Maximal Function Methods for Sobolev Spaces

This book concerns maximal functions and their use in the theory of Sobolev spaces and partial differential equations. Maximal functions have well-known uses in harmonic analysis and function theory, but their use in Sobolev-type inequalities and estimates is less familiar, which provides the motivation for this work. This book makes a valuable and expansive contribution to the literature in this direction and will be valuable for researchers due to the large number and variety of results throughout the text. Let me describe the contents in some detail.

 

The first two chapters introduce the tandem objects in the title that will be discussed throughout the book: the definition and basic properties of the Hardy-Littlewood maximal function \( M f \) and the Sobolev spaces \( W^{1,p}(\Omega) \) and \( W_{0}^{1,p}(\Omega) \). In many books, Hölder and Lipschitz norms are treated separately from Sobolev norms, but I found it particularly interesting and useful that these authors treat \( \mathop{Lip}(\Omega) \) and \( W^{1,p}(\Omega) \) simultaneously. Chapters 3 and 4 cover a number of inequalities, including various versions of Sobolev inequalities, Poincaré inequalities, and pointwise estimates using the maximal function of the gradient. Throughout these chapters, the authors do a good job of introducing the familiar theory of Sobolev spaces but from their unique perspective.  A lot of basic material is included with detailed proofs, but the authors also reference (without proof) more advanced results like Rademacher’s theorem.

 

The next several chapters turn to finer points of the theory. Chapter 5 is devoted to the capacity of sets, both the Sobolev p-capacity and the variational p-capacity for \( 1 < p < \infty \). The authors use the former to study pointwise behavior such as the p-quasicontinuity of functions in Sobolev spaces and relate the latter to Hausdorff content. In Chapter 6 the authors turn to Hardy’s inequalities in a domain \( \Omega \) with zero boundary values. They consider both the standard integral p-Hardy inequality and a stronger pointwise p-Hardy inequality; the maximal function of \( | \nabla u | p \) is used for the latter. The authors also formulate a p-capacity density condition on \( \Omega^{c} \), which is necessary and sufficient for the pointwise p-Hardy inequality to hold in \( \Omega \). The authors explore in Chapter 7 other geometric conditions that imply this p-capacity density condition. The

ideas in Chapters 6 & 7 have complicated inter-relationships, but the authors do a good job of sorting them out.

 

The next several chapters focus on various weight functions. Muckenhoupt weights and related weighted norm inequalities for maximal functions are discussed in Chapter 8: a theme is showing the “self-improving” phenomenon that if a weighted norm inequality holds for some \( p \in (1, \infty) \), then it also holds for \( p- \epsilon \) for some \( \epsilon in ( 0, p-1 ) \). Chapter 9 concerns weighted versions of the Poincaré inequality on cubes, including when different weight functions can be used on the two sides of the inequality; weighted estimates on a maximal function play a role in this. Chapter 10 concerns using a power of the distance function as a weight in Poincaré and Hardy-Sobolev inequalities.

 

Finally, Chapters 11 and 12 illustrate the application of Sobolev space theory to a particular partial differential equation, the p-Laplace equation. This includes the existence and uniqueness of weak solutions, properties of weak sub- and supersolutions, energy estimates and Hölder continuity for weak solutions, and Harnack’s inequality for nonnegative weak solutions, all for \( 1 < p < \infty) \). Then they turn to less standard “stability results”: higher integrability of the gradient of a weak solution, stability with respect to p of weak solutions with given boundary values, and “very weak” solutions which belong to a Sobolev space with exponent smaller than p. There is little to no mention of maximal functions in these last two chapters, but we know by now that they have played a role in establishing the Sobolev space methods that are being used.

 

In summary, this book gathers together a variety of results that are difficult to find in one place, and includes the familiar theory of Sobolev spaces, but often presented in the less familiar context of maximal functions. Moreover, the book is well-written and there are notes at the end of each chapter, which provide very useful references for additional background and details.  There is a lot of notation in the book, but a notation list appears at the beginning of the book and notation that involves letters is included in the Index at the end of the book. For all these reasons, I conclude that this book makes an important and valuable contribution to the literature on maximal functions, Sobolev spaces, and partial differential equations.

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Robert McOwen is a Professor of Mathematics at Northeastern University in Boston, Massachusetts.