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Title: Fundamentals of a generalized Wiman-Valiron theory for solutions to the Dirac-Hodge equation on upper half-space of $R^{n+1}$
Authors: Constales, Denis
De Almeida, Regina
Krausshar, Rolf Sören # ×
Issue Date: 2009
Publisher: J. Springer
Series Title: Mathematische Zeitschrift issue:submitted
Abstract: In this paper we study questions related to the growth behavior of paravector valued functions $f= f_0+\sum_{i=1}^n e_i f_i$ that satisfy the Dirac-Hodge equation $x_n (\sum_{i=0}^n \frac{\partial }{\partial x_i} e_i)f + (n-1) f_n = 0$ on the upper half-space of $\mathbb{R}^{n+1}$. The Dirac-Hodge operator provides a linearization of the Laplace-Beltrami operator on the upper half-space. First we prove a Cauchy type estimate for solutions to the Dirac-Hodge equation. This gives us a lower bound estimate of the maximum modulus of such a solution in any ball that is contained in the upper half-space.

By means of the Fourier transform we
introduce lower and upper growth orders and generalizations of the
maximum term and central index in the context of hypermonogenic
functions on the upper half-space. In this paper we
put the main focus on the study of $n$-fold periodic solutions to
the Dirac-Hodge equation for which one has a discrete Fourier
series representation. For every arbitrary
growth order we construct non-trivial examples of $n$-fold
periodic hypermonogenic functions. As main result we establish a
generalization of the Valiron inequality for $n$-fold periodic
solutions to the Dirac-Hodge equation on the upper half-space.
This provides us with an explicit upper bound estimate
of the maximum modulus $M(x_n,f)$ in terms of the maximum term and
the central index associated to the Fourier series of $f$.
ISSN: 0025-5874
Publication status: submitted
KU Leuven publication type: IT
Appears in Collections:Analysis Section
× corresponding author
# (joint) last author

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