Harmonic functions in non-locally convex spaces
Let X be a topological vector space which is p normable for some \(0<p\leq 1\) i.e. X is a complex complete metrisable topological vector space whose topology is given by a quasi-norm such that \(\| x+y\|^ p\leq \| x\|^ p+\| y\|^ p.\) One defines a harmonic function h on an open set \(\Omega\) of \({\mathbb{C}}\) as the sum \(h=f_ 1(z)+f_ 2(\bar z)\) where \(f_ 1\) and \(f_ 2\) are analytic functions from \(\Omega\) to X. It is well known that in the case of a Banach space X, the maximum modulus principle for X valued analytic functions holds. In particular let f be a harmonic function in the open unit disk D of \({\mathbb{C}}\) which can be continuously extended to the closed unit disk and such that f vanishes on the boundary of D then f is identically zero in D. This property is no longer true for non locally convex topological spaces and an explicit example is given by the Poisson kernel with values in \(L^ p\), \(0<p<1.\) Hence for a p normable space one can ask at what rate one can have f(z)\(\to 0\) when \(| z| \to 1\) for an non trivial harmonic function h. The main result of this paper is the following: Let \(\beta >1/p-1\). If one has \(\| h(z)\| =O(1-| z|)^{\beta}\) then \(h\equiv 0\) on D. Assume that \(<p<1\) and let us set \(\beta =1/p-1\). If one has \(\| h(z)\| =o(1-| z|)^{\beta}\) then \(h\equiv 0\) on D.
scientific article; zbMATH DE number 4109102
scientific article