An analytic Radon-Nikodym property related to systems of vector-valued conjugate harmonic functions and Clifford Analysis (original) (raw)

On Bergman Spaces in Clifford Analysis

In this paper, we give the definition of Bergman spaces in Clifford Analysis, then we give some basic and important properties of functions belonging to the Clifford Bergman spaces . Moreover, we give properties of uniformly discrete sequences pertaining to the Cliffordean Bergman spaces of hyperholoorphic functions in the unit ball. Our results in this paper generalize and improve a lot of previous results.

Boundary values of functions in vector-valued Hardy spaces and geometry on Banach spaces

Journal of Functional Analysis, 1988

The spaces of boundary values of vector-valued functions in Hardy spaces defined by either holomorphic functions on the disk or harmonic functions with maximal function in LJ' are characterized in terms of vector-valued measures of bounded p-variation. We extend to the case p = 1 a characterization of the Radon Nikodym property based on the existence of limits at the boundary for harmonic functions with maximal function in L'. In the case 0 <p< 1 we find the UMD property as the necessary and sufficient condition to make the spaces defined by maximal function and hy conjugate Poisson kernel coincide. '('1 1988 Academc Press, Inc. In this paper we are concerned with Hardy spaces of vector-valued functions on the disk. Our main objectives are: To extend several definitions of these spaces to the vector-valued setting, to study their relationships depending on the geometry of the Banach space, and to find representations for the boundary values of functions in these different Hardy spaces when we do not require any condition on the Banach space. The paper is divided into three sections. In the first one we show that the Hardy space of B-valued holomorphic functions H$(D) is isometric (via Poisson integral) to certain space of B-valued measures, the so-called measures of bounded p-variation. With this result we can regard the analytic RadonNikodym property in an equivalent way, which allows us to give another formulation of this property. The second section is devoted to solving the same question but for B-valued harmonic functions whose maximal function belongs to Lp (1 < p < co). We find now the Radon Nikodym property as the right condition to make the classical result remain * Supported by the C.

On conjugate harmonic functions in Euclidean space

Mathematical Methods in the Applied Sciences, 2002

In this paper we consider the problem of constructing in domains Ω of R m+1 with a specific geometric property, a conjugate harmonic V to a given harmonic function U , as a direct generalization of the complex plane case. This construction is carried out in the framework of Clifford analysis which focusses on so-called monogenic functions, i.e. null solutions of the Dirac operator. An explicit formula of the associated monogenic function F = U + e 0 V in terms of a harmonic potential is constructed and the interconnection with the Stein-Weiss notion of conjugate harmonicity will be shown.

An abstract Radon-Nikodym theorem

Journal of Mathematical Analysis and Applications, 1973

Introduction. Let J be a <r-field of subsets of an abstract set M and let m(e) be a non-negative measure function defined on J. The classical Radon-Nikodym theorem [17, p. 36](1) states that, if M is the union of a countable number of sets of finite measure, then a necessary and sufficient condition for a completely additive real function R(e), defined over J, to be a Lebesgue integral (with respect to m(e)) is that R(e) be absolutely continuous relative to m(e). Our purpose is to extend this theorem to functions with values in an arbitrary Banach space and apply the resulting theorem to obtain an integral representation for the general bounded linear transformation on the space of summable functions to an arbitrary Banach space. A number of writers [4, 6, 7, 8, 11, 12, 13, 14] have obtained similar extensions; however they have all imposed restrictions either on the Banach space or on the completely additive functions considered. The theorem proved here is free of all such restrictions. It is evident that any such generalization of the Radon-Nikodym theorem will involve a corresponding generalization of the Lebesgue integral, of which there are many. A variation of an integral studied in detail by B. J. Pettis(2) will be used here. A point function x(p) defined on ¥ to a Banach space X is said to be Pettis integrable [12] provided there exists a function X(e) on J to Ï such that, for each element x of the space 3-adjoint to ï and each element e of J, the function x(x(p)) is Lebesgue integrable on the set e to the value x(X(e)). Whenever X(e) exists, it is completely additive and absolutely continuous relative to m(e). On the other hand, Pettis [12, p. 303] gave an example of a completely additive function which is absolutely continuous but is not an integral in his sense. This shows that the ordinary Pettis integral cannot appear in a general Radon-Nikodym theorem. However, without changing essentially the definition or general properties of the integral, we can enlarge the class of functions admissable for integration (so that it contains certain functions other than point functions) and thus obtain an integral which will serve our purposes. The class of functions which we will admit for integration consists of all multivalued set functions x(e) defined for elements of J having finite, nonzero Except for §5, the contents of this paper were presented to the Society, September 12, 1943. The results in §5 were presented February 27, 1944, under the title Representation of linear transformations on summable functions.

Decomposition of first order Lipschitz functions by Clifford algebra-valued harmonic functions

arXiv (Cornell University), 2023

In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.

Almansi-type theorems in Clifford analysis

Mathematical Methods in the Applied Sciences, 2002

In this paper, we consider functions deÿned in a star-like domain ⊂ R n with values in the Cli ord algebra C' 0; n which are polymonogenic with respect to the (left) Dirac operator D = n j=1 e j @=@x j , i.e. they belong to the kernel of D k. We prove that any polymonogenic function f has a decomposition of the form f = f 1 + xf 2 + • • • + x k−1 f k ; where x = x 1 e 1 + • • • + xnen and f j ; j = 1; : : : ; k; are monogenic functions. This generalizes the classical Almansi theorem for polyharmonic functions as well as the Fischer decomposition of polynomials. Similar results are obtained for the powers of weighted Dirac operators of the formD = |x| − xD; ∈ R\{0}.

Biregular extendability via isotonic Clifford analysis

Mathematical Methods in the Applied Sciences, 2009

We use the so-called isotonic functions to obtain extension theorems in the framework of biregular functions of Clifford analysis. In this context we also prove the Plemelj-Sokhotski formulae for the Bochner-Martinelli integral and an expression for the square of its singular version.

A Generalization of Wiman and Valiron’s theory to the Clifford analysis setting

Cubo, 2009

CUBO A Mathematical Journal Vol. 11, No 01,(1–20). March 2009 A Generalization of Wiman and Valiron's theory to the Clifford analysis setting D. Constales 1 Department of Mathematical Analysis, Ghent University, Building S-22, Galglaan 2, B-9000 Ghent, Belgium. email: dc@ cage. UGent. be R. De Almeida 2 Departamento de Matemática, Universidade de Trás-os-Montes e Alto Douro, P-5000-911 Vila Real, Portugal. email: ralmeida@ utad. pt and RS Kraußhar 3 Department of Mathematics, Section of Analysis, Katholieke Universiteit Leuven, ...

A Note on Radon-Nikodym Theorem for Operator Valued Measures and Its Applications

Communications of the Korean Mathematical Society

In this note we present sufficient conditions for the existence of Radon-Nikodym derivatives (RND) of operator valued measures with respect to scalar measures. The RND is characterized by the Bochner integral in the strong operator topology of a strongly measurable operator valued function with respect to a nonnegative finite measure. Using this result we also obtain a characterization of compact sets in the space of operator valued measures. An extension of this result is also given using the theory of Pettis integral. These results have interesting applications in the study of evolution equations on Banach spaces driven by operator valued measures as structural controls.