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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].
##### MSC:
 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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