"The book is written in a very concise, clear, and elegant way. Misprints are rare... There are no exercises, but the book is well equipped with examples, which help to understand the assertions and are, as a rule, of independent interest. To sum up, the text presents an extremely interesting account of some of the most important developments in the chosen direction." -MATHEMATICAL REVIEWS "The boal of the book under review is to present a method for examing the surprising connection between invariant differential operators and almost periodic operators on Lie groups with polynomial growth... The book is deveoted to a very interesting topic. It is aimed to graduate studetns as well as researchers, and it can be highly recommended." ---ZAA

• 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.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.

"The book is written in a very concise, clear, and elegant way. Misprints are rare... There are no exercises, but the book is well equipped with examples, which help to understand the assertions and are, as a rule, of independent interest. To sum up, the text presents an extremely interesting account of some of the most important developments in the chosen direction." -MATHEMATICAL REVIEWS "The boal of the book under review is to present a method for examing the surprising connection between invariant differential operators and almost periodic operators on Lie groups with polynomial growth... The book is deveoted to a very interesting topic. It is aimed to graduate studetns as well as researchers, and it can be highly recommended." ---ZAA

• 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.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.