Distributional Boundary Values Research Articles

Distributional Boundary Values of Analytic Functions

Let D be a connected bounded domain in R2, S be its boundary which is closed, connected and smooth. Let Φ(z) = 1 2πi R S ϕ(s)ds s−z , ϕ ∈ X, z = x + iy, X is a Banach space of linear bounded functionals on Hμ, a Banach space of distributions, and Hμ is the Banach space of Hoelder-continuous functions on S with the usual norm. As X one can use also the space Hoelder continuous of bounded linear functionals on the Sobolev space Hℓ on S. Distributional boundary values of Φ(z) on S are studied in detail. The function Φ(t), t ∈ S, is defined in a new way. Necessary and sufficient conditions are given for ϕ ∈ X to be a boundary value of an analytic in D function. The Cauchy formula is generalized to the case when the boundary values of an analytic function in D are tempered distributions. The Sokhotsky-Plemelj formulas are derived for ϕ ∈ X.

Vector-Valued Analytic Functions Having Vector-Valued Tempered Distributions as Boundary Values

Vector-valued analytic functions in Cn, which are known to have vector-valued tempered distributional boundary values, are shown to be in the Hardy space Hp,1≤p<2, if the boundary value is in the vector-valued Lp,1≤p<2, functions. The analysis of this paper extends the analysis of a previous paper that considered the cases for 2≤p≤∞. Thus, with the addition of the results of this paper, the considered problems are proved for all p,1≤p≤∞.

Representations of generalized Hardy functions in Beurling’s tempered distributions

Let B be a proper open subset in $${{\mathbb {R}}}^N$$ and C be a regular cone in $${{\mathbb {R}}}^N$$ . On our previous paper of Acta Scientiarum Mathematicarum 85, 595–611 (2019), we have defined the space of generalized Hardy functions, $$G_{\omega ^*,A}^p(T^B)$$ , $$1< p \le 2,$$ and $$A \ge 0$$ , and have shown that the functions in $$G_{\omega ^*,A}^p(T^B)$$ have distributional boundary values in the weak topology of Beurling tempered distributions, $${\mathcal {S}}_{(\omega )}^\prime $$ . In this paper we show that if the distributional boundary values are convolutors in Beurling ultradistributions of $$L_2$$ -growth, then the functions in $$G_{\omega ^*,0}^p(T^C)$$ , $$1< p \le 2,$$ can be represented as Cauchy and Poisson integral of the boundary values in $${\mathcal {S}}_{(\omega )}^\prime $$ .

Crofton formulas in pseudo-Riemannian space forms

Crofton formulas on simply connected Riemannian space forms allow the volumes, or more generally the Lipschitz–Killing curvature integrals of a submanifold with corners, to be computed by integrating the Euler characteristic of its intersection with all geodesic submanifolds. We develop a framework of Crofton formulas with distributions replacing measures, which has in its core Alesker's Radon transform on valuations. We then apply this framework, and our recent Hadwiger-type classification, to compute explicit Crofton formulas for all isometry-invariant valuations on all pseudospheres, pseudo-Euclidean and pseudohyperbolic spaces. We find that, in essence, a single measure which depends analytically on the metric, gives rise to all those Crofton formulas through its distributional boundary values at parts of the boundary corresponding to the different indefinite signatures. In particular, the Crofton formulas we obtain are formally independent of signature.

Boundary values of zero solutions of hypoelliptic differential operators in ultradistribution spaces

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator P(D) = P(D_x, D_t) on {mathbb {R}}^{d+1}. Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of several results of Langenbruch (Manuscripta Math. 26:17–35, 1978/79) about distributional boundary values of zero solutions of P(D).

Vector valued Hardy spaces related to analytic functions having distributional boundary values

ABSTRACTThe Hardy space of vector valued analytic functions in tube domains in and with values in Banach space are defined. Analytic functions in tube domains in with values in Hilbert space which have functions with values in Hilbert space as distributional boundary value are proved to be in . A Poisson integral representation for such vector valued analytic functions is obtained.

The Cauchy integral formula and distributional integration

ABSTRACTWe study the Cauchy representation formula for analytic functions on the unit disc whose pointwise boundary value function is distributionally integrable. We prove that the formula holds when the distributional boundary values exist, and give examples that show that it may not be true when that is not the case. We also prove a maximum principle for pointwise boundary values valid for functions with distributional boundary values.

Distributional boundary values of holomorphic functions on product domains

We show that holomorphic functions of polynomial growth on domains with corners have distributional boundary values in an appropriate sense, provided the corners are generic CR manifolds. We also prove an analog of the Bochner–Hartogs theorem for these boundary values for the simplest such domains, the product domains.

Distributional boundary values of analytic functions and positive definite distributions

We propose necessary and sufficient conditions for a distribution (generalized function) fof several variables to be positive definite. For this purpose, certain analytic extensions of f to tubular domains in complex space Cn are studied. The main result is given in terms of the Cauchy transform and completely monotonic functions.

Generalized axially symmetric potentials with distributional boundary values

We study a counterpart of the classical Poisson integral for a family of weighted Laplace differential equations in Euclidean half space, solutions of which are known as generalized axially symmetric potentials. These potentials appear naturally in the study of hyperbolic Brownian motion with drift. We determine the optimal class of tempered distributions which by means of the so-called S′-convolution can be extended to generalized axially symmetric potentials. In the process, the associated Dirichlet boundary value problem is solved, and we obtain sharp order relations for the asymptotic growth of these extensions.

Representation of Distributions by Harmonic and Monogenic Potentials in Euclidean Space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space \({\mathbb{R}^{m+1}_+}\) was recently constructed, including a higher dimensional analogue of the logarithmic function in the complex plane, and their distributional boundary values were computed. In this paper we determine these potentials in lower half–space \({\mathbb{R}^{m+1}_-}\) and investigate whether they can be extended through the boundary \({\mathbb{R}^m}\) . This is a stepping stone to the representation of a doubly infinite sequence of distributions in \({\mathbb{R}^m}\) , consisting of positive and negative integer powers of the Dirac and the Hilbert–Dirac operator, as the jump across \({\mathbb{R}^m}\) of monogenic functions in the upper and lower half–spaces, in this way providing a sequence of interesting examples of Clifford hyperfunctions.

Characteristic functions as distributional boundary values of analytic functions in tube domains

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 \( {{\mathbb{R}}^n} \).

Harmonic and monogenic potentials in low dimensional Euclidean half-space

In the framework of Clifford analysis, a chain of harmonic and monogenic potentials in the upper half of Euclidean space was constructed recently, including a higher dimensional analogue of the logarithmic function in the complex plane. Their distributional limits at the boundary were also determined. In this paper, the potentials and their distributional boundary values are calculated in dimensions 3 and 4, dimensions for which the expressions in general dimension break down. Copyright © 2013 John Wiley & Sons, Ltd.

On singular nonlinear distributional and impulsive initial and boundary value problems

To derive existence and comparison results for extremal solutions of nonlinear singular distributional initial value problems and boundary value problems. Fixed point results in ordered function spaces and recently introduced concepts of regulated and continuous primitive integrals of distributions. Maple programming is used to determine solutions of examples. New existence results are derived for the smallest and greatest solutions of considered problems. Novel results are derived for the dependence of solutions on the data. The obtained results are applied to impulsive differential equations. Concrete examples are presented and solved to illustrate the obtained results. MSC: 26A24, 26A39, 26A48, 34A12, 34A36, 37A37, 39B12, 39B22, 47B38, 47J25, 47H07, 47H10, 58D25

Littlewood–Paley decompositions related to symmetric cones and Bergman projections in tube domains

Starting from a Whitney decomposition of a symmetric cone $\Omega$, analogous to the dyadic partition $[2^j, 2^{ j + 1})$ of the positive real line, in this paper we develop an adapted Littlewood?Paley theory for functions with spectrum in $\Omega$. In particular, we define a natural class of Besov spaces of such functions, $B^{p, q}_\nu$, where the role of the usual derivation is now played by the generalized wave operator of the cone $\Delta(\frac{\partial}{\partial x})$. We show that $B^{p, q}_\nu$ consists precisely of the distributional boundary values of holomorphic functions in the Bergman space $A^{p, q}_\nu (T_\Omega)$, at least in a 'good range' of indices $1 \leq q 2$. Moreover, we show the equivalence of this problem with the boundedness of Bergman projectors $P_\nu \colon L^{p, q}_\nu \to A^{p, q}_\nu$, for which our result implies a positive answer when $q_{\nu, p}' < q < q_{\nu, p}$. This extends, to general cones, previous work of the authors on the light-cone. Finally, we focus on light-cones and introduce new necessary and sufficient conditions for our conjecture to hold in terms of inequalities related to the cone multiplier problem. In particular, using recent work by Laba and Wolff, we establish the validity of our conjecture for light-cones when $p$ is sufficiently large.

Boundary values of integral operators

AbstractConsider a differential form u in the image of an integral operator K on a smooth domain D in a Riemannian manifold, i.e. u(y) = Kf(y) = ∫x∈D f(x) ∧ K(x, y) for y ∈ D. If the kernel K of the integral operator is sufficiently regular and defined also for y in the boundary bD of D, one may define two different kinds of boundary values of u on bD. Firstly, u may have boundary values in the sense of distributions, i.e. boundary values satisfying a Stokes' formula in a suitably weak sense. Secondly, one can simply restrict (pull back) y in the kernel K(x, y) to the boundary of D, then integrate with respect to x in the above formula. It is interesting to know under which hypotheses on f and K both types of boundary values agree, because the boundary values defined by restricting the kernel can often be estimated by direct methods, whereas the abstractly given distributional boundary values are less tractable but analytically interesting objects linked to u. We will give counter‐examples to show that even in quite regular situations the two notions do not necessarily agree. Then we study conditions implying equality. We also mention some interesting applications, e.g. generalizations of Stokes' formula and applications in the theory of integral representations in several complex variables. (© 2003 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Regularization of distributions

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.

Classical Theta Functions and Quantum Tori

The Schwartz kernel of the multiplication operation on a quantum torus is shown to be the distributional boundary value of a classical multivariate theta function. The kernel satisfies a Schrödinger equation in which the role of time is played by the deformation parameter ℏ and the role of the hamiltonian by a Poisson structure. At least in some special cases, the kernel can be written as a sum of products of single variable theta functions.

Generalized HPspaces on tubes

Spaces of holomorphic functions on tubes in which generalise the Hardy Hpspaces on tubes are defined and studied using the theory of ultradistributions. The holomorphic functions are of arbitrary growths both at the origin and at infinity. Representations of these holomorphic functions in terms of Fourier-Laplace transforms, Cauchy and Poisson integrals are given. Distributional boundary value results are obtained. Converse results are given. Various results on ultradistributions are also obtained which are relevant to the present investigation.

Boundary values of generalizations of Hρ functions in tubes

In previous work we have defined and studied holomorphic functions in tubes which generalize the Hardy Hρ (TB) spaces; the base B of the tube TB is a proper open subset of If B=C, an open convex cone in and 1 < ρ ⩽ 2, we have proved that our new functions have distributional boundary values in the strong topology of p as . In this paper we continue to study these new functions and extend the existence of boundary values to the cases 2 < ρ < ∞ We prove that our functions obtain boundary values in the strong topology of P as , where C is a polygonal cone or a regular cone, for the cases 2 < ρ < ∞.