riesz measure
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2020 ◽  
Vol 54 (2) ◽  
pp. 188-192
Author(s):  
M.V. Zabolotskyi

For $\delta$-subharmonic in $\mathbb{R}^m$, $m\geq2$, function $u=u_1-u_2$ of finite positiveorder we found the asymptotical representation of the form\[u(x)=-I(x,u_1)+I(x,u_2) +O\left(V(|x|)\right),\ x\to\infty,\]where $I(x,u_i)=\int\limits_{|a-x|\leq|x|}K(x,a)d\mu_i(a)$, $K(x,a)=\ln\frac{|x|}{|x-a|}$ for $m=2$,$K(x,a)=|x-a|^{2-m}-|x|^{2-m}$ for $m\geq3,$$\mu_i$ is a Riesz measure of the subharmonic function $u_i,$ $V(r)=r^{\rho(r)},$ $\rho(r)$ is a proximate order of $u$.The obtained result generalizes one theorem of I.F. Krasichkov for entire functions.



2004 ◽  
Vol 56 (2) ◽  
pp. 225-245 ◽  
Author(s):  
Gordon Blower ◽  
Thomas Ransford

AbstractThe norm on a Banach space gives rise to a subharmonic function on the complex plane for which the distributional Laplacian gives a Riesz measure. This measure is calculated explicitly here for LebesgueLpspaces and the von Neumann-Schatten trace ideals. Banach spaces that areq-uniformly PL-convex in the sense of Davis, Garling and Tomczak-Jaegermann are characterized in terms of the mass distribution of this measure. This gives a new proof that the trace idealscpare 2-uniformly PL-convex for 1 ≤p≤ 2.



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