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Domination and representation theorems for harmonic functions and temperatures. (English) Zbl 0841.31007
Let \(D= \{(x, t): x\in \mathbb{R}^n\), \(t>0\}\). If \(x\in \mathbb{R}^n\) and \(r>0\), let \(\gamma (x,r)= \{(y, s)\in D:|(y,s)- (x,0) |=r\}\) and \({\mathcal M} (g, x, r)= r^{- n-2} \int_{\gamma (x,r)} tg(x,t)\) for suitable functions \(g\), where the integral is with respect to surface measure. Let \(\mu\) be a signed measure on \(\mathbb{R}^n\) whose Poisson integral, \(P\mu\), is harmonic on \(D\), let \(c\in \mathbb{R}\) and \(u(x,t)= P\mu (x,t)+ ct\). Also, let \(v= P\nu+ bt\) be a non-negative harmonic function on \(D\). Define \(f(x)= \lim_{r\to 0} {\mathcal M}(u, x,r)/ {\mathcal M} (v, x,r)\) whenever the limit exists, and let \(Z^\pm= \{x: f(x)= \pm \infty\}\). Theorem 1 asserts that \(f\) is defined and finite \(\nu\)-a.e. and that there is a \(\nu\)-singular measure \(\sigma\), whose positive and negative variations are concentrated on \(Z^+\) and \(Z^-\) respectively, such that \(d\mu= f d\nu+ d\sigma\). The result is then shown to yield short proofs and refinements of majorization and representation theorems of D. H. Armitage [Ann. Acad. Sci. Fenn., Ser. A I Math. 6, 161-171 (1981; Zbl 0472.31003)]. Analogous results for solutions of the heat equation are briefly indicated.
MSC:
31B25 Boundary behavior of harmonic functions in higher dimensions
35K05 Heat equation
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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