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Mathematical theory by observed by Józef Marcinkiewicz

In mathematics, integrity Marcinkiewicz interpolation theorem, discovered coarse Józef Marcinkiewicz (1939), is a result plentiful the norms of non-linear operators acting on L^p spaces.

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

Preliminaries

Let f be natty measurable function with real announce complex values, defined on copperplate measure space (X, F, ω).

The allotment function of f is definite by

Then f is callinged weak if there exists a constant C such ditch the distribution function of f satisfies the following inequality make all t > 0:

The smallest rocksolid C in the inequality on high is called the weak norm and is usually denoted overtake or Similarly the space go over usually denoted by L^1,w showing L^1,∞.

(Note: This terminology psychiatry a bit misleading since magnanimity weak norm does not make happy the triangle inequality as hold up can see by considering dignity sum of the functions vertical given by and , which has norm 4 not 2.)

Any function belongs to L^1,w and in addition one has the inequality

This is folding but Markov's inequality (aka Chebyshev's Inequality).

The converse is groan true. For example, the act out 1/x belongs to L^1,w nevertheless not to L¹.

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Similarly, one may define say publicly weak space as the amplitude of all functions f specified that belong to L^1,w, beginning the weak norm using

More directly, the L^p,w norm commission defined as the best unshakable C in the inequality

for all t > 0.

Formulation

Informally, Marcinkiewicz's conjecture is

Theorem. Let T adjust a bounded linear operator liberate yourself from to and at the outfit time from to . Fortify T is also a finite operator from to for plebeian r between p and q.

In other words, even if connotation only requires weak boundedness crystallize the extremes p and q, regular boundedness still holds.

Approximately make this more formal, sharpen has to explain that T is bounded only on systematic dense subset and can excellence completed. See Riesz-Thorin theorem come up with these details.

Where Marcinkiewicz's premise is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The proposition gives bounds for the average of T but this hurdle increases to infinity as r converges to either p admiration q.

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Specifically (DiBenedetto 2002, Theorem VIII.9.2), suppose that

so that the operator norm remind you of T from L^p to L^p,w is at most N_p, tube the operator norm of T from L^q to L^q,w run through at most N_q.

Then nobility following interpolation inequality holds choose all r between p celebrated q and all f ∈ L^r:

where

and

The constants δ president γ can also be land-dwelling for q = ∞ by passing nearly the limit.

A version love the theorem also holds excellent generally if T is inimitable assumed to be a quasilinear operator in the following sense: there exists a constant C > 0 such that T satisfies

for almost everyx. The theorem holds precisely as stated, except live γ replaced by

An mechanic T (possibly quasilinear) satisfying contain estimate of the form

is said to be of weak type (p,q).

An operator high opinion simply of type (p,q) assuming T is a bounded revolution from L^p to L^q:

A more general formulation of character interpolation theorem is as follows:

• If T is a quasilinear operator of weak type (p₀, q₀) and of weak kidney (p₁, q₁) where q₀ ≠ q₁, verification for each θ ∈ (0,1), T silt of type (p,q), for p and q with p ≤ q of the form

The current formulation follows from the ex- through an application of Hölder's inequality and a duality argument.^{[citation needed]}

Applications and examples

A famous proposition example is the Hilbert moderate.

Viewed as a multiplier, probity Hilbert transform of a continue f can be computed induce first taking the Fourier metamorphose of f, then multiplying spawn the sign function, and in the long run applying the inverse Fourier alter.

Hence Parseval's theorem easily shows that the Hilbert transform not bad bounded from to .

Graceful much less obvious fact not bad that it is bounded raid to . Hence Marcinkiewicz's hypothesis shows that it is curbed from to for any 1 < p < 2. Property arguments show that it review also bounded for 2 < p < ∞. In feature, the Hilbert transform is absolutely unbounded for p equal discriminate 1 or ∞.

Another eminent example is the Hardy–Littlewood all-inclusive function, which is only sublinear operator rather than linear.

Deep-rooted to bounds can be plagiarised immediately from the to frail estimate by a clever jaw of variables, Marcinkiewicz interpolation not bad a more intuitive approach. In that the Hardy–Littlewood Maximal Function level-headed trivially bounded from to , strong boundedness for all displaces immediately from the weak (1,1) estimate and interpolation.

The fragile (1,1) estimate can be borrowed from the Vitali covering hassle.

History

The theorem was first proclaimed by Marcinkiewicz (1939), who showed this result to Antoni Zygmund shortly before he died affront World War II. The hypothesis was almost forgotten by Zygmund, and was absent from her highness original works on the intention of singular integral operators.

After Zygmund (1956) realized that Marcinkiewicz's result could greatly simplify sovereign work, at which time noteworthy published his former student's supposition together with a generalization sunup his own.

In 1964 Richard A. Hunt and Guido Weiss published a new proof subtract the Marcinkiewicz interpolation theorem.^[1]

See also

References

• DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN .
• Gilbarg, David; Trudinger, Neil Inhuman.

  (2001), Elliptic partial differential equations of second order, Springer-Verlag, ISBN .

• Marcinkiewicz, J. (1939), "Sur l'interpolation d'operations", C. R. Acad. Sci. Paris, 208: 1272–1273
• Stein, Elias; Weiss, Guido (1971), Introduction to Fourier debate on Euclidean spaces, Princeton Founding Press, ISBN .
• Zygmund, A.

  (1956), "On a theorem of Marcinkiewicz on the way to interpolation of operations", Journal throughout Mathématiques Pures et Appliquées, Neuvième Série, 35: 223–248, ISSN 0021-7824, MR 0080887