Marcinkiewicz interpolation theorem: Difference between revisions

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{{Short description|Mathematical theory by discovered by Józef Marcinkiewicz}}
In [[mathematics]], particularly in [[functional analysis]], the '''Marcinkiewicz interpolation theorem''', discovered by {{harvs|txt|authorlink=Józef Marcinkiewicz|first=Józef |last=Marcinkiewicz|year=1939}}, is a result bounding the norms of non-linear operators acting on [[lp space|''L''^p spaces]].

Marcinkiewicz' theorem is similar to the [[Riesz–Thorin theorem]] about [[linear operators]], but also applies to non-linear operators.

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:'''Theorem.''' Let ''T'' be a [[bounded linear operator]] from <math>L^p</math> to <math>L^{p,w}</math> and at the same time from <math>L^q</math> to <math>L^{q,w}</math>. Then ''T'' is also a bounded operator from <math>L^r</math> to <math>L^r</math> for any ''r'' between ''p'' and ''q''.

In other words, even if youone only requirerequires weak boundedness on the extremes ''p'' and ''q'', you still get regular boundedness insidestill holds. To make this more formal, one has to explain that ''T'' is bounded only on a [[Dense set|dense]] subset and can be completed. See [[Riesz-Thorin theorem]] for these details.

Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the <math>L^r</math> norm of ''T'' but this bound increases to infinity as ''r'' converges to either ''p'' or ''q''. Specifically {{harv|DiBenedetto|2002|loc=Theorem VIII.9.2}}, suppose that

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==History==

The theorem was first announced by {{harvtxt|Marcinkiewicz|1939}}, who showed this result to [[Antoni Zygmund]] shortly before he died in World War II. The theorem was almost forgotten by Zygmund, and was absent from his original works on the theory of [[singular integral operator]]s. Later {{harvtxt|Zygmund|1956}} realized that Marcinkiewicz's result could greatly simplify his work, at which time he published his former student's theorem together with a generalization of his own.

In 1964 [[Richard Allen Hunt|Richard A. Hunt]] and [[Guido Weiss]] published a new proof of the Marcinkiewicz interpolation theorem.<ref name="HuntWeiss1964">{{cite journal|last1=Hunt|first1=Richard A.|last2=Weiss|first2=Guido|title=The Marcinkiewicz interpolation theorem|journal=Proceedings of the American Mathematical Society|volume=15|issue=6|year=1964|pages=996–998|issn=0002-9939|doi=10.1090/S0002-9939-1964-0169038-4|doi-access=free}}</ref>

== See also ==

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==References==

{{reflist}}

* {{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| url=https://books.google.com/books?id=eoiGTf4cmhwC}}.

*{{Citation | last1=Marcinkiewicz | first1=J. | title=Sur l'interpolation d'operations | year=1939 |url=https://gallica.bnf.fr/ark:/12148/bpt6k6238835z/f16.item#| journal=C. R. Acad. 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|url-access=registration|url=https://archive.org/details/introductiontofo0000stei}}.

*{{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}}

{{Template:Lp spaces}}
{{Functional analysis}}

[[Category:Fourier analysis]]

[[Category:Theorems in functional analysis]]

[[Category:Lp spaces]]