In mathematics, the Marcinkiewicz interpolation theorem, discovered by Józef Marcinkiewicz (1939), is a result bounding the norms of non-linear operators acting on Lp spaces.
The distribution function of f is defined by Then f is called weak
if there exists a constant C such that the distribution function of f satisfies the following inequality for all t > 0: The smallest constant C in the inequality above is called the weak
(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
Informally, Marcinkiewicz's theorem is In other words, even if one only requires weak boundedness on the extremes p and q, regular boundedness still holds.
To make this more formal, one has to explain that T is bounded only on a dense subset and can be completed.
norm of T but this bound increases to infinity as r converges to either p or q.
Specifically (DiBenedetto 2002, Theorem VIII.9.2), suppose that so that the operator norm of T from Lp to Lp,w is at most Np, and the operator norm of T from Lq to Lq,w is at most Nq.
Then the following interpolation inequality holds for all r between p and q and all f ∈ Lr: where and 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 in the following sense: there exists a constant C > 0 such that T satisfies for almost every x.
The theorem holds precisely as stated, except with γ replaced by An operator T (possibly quasilinear) satisfying an estimate of the form is said to be of weak type (p,q).
An operator is simply of type (p,q) if T is a bounded transformation from Lp to Lq: A more general formulation of the interpolation theorem is as follows: The latter formulation follows from the former through an application of Hölder's inequality and a duality argument.
[citation needed] A famous application example is the Hilbert transform.
Viewed as a multiplier, the Hilbert transform of a function f can be computed by first taking the Fourier transform of f, then multiplying by the sign function, and finally applying the inverse Fourier transform.
Hence Parseval's theorem easily shows that the Hilbert transform is bounded from
Duality arguments show that it is also bounded for 2 < p < ∞.
In fact, the Hilbert transform is really unbounded for p equal to 1 or ∞.
Another famous example is the Hardy–Littlewood maximal function, which is only sublinear operator rather than linear.
estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach.
The weak (1,1) estimate can be obtained from the Vitali covering lemma.
The theorem was first announced by 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 operators.
Later 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.
Hunt and Guido Weiss published a new proof of the Marcinkiewicz interpolation theorem.