×

The Hölder duality for harmonic functions. (English) Zbl 0651.46035

Summary: It is proved that if D is a bounded domain with smooth boundary in \({\mathbb{R}}^ n,\) then the space of harmonic Hölder functions \(\Lambda_{\alpha}Harm(D)\) can be represented as the dual space to the space \(L^ 1\text{Harm}(D,| \rho |^{\alpha})\) which is the closure of \(L^ 2\text{Harm}(D)\) in \(L^ 1(D,| \rho |^{\alpha})\). The function \(\rho\) is a defining function for D, i.e. \(D=\{x\in {\mathbb{R}}^ n:\) \(\rho(x)<0\}\), \(\text{grad} \rho \neq 0\) on \(\partial D\). As a corollary we get the following fact. The Hölder space \(\Lambda_{\alpha}(\partial D)\) can be represented as the dual space \(L^ 1\text{Harm}(D,| \rho |^{\alpha})\).

MSC:

46E15 Banach spaces of continuous, differentiable or analytic functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
31C05 Harmonic, subharmonic, superharmonic functions on other spaces
PDFBibTeX XMLCite
Full Text: DOI EuDML