Analysis on Lie Groups with Polynomial Growth is the first book to present a method for examining the surprising connection between invariant differential operators and almost periodic operators on a suitable nilpotent Lie group. It deals with the theory of second-order, right invariant, elliptic operators on a large class of manifolds: Lie groups with polynomial growth. In systematically developing the analytic and algebraic background on Lie groups with polynomial growth, it is possible to describe the large time behavior for the semigroup generated by a complex second-order operator with the aid of homogenization theory and to present an asymptotic expansion. Further, the text goes beyond the classical homogenization theory by converting an analytical problem into an algebraic one.

This work is aimed at graduate students as well as researchers in the above areas. Prerequisites include knowledge of basic results from semigroup theory and Lie group theory.

Inhalt

I Introduction.- II General Formalism.- II.1 Lie groups and Lie algebras.- II.2 Subelliptic operators.- II.3 Subelliptic kernels.- II.4 Growth properties.- II.5 Real operators.- II.6 Local bounds on kernels.- II.7 Compact groups.- II.8 Transference method.- II.9 Nilpotent groups.- II.10 De Giorgi estimates.- II.11 Almost periodic functions.- II.12 Interpolation.- Notes and Remarks.- III Structure Theory.- III.1 Complementary subspaces.- III.2 The nilshadow; algebraic structure.- III.3 Uniqueness of the nilshadow.- III.4 Near-nilpotent ideals.- III.5 Stratified nilshadow.- III.6 Twisted products.- III.7 The nilshadow; analytic structure.- Notes and Remarks.- IV Homogenization and Kernel Bounds.- IV.1 Subelliptic operators.- IV.2 Scaling.- IV.3 Homogenization; correctors.- IV.4 Homogenized operators.- IV.5 Homogenization; convergence.- IV.6 Kernel bounds; stratified nilshadow.- IV.7 Kernel bounds; general case.- Notes and Remarks.- V Global Derivatives.- V.1 L2-bounds.- V.1.1 Compact derivatives.- V.1.2 Nilpotent derivatives.- V.2 Gaussian bounds.- V.3 Anomalous behaviour.- Notes and Remarks.- VI Asymptotics.- VI. 1 Asymptotics of semigroups.- VI.2 Asymptotics of derivatives.- Notes and Remarks.- Appendices.- A.1 De Giorgi estimates.- A.2 Morrey and Campanato spaces.- A.3 Proof of Theorem II.10.5.- A.4 Rellich lemma.- Notes and Remarks.- References.- Index of Notation.