Volume 2,  Issue 2, 2001

Article 16

SUBHARMONIC FUNCTIONS AND THEIR RIESZ MEASURE

RAPHAELE SUPPER

UNIVERSITÉ LOUIS PASTEUR,
UFR DE MATHÉMATIQUE ET INFORMATIQUE, 
URA CNRS 001, 7 RUE RENÉ DESCARTES,
F- 67 084 STRASBOURG CEDEX, FRANCE
 E-Mail: supper@math.u-strasbg.fr

 

Received 30 August, 2000; accepted 22 January, 2001.
Communicated by: A. Fiorenza

ABSTRACT.   

For subharmonic functions $u$ in ${ I\!\! R}^N$, of Riesz measure $\mu$, the growth of the function $s \mapsto \mu (s) =\int_{\vert \zeta \vert \leq s } d\mu (\zeta) $ ($s\geq 0$) is described and compared with the growth of $u$. It is also shown that, if $ \int_{{ I\!\! R}^N} u^+ (x) [-\varphi ' (\vert x\vert ^2)] dx <+\infty$ for some decreasing $ C ^1$ function $\varphi \geq 0$, then $ \int_{{ I\!\! R}^N} {1\over{\vert \zeta \vert ^2 }} \varphi(\vert \zeta \vert ^2 +1) d\mu (\zeta ) <+\infty .$ Given two subharmonic functions $u_1$ and $u_2$, of Riesz measures $\mu_1$ and $\mu_2$, with a growth like $u_i (x)\leq A+ B \vert x\vert ^\gamma$ $\forall x \in { I\!\! R}^N$ ($i=1,2$), it is proved that $\mu_1 +\mu_2$ is not necessarily the Riesz measure of any subharmonic function $u$ with such a growth as $u (x)\leq A'+ B' \vert x\vert ^\gamma$ $\forall x \in { I\!\! R}^N$ (here $A>0$, $A'>0$ and $0<B'<2B$).

Key words:
Subharmonic functions, Order of growth, Riesz measure.

2000 Mathematics Subject Classification:
31A05, 31B05, 26D15, 28A75.

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Power-Monotone Sequences and Fourier Series with Positive Coefficients
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On Some Fundamental Integral Inequalities and their Discrete Analogues
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Subharmonic Functions and their Riesz Measure
Raphaele Supper

A Priori Estimate for a System of Differential Operators
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Improved Inclusion-Exclusion Inequalities for Simplex and Orthant Arrangements
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Monotonic Refinements of a Ky Fan Inequality
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Sub-super Solutions and the Existence of Extremal Solutions in Noncoercive Variational Inequalities
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Refinements of Carleman's Inequality
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Inequalities related to the Chebychev Functional Involving Integrals Over Different Intervals
I. Budimir, P. Cerone, and  J. Pecaric

Some Distortion Inequalities Associated with the Fractional Derivatives of Analytic and Univalent Functions
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Necessary and Sufficient Condition for Existence and Uniqueness of the Solution of Cauchy Problem for Holomorphic Fuchsian Operators
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Bounds for Entropy and Divergence for Distributions over a Two-Element Set
Flemming Topsoe

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