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In [[mathematics]], 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''<sup>p</sup> spaces]].
Marcinkiewicz' theorem is similar to the [[Riesz–Thorin theorem]] about linear operators, but also applies to non-linear operators.
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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 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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