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Non-linear potential theory in Lebesgue spaces with mixed norm. (English) Zbl 0693.31003

Potential theory, Proc. Conf., Prague/Czech. 1987, 325-331 (1988).
[For the entire collection see Zbl 0675.00009.]
A Lebesgue space with mixed norm over \(R^ d=R^ m\times R^ n\) consists of functions f in \(R^ d\) under the norm \[ \| f\|^ q_{p,q}=\int (\int | f(x,s)|^ p dx)^{q/p} ds, \] where \((x,s)\in R^ m\times R^ n\) and \(1<p<\infty\), \(l<q<\infty\). Following N. G. Meyers [Math. Scand. 26, 255-292 (1970; Zbl 0242.31006)] and others, the author introduces very general kernels k on \(R^ d\times R^ d\) and corresponding set-functions \(C_{k,p,q}\) and studies the basic properties of corresponding capacitary distributions that realise the infimum in question, and their potentials. An explicit formula is given for the extremal capacitary distribution of a compact set. These results are applied to the Bessel kernel. No proofs.
Reviewer: E.J.Akutowicz

MSC:

31B15 Potentials and capacities, extremal length and related notions in higher dimensions