Distributions as Boundary Values of Analytic Functions

Tests are described for determining whether certain functions are the boundary values of analytic functions taking the upper half plane into itself. In one theorem the functions are real-valued and defined on an open interval and a single point. In other theorems, the functions are defined on a real interval taking values in an arc γ in the upper half plane. In every case these results are closely related to Loewner's theorem in which the putative boundary values are required to be real-valued. As in Loewner's theory, the quadratic inequalities are finite sums involving at most the value of the function and its first derivative at points. These results are closely related to the Pick-Nevanlinna inequalities for functions defined on a sequence of points in the upper half plane; the Pick-Nevanlinna inequalities characterize those functions that are restrictions of analytic functions taking the upper half plane into itself.

This theorem of F. Riesz' has the distinction of being true only for analytic functions and not for harmonic functions in general as many similar theorems are. Correspondingly, the proof is very analytic. It is based on the canonical decomposition f(z) = g(z).B(z) where g(z) $ 0 and B(z) has boundary values 1 almost everywhere (Blaschke product). Now, for functions of several complex variables no comparable decomposition exists, and yet we will succeed in generalizing the theorem of Riesz proper. If f(z, ***,z) is analytic for O < I zj I < 1,j = 1, ... k, and if

We give a new admissible maximal inequality for analytic functions of several complex variables, and also show that an analytic function which is admissibly bounded at almost all boundary points of a domain D must have an admissible limit a.e.These results are then applied to the boundary behaviour of the Nevanlinna class of D.We shall be concerned with the boundary behaviour of analytic functions in a smooth bounded domain in C. For background and motivation, the reader is referred to Chapter 1 of the book by Stein [3].The aim of this paper is to present two new results, as a corollary of which we shall be able to give very quick and simple proofs of the results in [3, Chapter 2].The author would very much like to express thanks to his adviser Dr. D. A. Edwards for encouragement, to Professor E. M. Stein for pointing out some ambiguity in the proof of Theorem 2 in the original manuscript, and to Mrs. V. Shires for performing the typing.Recall the definition [3, p. 32] of the admissible approach regions &a () at G3Z): &a(0 ={ZED: |(Z -Qv(\ < (1 + a)8e(Z), |Z -||2 < arS(Z)}, (Z) = min(dist of Z from dD, dist of Z from tangent plane at ), Vf = outward drawn normal at .The regions $"() have conical sections in the direction of the "classical tangent" {/uf}, but "tangential parabolic" sections in the 2n -2 complementary real directions (which are n -1 complex directions, of course).Recall also the two types of ball on dD [3, p. 33].The balls Bx are the usual ones of radius p centered at G dD, and the balls B2 are of radius ~p in the classical direction, but of radius ~pl//2 in the complex directions.We consider also the corresponding maximal functions, denoted * and *.Theorem 1.Let u be nonnegative, continuous, andplurisubharmonic in D (we do not require u to be continuous in D), and suppose u has a harmonic majorant, i.e. there is a finite positive measure p on dD so that u(Z) <

We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε–m) with m&gt;2. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. For the critical size ε=O(η2), the asymptotic behavior of the eigenvalues of order O(εm−2) is described via a Steklov problem, where the ‘mass’ is localized on the boundary, or through the eigenvalues of a local problem obtained from the micro-structure of the problem. We use the techniques of the formal asymptotic analysis in homogenization to determine both problems. We also use techniques of convergence in homogenization, Semigroups theory, Fourier and Laplace transforms and boundary values of analytic functions to prove spectral convergence. In the same framework we study the case m=2 as well as the case when other boundary conditions are imposed on ∂Ω.

We propose necessary and sufficient conditions for a complex-valued function f on \( {{\mathbb{R}}^n} \) to be a characteristic function of a probability measure. Certain analytic extensions of f to tubular domains in \( {{\mathbb{C}}^n} \) are studied. In order to extend the class of functions under study, we also consider the case where f is a generalized function (distribution). The main result is given in terms of completely monotonic functions on convex cones in \( {{\mathbb{R}}^n} \).

It is shown that for every 2-concave Banach lattice X of measurable fuctions on the circle, the quotient space X/XA has cotype 2. Here XA denotes the subclass of X consisting of the boundary values of analytic functions. It is also shown that, under slight additional assumptions, a p-concave operator defined on XA factors through L A p = Hp and extends to X, provided that X is p-convex. Bibliography: 10 titles.

Let D be a connected bounded domain in R2 , S be its boundary which is closed, connected and smooth or S = (−∞, ∞). Let Φ(z) = 1 2πi R S f(s)ds s−z , f ∈ L1 (S), z = x+iy. The singular integral operator Af := 1 iπ R S f(s)ds s−t , t ∈ S, is defined in a new way. This definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f ∈ L1 (S) to be boundary value of an analytic in D function. The Sokhotsky-Plemelj formulas are derived for f ∈ L1 (S). Our new definition allows one to treat singular boundary values of analytic functions.

SynopsisUltradistributions of compact support are represented as the boundary values of Cauchy and Poisson integrals corresponding to tubular radial domains Tc' =ℝn + iC', C'⊂⊂C, where C is an open, connected, convex cone. The Cauchy integral of is shown to be an analytic function in TC' which satisfies a certain boundedness condition. Analytic functions which satisfy a specified growth condition in TC' have a distributional boundary value which can be used to determine an distribution.

Let C be an open convex cone in ${\bf R}^n $, n-dimensional real space, such that $\bar C$ does not contain any entire straight line. We consider holomorphic functions in tubes $T^C = {\bf R}^n + iC \subset {\bf C}^n $, n-dimensional complex space, which are known to have $\mathfrak{S}'$ distributional boundary values. We prove that if the $\mathfrak{S}'$ boundary value of such a holomorphic function is an $L^p $ function, $1 \leq p \leq \infty $, then the holomorphic function is in the Hardy space $H^p (T^C )$, $1 \leq p \leq \infty $, corresponding to the tube $T^C $ and can be recovered as the Poisson integral of the distributional boundary value. A similar result is proved wih respect to the holomorphic functions in tubes $T^C $ which are known to have boundary values in the distribution spaces $\mathfrak{Z}',K'_r ,(\mathfrak{S}^\alpha )'$, and $(W^\Omega )'$ and which do not necessarily have $S'$ distributional boundary values. In all cases we prove converse theorems. Our basic results are motivated ...

Vacuum expectation values of products of neutral scalar field operators are discussed. The properties of these distributions arising from Lorentz invariance, the absence of negative energy states and the positive definiteness of the scalar product are determined. The vacuum expectation values are shown to be boundary values of analytic functions. Local commutativity of the field is shown to be equivalent to a symmetry property of the analytic functions. The problem of determining a theory of a neutral scalar field given its vacuum expectation values is posed and solved.

It is shown that the n-point functions of scalar massive free fields on the noncommutative Minkowski space are distributions which are boundary values of analytic functions. Contrary to what one might expect, this construction does not provide a connection to the popular traditional Euclidean approach to noncommutative field theory (unless the time variable is assumed to commute). Instead, one finds Schwinger functions with twistings involving only momenta that are on the mass-shell. This explains why renormalization in the traditional Euclidean noncommutative framework crudely differs from renormalization in the Minkowskian regime.

Elements of[formula omitted]′Ls(Mp)and[formula omitted]′Ls{Mp}as Boundary Values of Holomorphic Functions

A generalization of the Hardy Hp functions corresponding to tubes in and of other related spaces of functions is given. The holomorphic functions in our new spaces are represented as Fourier-Lapiace integrals and properties of the representing functions are obtained. In certain cases the holomorphic functions have distributional boundary values in the strong topology of the tempered distributions, and when the distributional boundary values are of a certain type the holomorphic functions can be represented as the Cauchy and Poisson integrals of the boundary values. All theorems have converse results. The corresponding results for the Hp and previously known related spaces are special cases of the theorems presented here.

We give a simple necessary and sufficient condition for the existence of distributional regularizations. Our results apply to functions and distributions defined in the complement of a point, in one or several variables. We also consider functions defined in the complement of a hypersurface. We apply these results to the existence of distributional boundary values of harmonic and analytic functions.

Holomorphic functions in tubes which generalize the Hardy Hp functions are considered. In previous work the existence of strong l′ distributional boundary values for the holomorphic functions has been established. In this paper we show that if these boundary values are in the distribution space then the holomorphic functions can be recovered by the Cauchy and Poisson integrals of the boundary values.