Marcinkiewicz interpolation theorem: Difference between revisions

Revision as of 12:19, 5 October 2017

Revision as of 03:50, 6 March 2018

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The constants δ and γ can also be given for ''q'' = ∞ by passing to the limit.

A version of the theorem also holds more generally if ''T'' is only assumed to be a [[quasilinear]]{{dndisambiguation needed|date=March 2017}} operator. That is, there exists a constant ''C'' > 0 such that ''T'' satisfies

:<math>|T(f+g)(x)| \le C(|Tf(x)|+|Tg(x)|)</math>

for [[almost everywhere|almost every]] ''x''. The theorem holds precisely as stated, except with γ replaced by

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* {{citation | last=DiBenedetto|first=Emmanuele|title=Real analysis|publisher=Birkhäuser|year=2002|isbn=3-7643-4231-5}}.

* {{citation|title=Elliptic partial differential equations of second order|first1=David|last1=Gilbarg|authorlink1=<!--David Gilbarg-->|first2=Neil S.|last2=Trudinger|authorlink2=Neil Trudinger|publisher=Springer-Verlag|year=2001|isbn=3-540-41160-7}}.

*{{Citation | last1=Marcinkiewicz | first1=J. | title=Sur l'interpolation d'operations | year=1939 | journal=C. R. Acad. des Sciences,Sci. Paris | volume=208 | pages=1272–1273}}

* {{citation|title=Introduction to Fourier analysis on Euclidean spaces|first1=Elias|last1=Stein|authorlink1=Elias Stein|first2=Guido|last2=Weiss|publisher=Princeton University Press|year=1971|isbn=0-691-08078-X}}.

*{{Citation | last1=Zygmund | first1=A. | title=On a theorem of Marcinkiewicz concerning interpolation of operations |mr=0080887 | year=1956 | journal=[[Journal de Mathématiques Pures et Appliquées]]|series= Neuvième Série | issn=0021-7824 | volume=35 | pages=223–248}}