This book offers a unified framework for the analysis of function spaces associated with non-additive measures. Motivated by questions arising in nonlinear potential theory and super-critical PDEs, it develops the calculus of Choquet integration and studies the structural properties of Lorentz-type spaces defined via general capacities. Particular attention is given to Bessel capacities, including their role in describing fine properties of Sobolev functions and the normability of Choquet integral spaces. Building on this foundation, the second part introduces and characterizes Sobolev multiplier spaces defined through capacities, establishing their preduals and embedding properties. The boundedness of maximal operators is analyzed using tools from nonlinear potential theory, yielding vector-valued estimates in this setting. The monograph is intended for researchers in analysis and PDEs interested in the interplay between capacity theory, function spaces, and operator estimates.

Autorentext

Keng Hao Ooi is a researcher in analysis who was formerly affiliated with the Department of Mathematics at National Central University, Taiwan. His work focuses on capacities, Choquet integrals, Lorentz type spaces, Sobolev multiplier spaces, and nonlinear potential theory. He has published single author and coauthored papers in leading journals such as the Journal of Functional Analysis, Mathematische Annalen, and the Annali della Scuola Normale Superiore di Pisa. His recent research develops new techniques for the boundedness of maximal operators in capacity based settings, the structure and preduals of Lorentz type and Sobolev multiplier spaces, and weighted inequalities in nonlinear potential theory.