratiev spaces with fractional smoothness related to refined localization spaces, explores their interpolation properties, revisits Stein's extension operator, and studies Sobolev embeddings on polyhedral domains.

Contribution

It develops a new framework for Kondratiev spaces with fractional smoothness and analyzes their interpolation and extension properties, enhancing the understanding of these function spaces.

Findings

Kondratiev spaces with fractional smoothness are closely related to refined localization spaces.

Complex interpolation extends the scale of Kondratiev spaces with integer smoothness.

Revisiting Stein's extension operator provides new insights for polyhedral domains.

Abstract

In this paper, we introduce Kondratiev spaces of fractional smoothness based on their close relation to refined localization spaces. Moreover, we investigate relations to other approaches leading to extensions of the scale of Kondratiev spaces with integer order of smoothness, based on complex interpolation, and give further results for complex interpolation of those function spaces. As it turns out to be one of the main tools in studying these spaces on domains of polyhedral type, certain aspects of the analysis of Stein's extension operator are revisited. Finally, as an application, we study Sobolev-type embeddings.