Contemporary Mathematics
The Schwarz Kernel in Clifford Analysis
Lev Aizenberg and Nikolai Tarkhanov
Abstract. For a Dirac type operator, we study the problem of characterising
solutions of the homogeneous system which take their values in a cone. This
problem goes back at least as far as the classical formula of Riesz-Herglotz
in function theory. We give a proof of this formula in the context of Dirac
operators.
1. Introduction
It is well-known that the nonnegative harmonic functions u in an open ball
B(0, R) in
Rn
have very particular properties. They are integrable over the entire
ball, can be represented by the Poisson-Stieltjes integral of a nonnegative mass
on the boundary ∂B(0, R), and satisfy Harnack’s inequality u(x) C(r, R)u(0)
for all x in a smaller ball B(0, r), with C(r, R) a constant independent of u. As
but one consequence of these properties we mention the classical Liouville theorem
which states that any nonnegative harmonic function in Rn is constant. The paper
[KE74] gives a profound exposition of these results in the context of solutions of
elliptic systems.
When dealing with solutions of elliptic systems which take their values in Ck,
one has to appropriately interpret the notion of “nonnegative” solutions. To this
end, we observe that the property of being nonnegative for a function y = u(x) is
equivalent to the fact that the values of u belong to the half-axis {y 0} which
is a cone with vertex at the origin. In [KE74], by “nonnegative” vector-valued
functions are meant those functions which take their values in a closed cone K with
vertex at the origin. The cone is assumed to have the only common point y = 0 with
a closed half-space of
Ck.
The study of nonnegative solutions to elliptic systems in
[KE74] is based on estimates in the
L1
-norm which answers better the essence of
the problem.
Holomorphic functions with nonnegative real part can be thought of as solutions
of the Cauchy-Riemann system which take their values in the closed upper half-
plane. A classical result here is due to Riesz and Herglotz who gave an explicit
2010 Mathematics Subject Classification. Primary 42C15; Secondary 41A44, 32A05.
Key words and phrases. Holomorphic functions with nonnegative real part, nonnegative har-
monic functions, solutions of elliptic systems with values in a cone, Dirac operators.
The first author gratefully acknowledges the financial support of the Deutsche Forschungs-
gemeinschaft.
2011 L. Aizenberg, N. Tarkhanov
1
Contemporary Mathematics
Volume 553, 2011
cc 2011 L. Aizenberg, N. Tarkhanov
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