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Nonlinear potentials and trace inequalities. (English) Zbl 0941.31001
Rossmann, Jürgen (ed.) et al., The Maz’ya anniversary collection. Vol. 2: Rostock conference on functional analysis, partial differential equations and applications, Rostock, Germany, August 31-September 4, 1998. Basel: Birkhäuser. Oper. Theory, Adv. Appl. 110, 323-343 (1999).
The author gives a good survey of the trace inequalities of the type $$\|Tf\|_{L^q(d\omega)}\leq C\|f\|_{L^p(dx)}$$. A number of well-known results are presented under the following unifying point of view: for different operators $$T$$ to find a condition (mainly in terms of capacity) on the weight $$\omega$$ for the trace inequality to be true. The survey is completed by a new result for the Riesz potential $$I_\alpha=(-\Delta)^{-\frac{\alpha}{2}}$$.
For the entire collection see [Zbl 0923.00035].

MSC:
 31B15 Potentials and capacities, extremal length and related notions in higher dimensions 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 31-02 Research exposition (monographs, survey articles) pertaining to potential theory
Keywords:
trace theorem; Riesz potential; capacity