maximum of two plurisubharmonic - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:26:14Z http://mathoverflow.net/feeds/question/82913 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/82913/maximum-of-two-plurisubharmonic maximum of two plurisubharmonic vu viet 2011-12-07T22:01:10Z 2012-08-05T16:19:16Z <p>Let $u,v$ be two plurisubharmonic functions in a domain $\Omega\subset \mathbb{C}^n.$ Then $w=\max$ {$u,v$} is plurisubharmonic. The support of $dd^c w$ is unclear in {$z: u(z)=v(z)$} </p> <p>How to estimate $dd^c w$ by $dd^c u$ and $dd^c v$? Generally, is there any information on $dd^c$ of a combination of $"max", "min", "+,-"$ operators of a family of plursisubharmonic functions?</p> http://mathoverflow.net/questions/82913/maximum-of-two-plurisubharmonic/83088#83088 Answer by Margaret Friedland for maximum of two plurisubharmonic Margaret Friedland 2011-12-09T23:18:52Z 2012-08-05T16:19:16Z <p>Under some additional assumptions one can prove that $dd^c \max (u,v) \geq \min (dd^cu,dd^cv)$ in $\Omega$. The proof seems long-winded for such a simple statement, but this is the best I can think of now. First of all, I will assume that $u$ and $v$ are continuous in $\Omega \subset \mathbb{C}^n$ and let $w:=\max(u,v)$. I will also use a criterion of positivity of currents of degree $(n-p,n-p), \quad 0 \leq p \leq n$ in $\Omega$. Let $L$ be a complex linear subspace of $\mathbb{C}^n$ of dimension $p$ and let $g \in \rm U(n)$ be such that $g(L)$ is defined by equations $u_{p+1}=...=u_n=0$. Define a $(p,p)$-form $\tau(L):= g^*\hat{\beta}$, where $\hat{\beta}=(i/2)du_1\wedge d\bar{u}_1\wedge...\wedge (i/2)du_p\wedge d\bar{u}_p$. This does not depend on the choice of orthonormal coordinates and gives a positive form. Now, Proposition 2.10 in: P. Lelong, L. Gruman: Entire functions of several complex variables, Springer-Verlag 1986 says the following: A $(n-p,n-p)$-current $T$ in $\Omega$ is positive if and only if for every linear subspace $L$ of dimension $p$ the current $T \wedge \tau(L)$ is a positive distribution (and hence a measure). Because of linearity of $dd^c$ and of the equality $dd^ch \wedge S = dd^c(hS)$, which holds when u is plurisubharmonic and locally integrable with respect to the mass of $S$, it is enough to prove that $dd^c w \wedge \tau(L) \geq \min(dd^cu,dd^cv) \wedge \tau(L)$ for every linear subspace $L$ of dimension $n-1$. Here is the proof, modeled on Proposition 2.8 in: Bedford, Eric; Taylor, B. A.: The Dirichlet problem for a complex Monge-Amp`{e}re equation. \emph{Invent. Math. 37} (1976), no. 1, 1-44. The authors of this paper study Monge-Ampere measures, not trace measures, but the argument goes through. I cannot resist reproducing it here in full, partly because of their cute application of the fact that an uncountable sum of positive real numbers is infinite (for the proof of it, see e.g. <a href="http://www.xamuel.com/uncountable-sums/" rel="nofollow">http://www.xamuel.com/uncountable-sums/</a>; there is also related discussion in <a href="http://math.stackexchange.com/questions/70194/does-uncountable-summation-with-a-finite-sum-ever-occur-in-mathematics" rel="nofollow">http://math.stackexchange.com/questions/70194/does-uncountable-summation-with-a-finite-sum-ever-occur-in-mathematics</a>). </p> <p>Claim: Let $\Omega,u,v,w$ be as above. If for every $L$ of complex dimension $n-1$ the measure $\mu_L := \min(dd^cu, dd^cv) \wedge \tau(L)$ is finite, then $dd^c w \wedge \tau(L)\geq \min(dd^cu, dd^cv)\wedge \tau(L)=\mu_L$. Proof: Consider first the case when $\mu_L({u=v})=0$. Let <code>$\Omega_1 = \{u&lt;v\}$,$\Omega_2=\{v&lt;u\}$</code>. Then $dd^cw \wedge \tau(L)=dd^cv \wedge \tau(L)\geq \mu_L$ in $\Omega_1$ and similarly $dd^cw \geq \min(dd^cu, dd^cv)$ in $\Omega_2$. And of course $dd^cw \wedge \tau(L)({u=v}) \geq 0 = \mu_L({u=v})$.\ Now let us consider $L$ such that $\mu_L({u=v})> 0$. Then replacing $v$ by $v+\varepsilon$ for small $\varepsilon$ we still have $\mu_L = \min(dd^cu, dd^c(v+\varepsilon)) \wedge \tau(L)$ and arguing as above we can prove that $dd^c \max(u,v+\varepsilon) \wedge \tau(L) \geq \mu_L$ for all $\varepsilon$ for which $\mu_L({z\in \mathbb{C}^n: u(z)=v(z)+\varepsilon })=0$. This happens for all but countably many $\varepsilon$ , which allows us to obtain the inequality in the general case by letting $\varepsilon \to 0$ in a way that avoids those countably many values. </p> <p>I do not know if and how the assumption of continuity can be relaxed. Bedford and Taylor give a counterexample of discontinuous plurisubharmonic functions for which the inequality $(dd^cw)^n \geq \min((dd^cu)^n,(dd^cv)^n)$ does not hold. If one wants to know which currents satisfy the assumption of finiteness of the trace measures, one should look at Fornaess, John Erik; Sibony, Nessim: Oka's inequality for currents and applications. \emph{Math. Ann. 301} (1995), no. 3, 399-419, where similar estimates were investigated.</p> http://mathoverflow.net/questions/82913/maximum-of-two-plurisubharmonic/103928#103928 Answer by Alexandre Eremenko for maximum of two plurisubharmonic Alexandre Eremenko 2012-08-04T10:05:14Z 2012-08-04T10:05:14Z <p>I am writing everything in dimension 1; $dd^c=\Delta$.</p> <p>You cannot estimate $\Delta\max( u,v)$ in terms of $\Delta u$ and $\Delta v$. Example $u(x,y)=x,\; v(x,y)=ax$.</p> <p>Relations between $\Delta$ of $u,v,$ their max and min were studied in the paper Eremenko, Fuglede, Sodin, On the Riesz charge of the lower envelope of delta-subharmonic functions. Potential Analysis, 1992, vol. 1, N 2, 191-204. Also available as No. 58 on my web site, <a href="http://www.math.purdue.edu/~eremenko/papers.html" rel="nofollow">http://www.math.purdue.edu/~eremenko/papers.html</a></p> <p>There are references to other results on the topic in this paper. See also my survey talk No. 98 on the same web page for applications of such relations.</p>