Dr. TAIRA, Kazuaki, born in Tokyo, Japan, on January 1, 1946, was a professor of mathematics at the University of Tsukuba, Japan (1998–2009). He received his Bachelor of Science degree in 1969 from the University of Tokyo, Japan, and his Master of Science degree in 1972 from Tokyo Institute of Technology, Japan, where he served as an assistant from 1972 to 1978. The Doctor of Science degree was awarded to him on June 21, 1976, by the University of Tokyo, and on June 13, 1978, the Doctorat d'Etat degree was given to him by Universit\'{e} de Paris-Sud (Orsay), France. He had been studying there on the French government scholarship from 1976 to 1978.Dr. TAIRA was also a member of the Institute for Advanced Study (Princeton), USA (1980–1981), was an associate professor at the University of Tsukuba (1981–1995), and a professor at Hiroshima University, Japan (1995–1998). In 1998, he accepted the offer from the University of Tsukuba to teach there again as a professor. He was a part-time professor at Waseda University (Tokyo), Japan, from 2009 to 2017.His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems, and Markov processes.

“This book is addressed to advanced undergraduate and beginning graduate students and also mathematicians with interest in (classical) real analysis, functional analysis, partial differential equations, and probability theory (in the form of Markov processes). ... this book is amply illustrated for easier understanding: 18 tables and 76 figures are provided with appropriate explanations, captions and examples.” (J. A. van Casteren, Mathematical Reviews, April, 2026)

• Introduction and Main Results.- Elements of Functional Analysis.- Elements of Measure Theory and Lp Spaces.- Elements of Real Analysis.- Harmonic Functions and Poisson Integrals.- Besov Spaces via Poisson Integrals.- Sobolev and Besov Spaces.- Maximum Principles in Sobolev Spaces.- Elements of Singular Integrals.- Calder´on–Zygmund Kernels and Their Commutators.- Calder´on–Zygmund Variable Kernels and Their Commutators.- Dirichlet Problems in Sobolev Spaces.- Calder´on–Zygmund Kernels and Interior Estimates.- Calder´on–Zygmund Kernels and Boundary Estimates.- Unique Solvability of the Homogeneous Dirichlet Problem.- Regular Oblique Derivative Problems in Sobolev Spaces.- Oblique Derivative Boundary Conditions.- Boundary Representation Formula for Solutions.- Boundary Regularity of Solutions.- Proof of Theorems 16.1 and 16.2.- Markov Processes and Feller Semigroups.- Feller Semigroups with Dirichlet Condition.- Feller Semigroups with an Oblique Derivative Condition.- Feller Semigroups and Boundary Value Problems.- Feller Semigroups with a First Order Ventcel’ Boundary Condition.- Concluding Remarks.

Dr. TAIRA, Kazuaki, born in Tokyo, Japan, on January 1, 1946, was a professor of mathematics at the University of Tsukuba, Japan (1998–2009). He received his Bachelor of Science degree in 1969 from the University of Tokyo, Japan, and his Master of Science degree in 1972 from Tokyo Institute of Technology, Japan, where he served as an assistant from 1972 to 1978. The Doctor of Science degree was awarded to him on June 21, 1976, by the University of Tokyo, and on June 13, 1978, the Doctorat d'Etat degree was given to him by Universit\'{e} de Paris-Sud (Orsay), France. He had been studying there on the French government scholarship from 1976 to 1978.Dr. TAIRA was also a member of the Institute for Advanced Study (Princeton), USA (1980–1981), was an associate professor at the University of Tsukuba (1981–1995), and a professor at Hiroshima University, Japan (1995–1998). In 1998, he accepted the offer from the University of Tsukuba to teach there again as a professor. He was a part-time professor at Waseda University (Tokyo), Japan, from 2009 to 2017.His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems, and Markov processes.

“This book is addressed to advanced undergraduate and beginning graduate students and also mathematicians with interest in (classical) real analysis, functional analysis, partial differential equations, and probability theory (in the form of Markov processes). ... this book is amply illustrated for easier understanding: 18 tables and 76 figures are provided with appropriate explanations, captions and examples.” (J. A. van Casteren, Mathematical Reviews, April, 2026)

• Introduction and Main Results.- Elements of Functional Analysis.- Elements of Measure Theory and Lp Spaces.- Elements of Real Analysis.- Harmonic Functions and Poisson Integrals.- Besov Spaces via Poisson Integrals.- Sobolev and Besov Spaces.- Maximum Principles in Sobolev Spaces.- Elements of Singular Integrals.- Calder´on–Zygmund Kernels and Their Commutators.- Calder´on–Zygmund Variable Kernels and Their Commutators.- Dirichlet Problems in Sobolev Spaces.- Calder´on–Zygmund Kernels and Interior Estimates.- Calder´on–Zygmund Kernels and Boundary Estimates.- Unique Solvability of the Homogeneous Dirichlet Problem.- Regular Oblique Derivative Problems in Sobolev Spaces.- Oblique Derivative Boundary Conditions.- Boundary Representation Formula for Solutions.- Boundary Regularity of Solutions.- Proof of Theorems 16.1 and 16.2.- Markov Processes and Feller Semigroups.- Feller Semigroups with Dirichlet Condition.- Feller Semigroups with an Oblique Derivative Condition.- Feller Semigroups and Boundary Value Problems.- Feller Semigroups with a First Order Ventcel’ Boundary Condition.- Concluding Remarks.

Dr. TAIRA, Kazuaki, born in Tokyo, Japan, on January 1, 1946, was a professor of mathematics at the University of Tsukuba, Japan (1998–2009). He received his Bachelor of Science degree in 1969 from the University of Tokyo, Japan, and his Master of Science degree in 1972 from Tokyo Institute of Technology, Japan, where he served as an assistant from 1972 to 1978. The Doctor of Science degree was awarded to him on June 21, 1976, by the University of Tokyo, and on June 13, 1978, the Doctorat d'Etat degree was given to him by Universit\'{e} de Paris-Sud (Orsay), France. He had been studying there on the French government scholarship from 1976 to 1978.Dr. TAIRA was also a member of the Institute for Advanced Study (Princeton), USA (1980–1981), was an associate professor at the University of Tsukuba (1981–1995), and a professor at Hiroshima University, Japan (1995–1998). In 1998, he accepted the offer from the University of Tsukuba to teach there again as a professor. He was a part-time professor at Waseda University (Tokyo), Japan, from 2009 to 2017.His current research interests are in the study of three interrelated subjects in analysis: semigroups, elliptic boundary value problems, and Markov processes.

“This book is addressed to advanced undergraduate and beginning graduate students and also mathematicians with interest in (classical) real analysis, functional analysis, partial differential equations, and probability theory (in the form of Markov processes). ... this book is amply illustrated for easier understanding: 18 tables and 76 figures are provided with appropriate explanations, captions and examples.” (J. A. van Casteren, Mathematical Reviews, April, 2026)

• Introduction and Main Results.- Elements of Functional Analysis.- Elements of Measure Theory and Lp Spaces.- Elements of Real Analysis.- Harmonic Functions and Poisson Integrals.- Besov Spaces via Poisson Integrals.- Sobolev and Besov Spaces.- Maximum Principles in Sobolev Spaces.- Elements of Singular Integrals.- Calder´on–Zygmund Kernels and Their Commutators.- Calder´on–Zygmund Variable Kernels and Their Commutators.- Dirichlet Problems in Sobolev Spaces.- Calder´on–Zygmund Kernels and Interior Estimates.- Calder´on–Zygmund Kernels and Boundary Estimates.- Unique Solvability of the Homogeneous Dirichlet Problem.- Regular Oblique Derivative Problems in Sobolev Spaces.- Oblique Derivative Boundary Conditions.- Boundary Representation Formula for Solutions.- Boundary Regularity of Solutions.- Proof of Theorems 16.1 and 16.2.- Markov Processes and Feller Semigroups.- Feller Semigroups with Dirichlet Condition.- Feller Semigroups with an Oblique Derivative Condition.- Feller Semigroups and Boundary Value Problems.- Feller Semigroups with a First Order Ventcel’ Boundary Condition.- Concluding Remarks.