Square Roots of Elliptic Systems in Locally Uniform Domains

This book establishes a comprehensive theory to treat square roots of elliptic systems incorporating mixed boundary conditions under minimal geometric assumptions. To lay the groundwork, the text begins by introducing the geometry of locally uniform domains and establishes theory for function spaces on locally uniform domains, including interpolation theory and extension operators. In these introductory parts, fundamental knowledge on function spaces, interpolation theory and geometric measure theory and fractional dimensions are recalled, making the main content of the book easier to comprehend. The centerpiece of the book is the solution to Kato's square root problem on locally uniform domains. The Kato result is complemented by corresponding L? bounds in natural intervals of integrability parameters.
This book will be useful to researchers in harmonic analysis, functional analysis and related areas.

Introduction.- Locally uniform domains.- A density result for locally uniform domains.- Sobolev extension operator.- A short account on sectorial and bisectorial operators.- Elliptic systems in divergence form.- Porous sets.- Sobolev spaces with a vanishing trace condition.- Hardy’s inequality.- Real interpolation of Sobolev spaces.- Higher regularity for fractional powers of the Laplacian.- First order formalism.- Kato’s square root property on thick sets.- Removing the thickness condition.- Interlude: Extension operators for fractional Sobolev spaces.- Critical numbers and Lp - Lq bounded families of operators.- Lp-bounds for the H1-calculus and Riesz transform.- Calder´on–Zygmund decomposition for Sobolev functions.- Lp bounds for square roots of elliptic systems.- References.- Index.

Sebastian Bechtel is a postdoctoral researcher in the analysis group of the Delft Institute of Applied Mathematics at Delft university of Technology. He obtained his PhD in Mathematics at the Technical University of Darmstadt, Germany in 2021. His PhD studies were supported by a scholarship of "Studienstiftung des Deutschen Volkes". His research interests include harmonic analysis, PDEs, function spaces, functional calculus, and related topics.