Nobuyuki Suita

Bio: Nobuyuki Suita is an academic researcher from Tokyo Institute of Technology. The author has contributed to research in topics: Riemann surface & Riemann sphere. The author has an hindex of 8, co-authored 22 publications receiving 251 citations.

Capacities and kernels on Riemann surfaces

Abstract: Let K(z, z), cp(z) and c~(z) be the values of the Bergman kernel, the capacity, and the analytic capacity, on an open Riemann surface 12 (with respect to a fixed local parameter z). The following problem was raised by SARIO & OIKAWA [9]: Find a relation between the magnitudes of the quantities ]/~---g~(z, z), c a (z) and cg (z). As to ]/n--K-~(z, z) and ce (z), HEMAL obtained an answer for finite Riemann surfaces f2, namely that ] / ~ > c B ( z ) if f2 is not simply connected. 1 In the present paper, we shall give the following complete answer to the latter problem: Vn--g~(z,z)>cB(z) with equality if and only if either (i) 12e0o or (ii) f2 is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero. Concerning the problem for ]/n--K~(z, z) and ca(z), we are led to conjecture that ]/n--g-(~,(z,-z))> ca(z); this will be verified for doubly-connected regions in w 4. 6 2 By proving a new identity ~ log c a (z)= n K(z, z), we show that the conjecture

Painlevé's problem and the semiadditivity of analytic capacity

Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

Painleve's problem and the semiadditivity of analytic capacity

Abstract: Let $\gamma(E)$ be the analytic capacity of a compact set $E$ and let $\gamma_+(E)$ be the capacity of $E$ originated by Cauchy transforms of positive measures. In this paper we prove that $\gamma(E)\approx\gamma_+(E)$ with estimates independent of $E$. As a corollary, we characterize removable singularities for bounded analytic functions in terms of curvature of measures, and we deduce that $\gamma$ is semiadditive, which solves a long standing question of Vitushkin.

Symmetrization in the geometric theory of functions of a complex variable

Abstract: CONTENTSIntroduction Chapter I. Symmetrization transformations ?1. Capacities 1. Functions satisfying the Lipschitz condition 2. Condenser capacity 3. Reduced modulus and inner radius ?2. Polarization ?3. Symmetrization 1. Schwarz symmetrization 2. Steiner symmetrization 3. Circular symmetrization 4. Symmetrization of functions 5. Elliptic symmetrization 6. Symmetrization with respect to a circle ?4. Piecewise separating symmetrization 1. Separating transformation 2. Other types of piecewise separating symmetrization ?5. Averaging transformations ?6. Dissymmetrization Comments on Chapter I Chapter II. Application of symmetrization transformations to some problems in geometric function theory ?7. Inequalities for moduli and capacities 1. Annuli 2. The Fekete problem3. Subsets of the unit circle4. Gonchar's problem5. Polygons ?8. Estimates of harmonic measure ?9. Problems of extremal partitioning1. Fixed poles2. Free poles ?10. Univalent functions 1. The Szeg? problem 2. The Bazilevich problem 3. Distortion theorems 4. Covering theorems ?11. Multivalent functions 1. Symmetrization of plane images 2. Symmetrization of a Riemann surface Unsolved problems References

Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)

Abstract: Preface Prologue: The Exponential Function Chapter 1: Abstract Integration Set-theoretic notations and terminology The concept of measurability Simple functions Elementary properties of measures Arithmetic in [0, ] Integration of positive functions Integration of complex functions The role played by sets of measure zero Exercises Chapter 2: Positive Borel Measures Vector spaces Topological preliminaries The Riesz representation theorem Regularity properties of Borel measures Lebesgue measure Continuity properties of measurable functions Exercises Chapter 3: Lp-Spaces Convex functions and inequalities The Lp-spaces Approximation by continuous functions Exercises Chapter 4: Elementary Hilbert Space Theory Inner products and linear functionals Orthonormal sets Trigonometric series Exercises Chapter 5: Examples of Banach Space Techniques Banach spaces Consequences of Baire's theorem Fourier series of continuous functions Fourier coefficients of L1-functions The Hahn-Banach theorem An abstract approach to the Poisson integral Exercises Chapter 6: Complex Measures Total variation Absolute continuity Consequences of the Radon-Nikodym theorem Bounded linear functionals on Lp The Riesz representation theorem Exercises Chapter 7: Differentiation Derivatives of measures The fundamental theorem of Calculus Differentiable transformations Exercises Chapter 8: Integration on Product Spaces Measurability on cartesian products Product measures The Fubini theorem Completion of product measures Convolutions Distribution functions Exercises Chapter 9: Fourier Transforms Formal properties The inversion theorem The Plancherel theorem The Banach algebra L1 Exercises Chapter 10: Elementary Properties of Holomorphic Functions Complex differentiation Integration over paths The local Cauchy theorem The power series representation The open mapping theorem The global Cauchy theorem The calculus of residues Exercises Chapter 11: Harmonic Functions The Cauchy-Riemann equations The Poisson integral The mean value property Boundary behavior of Poisson integrals Representation theorems Exercises Chapter 12: The Maximum Modulus Principle Introduction The Schwarz lemma The Phragmen-Lindelof method An interpolation theorem A converse of the maximum modulus theorem Exercises Chapter 13: Approximation by Rational Functions Preparation Runge's theorem The Mittag-Leffler theorem Simply connected regions Exercises Chapter 14: Conformal Mapping Preservation of angles Linear fractional transformations Normal families The Riemann mapping theorem The class L Continuity at the boundary Conformal mapping of an annulus Exercises Chapter 15: Zeros of Holomorphic Functions Infinite Products The Weierstrass factorization theorem An interpolation problem Jensen's formula Blaschke products The Muntz-Szas theorem Exercises Chapter 16: Analytic Continuation Regular points and singular points Continuation along curves The monodromy theorem Construction of a modular function The Picard theorem Exercises Chapter 17: Hp-Spaces Subharmonic functions The spaces Hp and N The theorem of F. and M. Riesz Factorization theorems The shift operator Conjugate functions Exercises Chapter 18: Elementary Theory of Banach Algebras Introduction The invertible elements Ideals and homomorphisms Applications Exercises Chapter 19: Holomorphic Fourier Transforms Introduction Two theorems of Paley and Wiener Quasi-analytic classes The Denjoy-Carleman theorem Exercises Chapter 20: Uniform Approximation by Polynomials Introduction Some lemmas Mergelyan's theorem Exercises Appendix: Hausdorff's Maximality Theorem Notes and Comments Bibliography List of Special Symbols Index

Suita conjecture and the Ohsawa-Takegoshi extension theorem

Abstract: We prove a conjecture of N. Suita which says that for any bounded domain D in ℂ one has $c_{D}^{2}\leq\pi K_{D}$ , where c D (z) is the logarithmic capacity of ℂ∖D with respect to z∈D and K D the Bergman kernel on the diagonal. We also obtain optimal constant in the Ohsawa-Takegoshi extension theorem.