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Hörmander´s operators are an important class of linear elliptic-parabolic degenerate partial differential operators with smooth coefficients, which have been intensively studied since the late 1960s and are still an active field of research. This text provides the reader with a general overview of the field, with its motivations and problems, some of its fundamental results, and some recent lines of development.

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1 Hörmander´s operators: what they are.- 2 Hörmander´s operators: why they are studied.- 3 A priori estimates in Sobolev spaces.- 4 Geometry of Hörmander´s vector fields.- 5 Beyond Hörmander´s operators.

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From the book reviews:

"It represents a detailed survey of the theory of hypoelliptic operators and Hörmander´s vector fields, containing all the main results, from the first contributions in the late 1960s up to the most recent results. ... a valuable guide to the literature of this field and will certainly be much appreciated by every Ph.D. student working in PDEs." (Fabio Nicola, Mathematical Reviews, February, 2015)

"The text is a good introduction to the subject for those interested in this branch of the theory of partial differential equations. ... It has plenty of examples that illustrate the topics being studied, mainly in the first chapters. ... The book finishes with some more recent results that are set, in some way, in an extended class of Hörmander´s operators." (Sérgio Luís Zani, zbMATH, Vol. 1298, 2014)

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Marco Bramanti is Associate Professor of Mathematical Analysis at the Department of Mathematics, Politecnico di Milano, Italy. Graduated in Mathematics in 1987, Ph.D. in Mathematics in 1993 at the University of Milano (Italy). He has been visiting University of Minnesota (USA) and Universidad Autnoma de Madrid (Spain). His current reasearch interests include linear second order partial differential equations with nonnegative characteristic form, geometry of Hörmander´s vector fields, real analysis, in particular singular integral theory. Author of about 30 research papers on refereed journals, one research monograph, some contributions to congress proceedings, 14 textbooks, more than 60 reviews for "Mathematical Reviews".

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