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An essay on the interpolation theorem of Józef Marcinkiewicz – Polish patriot. (English) Zbl 1243.46010
Nawrocki, Marek (ed.) et al., Marcinkiewicz centenary volume. Proceedings of the Józef Marcinkiewicz centenary conference, June 28–July 2, 2010. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-14-0/pbk). Banach Center Publications 95, 55-74 (2011).
Suppose \(\widetilde{p},\widetilde{q}\in[1,\infty]\) and \(p,q\in[1,\infty]\) are Hölder conjugate pairs: \(1/p+1/q=1/\widetilde{p}+1/\widetilde{q}=1\). Let \(({\mathbb X},dx)\) be a \(\sigma\)-finite measure space and \({\mathbb V}\) a finite dimensional vector space with an inner product \(\langle\cdot|\cdot\rangle\). Suppose \(\Pi_\pm\) are complementary projections on \(L^p({\mathbb X})\) and \(L^p_\pm({\mathbb X})=\{\alpha\in L^p({\mathbb X}):\Pi_\pm\alpha=\alpha\}\). Let \({\mathcal A}: {\mathbb X}\times {\mathbb V}\to{\mathbb V}\) be a given function satisfying the following requirements: (a) Carathéodory regularity: the function \(x\mapsto{\mathcal A}(x,b)\) is measurable for every \(b\in{\mathbb V}\) and the function \(b\mapsto{\mathcal A}(x,b)\) is continuous for almost every \(x\in{\mathbb X}\); (b) homogeneity: \({\mathcal A}(x,\lambda b)=\lambda^{p-1}\cdot{\mathcal A}(x,b)\) for all \(\lambda\geq 0\); (c) Lipschitz condition: \[ |{\mathcal A}(x,b_1)-{\mathcal A}(x,b_2)|\leq \text{const}(|b_1|^{p-2}+|b_2|^{p-2})|b_1-b_2|; \] (d) monotonicity condition: \[ \langle {\mathcal A}(x,b_1)-{\mathcal A}(x,b_2)|b_1-b_2\rangle \geq\text{const}(|b_1|^{p-2}+|b_2|^{p-2})|b_1-b_2|. \] Consider the following problem: Given a pair \((a,b)\in L^p({\mathbb X},{\mathbb V})\times L^q({\mathbb X},{\mathbb V})\), solve the equation \({\mathcal A}(x,a+\alpha)=b+\beta\) for \((\alpha,\beta)\in L^p_+({\mathbb X},{\mathbb V})\times L^q_-({\mathbb X},{\mathbb V})\). One can associate with this problem the non-linear operator \({\mathcal R}:L^p({\mathbb X})\times L^q({\mathbb X})\to L^p({\mathbb X})\times L^q({\mathbb X})\) defined by \({\mathcal R}(a,b)=(\alpha,\beta)\).
The main result of the paper is the following version of the Marcinkiewicz interpolation theorem for \({\mathcal R}\). Let \(\lambda_-\) and \(\lambda_+\) be exponents, \(\widetilde{q}/q\leq\lambda_-<1<\lambda_+\leq\widetilde{p}/p\), for which \({\mathcal R}\) is both of weak \(\lambda_-\)-type and weak \(\lambda_+\)-type. Then for every \(\tau\in(\lambda_-,\lambda_+)\) the operator \({\mathcal R}\) is of strong \(\tau\)-type, meaning that \[ \int_{\mathbb X}[{\mathcal R}(a,b)]^\tau\leq\text{const}\int_{\mathbb X}[(a,b)]^\tau\text{\;for\;all\;} a,b\in \bigcap_{1\leq s\leq\infty} L^s({\mathbb X},{\mathbb V}), \] where \([(a,b)]=|a|^p+|b|^q\).
For the entire collection see [Zbl 1234.00021].
46B70 Interpolation between normed linear spaces
46E40 Spaces of vector- and operator-valued functions
58A10 Differential forms in global analysis
35J60 Nonlinear elliptic equations
41A05 Interpolation in approximation theory
47B38 Linear operators on function spaces (general)
Biographic References:
Marcinkiewicz, Józef
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