##
**Quasiregular mappings.**
*(English)*
Zbl 0816.30017

Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 26. Berlin: Springer-Verlag. x, 213 p. (1993).

This monograph distinguishes among several important new books on quasiconformality and quasiregularity since it is not only an excellent presentation of the geometric theory of quasiregular maps but opens for the first time an accessible way to Rickman’s profound results on the \(n\)-dimensional Picard-type theorem and Nevanlinna-Ahlfors value distribution theory. As a matter of fact the author’s contribution at the foundation of the quasiregularity goes far beyond these difficult but significant themes because he initiated together with O. Martio and J. Väisälä the systematic study of these maps even in 1968, soon after Yu. G. Reshetnyak has introduced them as mappings in Sobolev spaces with bounded distortion. Thus the book has the privilege to be written by a specialist who lived step by step the development of the theory bringing to it brilliant contribution. This is reflected in the enormous information contained in the text as well as in the notes which end the sections and in the rich bibliography.

Lucidity is a special feature of the book: one feels that each assertion has been subject to a thorough analysis, the necessity of each condition has been examined. Many results are new or have simplified proofs due to the author or to other mathematicians most of them his students. In order to render the theorems more accessible the author points out the main ideas of the intricate proofs, postpones ones of them or even renounce to prove a few because of too complicated technical details. The book is self-contained except for some proofs which can be found in the reference monograph “Lectures on \(n\)-dimensional quasiconformal mappings” (1971; Zbl 0221.30031) by J. Väisälä.

Chapter I opens with preliminary results on \(\text{ACL}^ p\), Sobolev spaces, smoothing operations, the adjunct of a linear map, divergence free vector fields. Then the quasiregular maps are introduced by the analytic definition. The outer and inner dilatations \(K_ 0\) and \(K_ I\), and the branch set are defined. Open and discrete maps are thoroughly studied: their degree, the local topological index, normal domains and neighborhoods, the branch set. The transformation formula for integrals under quasiregular maps is established following a recent plan of M. Pesonen.

Chapter II begins with the theory of the \(p\)-moduli of path families. Various formulae and inequalities, among which the \(K_ 0\)-inequality, are deduced. The path lifting obtains a formalization very adequate for modulus estimates. Then the linear direct and inverse dilatations \(H\) and \(H^*\) are considered and Poletskij’s lemma is proved. The important Poletskij- and Väisälä-inequalities are given. Capacities inequalities follow as corollaries.

Chapter III contains applications of modulus inequalities. First by means of spherical symmetrisation and estimates for the capacity of the Grötzsch condenser, global distortion results: Schwarz’s lemma, Hölder’s continuity, Liouville’s theorem are established. Further, results on sets of capacity zero, on quasimeromorphic maps, properties of the local homeomorphisms, e.g. the injectivity radius and the famous Zorich theorem of rigidity, are proved; equicontinuity results, bounds for the local distortion, for \(H^*\), for the local index are presented.

After this deep insight in quasiregularity the author enters in the core of his value distribution theory, which is exposed in Chapters IV and V. In Chapter IV the counting function \(n(r,y)\) and the covering averages of \(n(r, y)\) over \((n- 1)\)-spheres are introduced. A comparison lemma for these averages and inequalities which bound using \(K_ I\) the average \(A(r)\) of \(n(r,y)\) over \(\mathbb{R}^ n\) with respect to the \(n\)-dimensional spherical measure are proved. By an ingenious device based on the path lifting a growth estimate for the covering average in terms of the number of the omitted values leads to Rickman’s theorem of Picard type (1980). For \(n= 3\) this is the best possible qualitative result, as shows another Rickman theorem (1985) (without proof in the book) assuring the existence of a nonconstant entire quasimeromorphic map which omits an arbitrary number of values. A big Picard-type theorem is also deduced. Interesting comments on recent work by Jormakka, Holopainen and Rickman, Eremenko and Lewis, Vuorinen, Järvi are given. Results on quasimeromorphic maps in a ball and refinements of previous results on equicontinuity are proved.

Chapter V is devoted to the renown Rickman defect relation (first version 1982, actual sharp form 1992). The other Rickman theorem (1992) assuring for \(n= 3\) the existence of quasimeromorphic maps with prescribed defects and the qualitative sharpness of the defect theorem is announced. The defect relation is further established for quasimeromorphic maps of the unit ball with a growth condition. The Ahlfors’ case \(n=2\) is also obtained. Chapter V is completed by results on the order of growth of quasimeromorphic maps (Rickman-Vuorinen), an extension of the comparison lemma in Chapter IV to Riemannian manifolds (Mattila-Rickman), and various relationships opposite to the defect relation.

Chapter VI develops another powerful method in quasiregularity: the variational integrals, used by Reshetnyak, Martio and his studens, Bojarski, Iwaniec and others. Starting from the integral which defines the capacity and replacing the kernel by general variational kernels \(F\) subject to specific conditions, \(F\)-extremals are defined and thoroughly studied. Then the author presents Reshetnyak’s fundamental theorems on quasiregular maps: a.e. differentiability, discreteness and openness, with simplified proofs. Pullbacks of kernels under quasiregular maps are discussed and, among the consequences, the quasiregular invariance of the \(\text{ACL}^ n\) class. The unicity of the continuous \(F\)-extremals, Harnack’s inequality, the existence of \(F\)-extremals, a comparison principle with general boundary values are established. The Reshetnyak limit theorem with Lindqvist’s proof applies to show that small dilatation implies local homeomorphism.

The boundary behavior is the subject of Chapter VII: removability results by Iwaniec and Martin, and Iwaniec alone, Rickman’s example of nonremovable Cantor set (without proof), the connection between radial and angular limits for quasimeromorphic maps in \(B^ n\), theorem of Iversen-, Fatou-, F. and M. Riesz-type, Rickman’s version of Lindelöf’s theorem replacing asymptotic values by limits along \((n- 1)\)-dimensional sets. The presentation of Granlund, Lindqvist and Martio nonlinear potential theory is continued: essential continuity of the \(F\)-extremals, their continuity up to the boundary under Wiener’s condition, the \(F\)- capacity and \(F\)-potential, the important notions of \(F\)-harmonic measures and sub-\(F\)-extremals, applications to an Phragmén-Lindelöf principle and to the growth of quasiregular maps.

The richness of the contents, the systematic form, numerous examples and open problems makes extremely valuable this book, which surely will strongly contribute to the further development of the quasiregularity.

Lucidity is a special feature of the book: one feels that each assertion has been subject to a thorough analysis, the necessity of each condition has been examined. Many results are new or have simplified proofs due to the author or to other mathematicians most of them his students. In order to render the theorems more accessible the author points out the main ideas of the intricate proofs, postpones ones of them or even renounce to prove a few because of too complicated technical details. The book is self-contained except for some proofs which can be found in the reference monograph “Lectures on \(n\)-dimensional quasiconformal mappings” (1971; Zbl 0221.30031) by J. Väisälä.

Chapter I opens with preliminary results on \(\text{ACL}^ p\), Sobolev spaces, smoothing operations, the adjunct of a linear map, divergence free vector fields. Then the quasiregular maps are introduced by the analytic definition. The outer and inner dilatations \(K_ 0\) and \(K_ I\), and the branch set are defined. Open and discrete maps are thoroughly studied: their degree, the local topological index, normal domains and neighborhoods, the branch set. The transformation formula for integrals under quasiregular maps is established following a recent plan of M. Pesonen.

Chapter II begins with the theory of the \(p\)-moduli of path families. Various formulae and inequalities, among which the \(K_ 0\)-inequality, are deduced. The path lifting obtains a formalization very adequate for modulus estimates. Then the linear direct and inverse dilatations \(H\) and \(H^*\) are considered and Poletskij’s lemma is proved. The important Poletskij- and Väisälä-inequalities are given. Capacities inequalities follow as corollaries.

Chapter III contains applications of modulus inequalities. First by means of spherical symmetrisation and estimates for the capacity of the Grötzsch condenser, global distortion results: Schwarz’s lemma, Hölder’s continuity, Liouville’s theorem are established. Further, results on sets of capacity zero, on quasimeromorphic maps, properties of the local homeomorphisms, e.g. the injectivity radius and the famous Zorich theorem of rigidity, are proved; equicontinuity results, bounds for the local distortion, for \(H^*\), for the local index are presented.

After this deep insight in quasiregularity the author enters in the core of his value distribution theory, which is exposed in Chapters IV and V. In Chapter IV the counting function \(n(r,y)\) and the covering averages of \(n(r, y)\) over \((n- 1)\)-spheres are introduced. A comparison lemma for these averages and inequalities which bound using \(K_ I\) the average \(A(r)\) of \(n(r,y)\) over \(\mathbb{R}^ n\) with respect to the \(n\)-dimensional spherical measure are proved. By an ingenious device based on the path lifting a growth estimate for the covering average in terms of the number of the omitted values leads to Rickman’s theorem of Picard type (1980). For \(n= 3\) this is the best possible qualitative result, as shows another Rickman theorem (1985) (without proof in the book) assuring the existence of a nonconstant entire quasimeromorphic map which omits an arbitrary number of values. A big Picard-type theorem is also deduced. Interesting comments on recent work by Jormakka, Holopainen and Rickman, Eremenko and Lewis, Vuorinen, Järvi are given. Results on quasimeromorphic maps in a ball and refinements of previous results on equicontinuity are proved.

Chapter V is devoted to the renown Rickman defect relation (first version 1982, actual sharp form 1992). The other Rickman theorem (1992) assuring for \(n= 3\) the existence of quasimeromorphic maps with prescribed defects and the qualitative sharpness of the defect theorem is announced. The defect relation is further established for quasimeromorphic maps of the unit ball with a growth condition. The Ahlfors’ case \(n=2\) is also obtained. Chapter V is completed by results on the order of growth of quasimeromorphic maps (Rickman-Vuorinen), an extension of the comparison lemma in Chapter IV to Riemannian manifolds (Mattila-Rickman), and various relationships opposite to the defect relation.

Chapter VI develops another powerful method in quasiregularity: the variational integrals, used by Reshetnyak, Martio and his studens, Bojarski, Iwaniec and others. Starting from the integral which defines the capacity and replacing the kernel by general variational kernels \(F\) subject to specific conditions, \(F\)-extremals are defined and thoroughly studied. Then the author presents Reshetnyak’s fundamental theorems on quasiregular maps: a.e. differentiability, discreteness and openness, with simplified proofs. Pullbacks of kernels under quasiregular maps are discussed and, among the consequences, the quasiregular invariance of the \(\text{ACL}^ n\) class. The unicity of the continuous \(F\)-extremals, Harnack’s inequality, the existence of \(F\)-extremals, a comparison principle with general boundary values are established. The Reshetnyak limit theorem with Lindqvist’s proof applies to show that small dilatation implies local homeomorphism.

The boundary behavior is the subject of Chapter VII: removability results by Iwaniec and Martin, and Iwaniec alone, Rickman’s example of nonremovable Cantor set (without proof), the connection between radial and angular limits for quasimeromorphic maps in \(B^ n\), theorem of Iversen-, Fatou-, F. and M. Riesz-type, Rickman’s version of Lindelöf’s theorem replacing asymptotic values by limits along \((n- 1)\)-dimensional sets. The presentation of Granlund, Lindqvist and Martio nonlinear potential theory is continued: essential continuity of the \(F\)-extremals, their continuity up to the boundary under Wiener’s condition, the \(F\)- capacity and \(F\)-potential, the important notions of \(F\)-harmonic measures and sub-\(F\)-extremals, applications to an Phragmén-Lindelöf principle and to the growth of quasiregular maps.

The richness of the contents, the systematic form, numerous examples and open problems makes extremely valuable this book, which surely will strongly contribute to the further development of the quasiregularity.

Reviewer: C.Andreian Cazacu (Bucureşti)

### MSC:

30C65 | Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations |