Chapter 8 studies symmetrization and convolution.The Riesz-Sobolev convolution theorem is first proved for functions in the unit circle, and then the real line, and finally in n-dimensional space. The Brunn-Minkowski inequality is proved as an application. The Brascamp-LIeb-Luttinger inequality, which extends the Riesz-Sobolev inequality to multiple integrals,is proved too. It implies that the Dirichlet heat kernel increases under symmetrization of the domain.The chapter includes a variation of the sharp Hardy-Littlewood-Sobolev inequality that implies Beckner's logarithmic Sobolev inequality. The latter result is used to establish hypercontractivity of the Poisson semigroup.

The star function was originally developed to prove the spread theorem, a problem dealing with meromorphic functions in the complex plane. The first sections prove the spread theorem, along with other applications to the study of these functions. Later sections center on analytic functions in the unit disk. The star function technique yields to sharp estimates for integral means of univalent functions and the (harmonic) conjugate function, along with the behavior of the Green function and harmonic measure under symmetrization. The final section extends some results to domains of arbitrary connectivity. The chapter includes the necessary background in Nevanlinna theory and the Poincaré metric on hyperbolic plane domains, and in almost all cases, the mappings which exhibitextremal behavior are identified.

Professor Walter K. Hayman, FRS, writesa special Foreword where he notes therole ofBaernstein's star function in comple analysis.The Introduction describes the content of each chapter is some detail, and serves as a guide to the reader.

This chapter presents the basic theory of rearrangements of functions, with special emphasis on the symmetric decreasing rearrangement. Another type of rearrangement central to thisbook ispolarization with respect to an affinehyperplane. Examples and graphs are included throught the chapter.

This chapter marks the debut of the star function in the book. Each type of rearrangement has an associated star function, which is an indefinite integral of the rearranged function. This chapter proves ``subharmonicity'' theorems for the star function, expressing the fact that if a function satisfies a Poisson-type partial differential equation then its star function satisfies a related differential inequality. In the simplest case of circular symmetrization in the plane, the result says that if a function is subharmonic then so is its star function. Subharmonicity is applied in the succeeding chapters to yield comparison theorems for solutions of partial differential equations and extremal results in complex analysis.

Chapter 10 establishes comparison principles for solutions of partial differential equations. The prototypical result says that the solution of Poisson's equation gets bigger in an integral sense when the data in the equation is rearranged. Such comparisons have been used in the literature for deriving sharp bounds on certain eigenvalues, obtaining a priori bounds on solutions, and comparing Green functions, among other uses. These integral norm comparisons follow from star function comparisons, and so the task is to prove that rearranging the data in Poisson's equation increases the star function of the solution. The key is a maximum principle argument applied to the difference of star functions, making use of subharmonicity results from the preceding chapter.

Chapter 6 discusses Steiner symmetrization. Basic properties of symmetric decreasing rearrangement and polarization that were developed in Chapter 1 are adapted to Steiner symmetrization, to show that it decreases the modulus of continuity and acts contractively in L-Infinity.The effect of Steiner symmetrization on various Dirichlet integrals is studied. It is shown that Steiner symmetrization decreases perimeter and Minkowski content, but in general it is not known whether it decreases the (n-1)-dimensional Hausdorff measure. Steiner symmetrization also decreases the principal frequency andvarious capacities, and increases the torsional rigidity and mean lifetime of a Brownian particle.

Chapter 3develops the basic Dirichlet integral inequalities for symmetric decreasing rearrangement. The main result is the decrease of the integral of the p-th power of the gradient(or p-Dirichlet integral) of a function under symmetric decreasing rearrangement. Background material on Sobolev spaces and functional analysis is included as needed to study the continuity of the symmetric decreasing rearrangement in various Sobolev spaces.

This chapter is devoted to the isoperimetric inequality and sharp Sobolev inequalities.

Itbegins with a review of tools from geometric measure theory(Hausdorff measures,area formula, and Gauss--Green theorem) used in this and later chapters. Three isoperimetric inequalities are presented: for perimeter,for Hausdorff measures, and for Minkowski content. Additional facts from geometric measure theory (the coarea formula, and polar coordinates) are included to showthat the coarea formula and the isoperimetric inequality for perimeter together imply decrease of the Dirichlet integral under symmetrization.The sharp Sobolev inequality for p = 1, and its equivalence to the isoperimeric inequality, are due to Federer and Fleming (1960). As discussed in the text, the sharp Sobolev inequality for 1 < p < n is due independently to Rodemich, Aubin and Talenti. The proof presented in this book is a hybrid using both the “classical” method of symmetrization and the recent mass transportation approach of Cordero-Erausquin et al.

Chapter 7 covers symmetrization in the sphere,hyperbolic space, and Gauss space, and includes as an application a landmark theorem of Gehring on quasiconformal mappings. Spheres and hyperbolic spaces have a canonical distance and measure, and possess rich isometry groups of measure preserving mappings. There are plenty of hyperplanes in which to polarize, and so most of the theoryfrom Chapters 2 and 6 can be extended.Sphericaland hyperbolic analogs of inequalities from Chapters 1 and 2 are developed., including the basic polarization inequalityand the foundational inequality for integrals of functions on the sphere under symmetric decreasing rearrangement.We also find a discussion on (k,n)-caps symmetrization.

Chapter2 covers the foundational inequalities for integrals of functions in Euclidean space. The two key results in this chapter are that symmetric decreasing rearrangement of a continuous function decreases the modulus of continuity, and that certain integral expressions increase when functions are replaced by their symmetric decreasing rearrangement. Other results include the Hardy-Littlewood inequalityand the contractivity of rearrangements in the L-infinity norm.

Chapter 5 covers three classical topics in symmetrization, and includes historical remarks as well as the needed background in physics to guide the reader. The first result is that symmetrizing a fixed membrane into a disk of the same area decreases its principal frequency (the first eigenvalue of the Laplacian with Dirichlet boundary conditions), as conjectured by Rayleigh in 1877 and proved independently by Faber and Krahn. The second result is that symmetrization increases the torsional rigidity of a planar domain, as conjectured by St. Venant in 1856 and proved by Pólya.Lastly, a closed ballin three dimensionsal space is shown to have the smallest Newtonian capacity among all compact sets with the same volume. This conjecture was raised by Poincaré in 1887 and proved by Szegö. The proofs depend on the decrease of the Dirichlet integral under symmetric decreasing rearrangement of the function