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

1

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

1