A First Course in Sobolev Spaces

I’m loath to admit that despite my having been honorably discharged from graduate school over twenty years ago, my contact with Sobolev spaces has heretofore been nigh on nil. As a number theorist I didn’t encounter these beasties except anecdotally or marginally, e.g., in graduate school colloquia in analysis which I attended as an interested (or coerced) outsider — but unquestionably an outsider — or leafing though a journal or a book, again generally far outside of my area of activity. To be sure, until recently, I have not had any need for Sobolev spaces in my research. But things have just changed for me in a rather dramatic manner and I find myself faced with having to learn functional analysis, and in particular certain things about Sobolev spaces, at a deeper level than what I have available as a bequest from my graduate studies. For this reason, admittedly a somewhat selfish one, Giovanni Leoni’s brand new A First Course in Sobolev Spaces appears on the scene as a welcome surprise. Says Leoni: “To my knowledge, this is one of the first books to follow the… approach… [of looking at Sobolev spaces] as the natural development and unfolding of [the theory of] monotone, absolutely continuous, and B[ounded] V[ariation] functions of one variable.”

And so we find on p. 279 (!) of this big (~ 600 pp.) book the definition that a Sobolev space is a subspace of an L^p-space on some open set in Rⁿ characterized by the condition that the members of the subspace are functions all of whose distributional first-order partial derivatives live in the given L^p-space. Fair enough, but two questions arise: what’s this distributions business, and what’s with the 278-page build-up?

Well, there’s a single answer to these questions available: Leoni opted to structure his book in accord with accommodations he had to make to a segment of nonplussed beginning graduate students he encountered in his 2006 and 2008 courses titled “Sobolev Spaces.” These latter-day youths manifestly knew not that a stiff dose of functional analysis were de rigeur for having a go at Sobolev spaces, and so Leoni had to change his selection of pitches if there was to be a game at all. Hence Leoni characterizes the first part of A First Course in Sobolev Spaces as amenable to being used for “an advanced undergraduate or beginning graduate course on real analysis or functions of one variable.”

The book’s second part “begins with [a] chapter on absolutely continuous transformations from a domains of R^N into R^N,” followed by a discussion of Laurent Schwartz’s distributions: the stage is set — at last. And this second part of the book is serious analysis indeed: coverage of weak derivatives, BV functions of several variables, Besov spaces, and more. Hard core hard analysis. The game’s afoot…

Leoni has worked hard to make A First Course in Sobolev Spaces maximally effective pedagogically: solid but accessible prose, great attention to detail, exercises, and a good sense of humor in evidence. This is a book that will see a lot of use. And, to get back to my earlier personal disclosures, I do propose to use it myself when the time is right — soon, I hope.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.