Distance and Measure in Analysis and Partial Differential Equations

This comprehensive exploration of a metric approach to relevant problems and techniques in analysis focuses on examining spaces of homogeneous type. Reflecting active research, the text includes examples and details appearing for the first time in book form.

This comprehensive exploration of a metric approach to some relevant problems and techniques in analysis focuses on examining spaces of homogeneous type. Both the structural and analytical problems under consideration reflect very active areas of current research. The exposition, motivated by examples and details of which many cannot be found elsewhere in book form, unfolds systematically in three parts, starting from such preliminaries as the basic structures of quasi-metric spaces, the homogeneity property and measure theory. More advanced topics are then covered, including Calderón-Zygmund decompositions, L^2 techniques, weighted norm inequalities, BMO, and Hardy spaces. The final aim is to encourage a well-developed way of thinking about some mathematical and even physical problems, i.e. Newtonian and Riesz potentials, fields and their derivatives, mean value and Harnack inequalities related to Hölder regularity, wavelet type analysis on non-euclidian contexts, etc. The book is suitable for self-study or as a classroom resource for graduate students in analysis and PDEs. Real analysis and topology, some basics of Fourier analysis and PDEs are requisite background for the reader.

This comprehensive exploration of a metric approach to some relevant analytical problems and techniques focuses on examining spaces of homogeneous type. Both the structural and analytical problems examined reflect very active areas of current research. The exposition is motivated by examples and details of which many cannot be found elsewhere in book form. The book is suitable for self-study or as a classroom resource for graduate students in analysis and PDEs. Real analysis and topology, some basics of Fourier analysis and PDEs are requisite background for the reader.

Contenu
Part I: POINT SPACES Quasi-Metric Spaces The Weak Homogeneity Property on Quasi-Metric Spaces Spaces of Homogeneous Type Examples Part II: FUNCTION SPACES The Hardy--Littlewood Maximal Function and the Differentiation Theorem Lipschitz Functions The Space L^2 Hardy and John--Birenberg Spaces BMO (f) Spaces Harnack Inequalities and Hölder--Lipschitz Regularity of Functions Besov--Taibleson and Triebel--Lizorkin Spaces Part III: OPERATORS Weighted Norm Inequalities or the Hardy--Littlewood Maximal Operator Electrostatic Potentials and Singular Integrals of Calderón--Zygmund Type Singular Integrals * Unconditional Haar Bases for L^p (1