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Some results in the theory of Orlicz spaces and applications to variational problems. (English) Zbl 0965.46017

Krbec, Miroslav (ed.) et al., Nonlinear analysis, function spaces and applications. Vol. 6. Proceedings of the spring school held in Prague, Czech Republic, May 31-June 6, 1998. Prague: Czech Academy of Sciences, Mathematical Institute. 50-92 (1999).
The author presents a deep and comprehensive survey of his recent results on interpolation of operators, Sobolev inequalities, and applications to nonlinear problems in the calculus of variations.
One of the main achievements of the paper is a fairly general construction of an optimal “range” Young function when a “domain” Young function is given, for a certain operator or embedding. This is first applied in connection with Sobolev-type embeddings. More precisely, given a Young function \(A\), its “Sobolev conjugate” Young function, \(A_n\) (\(n\) is the dimension of the underlying domain), is given by \( A_n(t)=A\circ H^{-1}(t), \;t\in (0,\infty),\) where \[ H(s)=\Big (\int _{0}^{s}\Big (\frac {t}{A(t)} \Big)^{n'-1} dt\Big)^{\frac {1}{n'}},\qquad s\in (0,\infty). \] Then the inequality \( \|u\|_{L_{A_n}}\leq C \|Du\|_{L_{A}}\) holds with some \(C>0\) for all weakly differentiable functions \(u\) on \(\mathbb R^n\) vanishing at infinity (with \(Du\) standing for the gradient of \(u\)). Moreover, the result is optimal in the sense that there is no smaller Orlicz space than \(L_{A_n}\) for which an analogous inequality holds. A key to this result is an interpolation theorem producing pairs of Orlicz spaces between which a certain linear operator is bounded provided that this operator is of strong type \((1,p')\) and of weak type \((p,\infty)\). Analogous results for functions which not necessarily vanish at infinity are given, too, and the theory is illustrated by plenty of examples.
As a next step, an interpolation theorem for quasilinear operators of two weak types is shown, again resulting in (sharp) boundedness of the operator between two Orlicz spaces. This result has a broad variety of applications, including boundedness of Riesz potentials between Orlicz spaces, a-priori estimates for solutions of uniformly elliptic boundary value problems, and \(n\)-dimensional Hardy inequalities in Orlicz norms.
In the final third of the paper, regularity properties of solutions to variational problems under general (non-power) growth conditions are proved.
For the entire collection see [Zbl 0952.00033].
Reviewer: L.Pick (Praha)

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B70 Interpolation between normed linear spaces
47B38 Linear operators on function spaces (general)
47G10 Integral operators
49N60 Regularity of solutions in optimal control
35B45 A priori estimates in context of PDEs
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