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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Marcinkiewicz interpolation theorem ： ウィキペディア英語版 Marcinkiewicz interpolation theorem In mathematics, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on ''L''p spaces. Marcinkiewicz' theorem is similar to the Riesz–Thorin theorem about linear operators, but also applies to non-linear operators. ==Preliminaries== Let ''f'' be a measurable function with real or complex values, defined on a measure space (''X'', ''F'', ω). The distribution function of ''f'' is defined by :$\backslash lambda\_f(t)\; =\; \backslash omega\backslash left\backslash .$ Then ''f'' is called weak $L^1$ if there exists a constant ''C'' such that the distribution of ''f'' satisfies the following inequality for all ''t'' > 0: :$\backslash lambda\_f(t)\backslash leq\; \backslash frac.$ The smallest constant ''C'' in the inequality above is called the weak $L^1$ norm and is usually denoted by ||''f''||1,''w'' or ||''f''||1,∞. Similarly the space is usually denoted by ''L''1,''w'' or ''L''1,∞. (Note: This terminology is a bit misleading since the weak norm does not satisfy the triangle inequality as one can see by considering the sum of the functions on $(0,1)$ given by $1/x$ and $1/(1-x)$, which has norm 4 not 2.) Any $L^1$ function belongs to ''L''1,''w'' and in addition one has the inequality :$\backslash |f\backslash |\_\backslash leq\; \backslash |f\backslash |\_1.$ This is nothing but Markov's inequality (aka Chebyshev's Inequality). The converse is not true. For example, the function 1/''x'' belongs to ''L''1,''w'' but not to ''L''1. Similarly, one may define the weak $L^p$ space as the space of all functions ''f'' such that $|f|^p$ belong to ''L''1,''w'', and the weak $L^p$ norm using :$\backslash |f\backslash |\_=\backslash |\backslash ,|f|^p\; \backslash |\_^.$ More directly, the ''L''''p'',''w'' norm is defined as the best constant ''C'' in the inequality :$\backslash lambda\_f(t)\; \backslash le\; \backslash frac$ for all ''t'' > 0. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Marcinkiewicz interpolation theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.