Beurling's criterion and extremal metrics for Fuglede modulus (Q2860890)

In this paper \((X, \mathcal{M},m)\) is a measure space. A measure system \(E\) is a collection of measures on \(\mathcal{M}\); a Borel function \(\phi: X \to [0, \infty]\) is called a metric and is said to be admissible for \(E\) if \(\int \phi d \mu \geq 1\) for any \(\mu \in E\). For \( 0<p < \infty\), the ``\(p\)-module'' of \(E\) is NEWLINENEWLINENEWLINE\[NEWLINE\text{mod}_{p} E= \inf \bigg\{ \int \phi^{p} d m : \phi \text{ admissible in } E \bigg\}.NEWLINE\]NEWLINE NEWLINENEWLINEIf for some admissible metric \(\phi\), \(\text{mod}_{p} E= \int \phi^{p} d m\), then \(\phi\) is said to be extremal for the \(p\)-module of \(E\). NEWLINENEWLINENEWLINENEWLINE In this paper the author extends some results of Beurling about the existence and uniqueness of extremal metrics. The main results are: NEWLINENEWLINENEWLINENEWLINE I. Suppose \(1 \leq p< \infty\), \(E\) a measure system, \(\phi\) an admissible metric for \(E\) and \(\phi \in L^{p}(m)\). Then \(\phi\) is extremal for the \(p\)-module of \( E\) iff there exists a measure system \(F\) satisfying the following three conditions:NEWLINENEWLINENEWLINENEWLINE (i) \(\text{mod}_{p} E \cup F =\text{mod}_{p} E\),NEWLINENEWLINENEWLINENEWLINE (ii) \( \int \phi d \nu=1 \; \forall \nu \in F\),NEWLINENEWLINENEWLINENEWLINE (iii) Case \(p>1\): if \(f: X \to [ - \infty, \infty]\) with \( f \in L^{p}(m) \text{ and } \int f d \nu \geq 0 \; \forall \nu \in F\), then \( \int\phi^{p-1} d m \geq 0\). NEWLINENEWLINENEWLINENEWLINE Case \(p=1\): if \(f: X \to [ - \infty, \infty]\) with \(f \in L^{1}(m)\) has the properties that \(\int f d \nu \geq 0 \; \forall \nu \in F\) and \(\phi(x)=0 \Rightarrow f(x) \geq 0\), then \(\int f d m \geq 0\). NEWLINENEWLINENEWLINENEWLINE II. If \(\phi: X \to [0, \infty] \) is a metric and \( \phi < \infty\) \(m\)-a.e., then \(\phi\) is extremal for the \(p\)-module of \(E_{\phi}= \{ \mu \text{ defined on } \mathcal{M} : \int \phi d \mu \geq 1 \}\) for all \( p\), \(0 < p < \infty \). A similar result for curve families in \(\mathbb R^{n}\) is also proved.NEWLINENEWLINESeveral additional results and corollaries are also obtained.