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Nonlinear harmonic measures on trees. (English) Zbl 1033.31004

The authors investigate averaging operators and nonlinear harmonic measures on a \(\kappa\)-regular forward branching tree \(T\). Central to their study is the observation, that such measures lack many nice properties of their counterparts of classical analysis.
An averaging operator in the sense of V. Alvarez et al. [ Mich. Math. J. 49, 47–64 (2001; Zbl 1006.31006)] is a continuous function \(F:\mathbb{R}^\kappa \to \mathbb{R}\) which satisfies (i) \(F(\underline 0) = 0, F(\underline 1) = 1\) (underlined letters \(\underline x\) denote vectors from \(\mathbb{R}^\kappa\)); (ii) \(F(t\underline x) = tF(\underline x)\) and \(F(\underline t + \underline x) = t+F(\underline x)\) for all \(t\in\mathbb{R}\); (iii) \(F(\underline x) < | \underline x| _\infty\) unless all coordinates of \(\underline x\) are equal; (iv) \(F\) is nondecreasing w.r.t. each variable. \(F\) satisfies the strong maximum principle if \(F(\underline x) = 0,\;\underline x \neq 0\) implies a sign change in the entries of \(\underline x\). If \(\nabla F(\underline 0)\) exists, \(F\) is necessarily a convex combination and, if \(F\) is also invariant under permutations of its entries, \(F\) is just the arithmetic mean. A function \(u\) on \(T\) is \(F\)-harmonic, if \(F(u(\varphi,1),\ldots, u(\varphi,\kappa)) = u(\varphi)\) for all \(\varphi\in T\), \((\varphi,j)\) denotes the \(j\)th successor of \(\varphi\). Sub- and superharmonic functions are defined analogously. The (\(F\)-)harmonic measure function is given by \[ \omega_F(E) := \omega_F(\varphi,E) := \inf_{u\in U} u(\varphi) \] where \(U\) is the collection of all \(F\)-superharmonic \(u\) on \(T\) such that \(\liminf_{\varphi\to b}u(\varphi) \geq 1_E(b)\) \(\forall b\in\partial T\).
Using rather elementary methods, the authors show that for every such averaging operator \(F\) and every compact \(E\subset T\) with \(\dim E < d(\kappa, F)\) (this is the critical dimension of \(F\)) one has \(\omega_F(E) = 0\). It is unclear if this result can be pushed through to all Borel sets, but the fact that \(\omega_F\) is (even for a permutation-invariant \(F\) other than the arithmetic mean) not a Choquet capacity, indicates that this is quite a profound question. Moreover, for \(F\) there is some \(E\subset\partial T\) with \(\omega_F(E)=0\), \(\omega_F(E^c) = 1\) but yet \(\dim E^c \leq d(\kappa, F)\). Even worse, for a permutation invariant \(F\) (not equalling the arithmetic mean) one can find congruent sets \(E_1, \ldots, E_\kappa\) with \(\bigcup_j E_j = \partial T\) and \(\omega_F(E_j) = 0\) for all \(j\); further, the union of two \(\omega_F\)-null sets may have strictly positive \(\omega_F\)-measure.
A nonlinear monotone elliptic operator is a map \(A:\mathbb{R}^\kappa\to\mathbb{R}^\kappa\) satisfying \(\langle A\underline x, \underline x\rangle \geq \alpha\| \underline x\| _p^p\), \(\| A \underline x\| _q \leq \beta \| \underline x\| _p^{p-1}\) (\(q\) is conjugate to \(p\)), \(\langle A\underline x- A\underline y, \underline x - \underline y\rangle > 0\) if \(\underline x\neq \underline y\) and \(A(t\underline x) = t| t| ^{p-2}A\underline x\) for all \(t\in\mathbb{R}\). If \(\langle A(\underline X - \underline t),\underline 1\rangle = 0\), the monotone operator \(A\) defines an averaging operator \(F_A\) with \(F_A(\underline x) = t\). In this setting it is shown that \(F_A\) satisfies the strong maximum principle if the ratio of the ellipticity constants \(\alpha/\beta\) is close to one; in this case, the ratio of the critical dimensions \(d(\kappa, A)\) for the operator \(A\) and for the \(p\)-Laplacian \(d(\kappa,p)\) also tend to 1.

MSC:

31C05 Harmonic, subharmonic, superharmonic functions on other spaces
31C45 Other generalizations (nonlinear potential theory, etc.)
31C20 Discrete potential theory
28A12 Contents, measures, outer measures, capacities

Citations:

Zbl 1006.31006
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