In mathematics, particularly in functional analysis, the Marcinkiewicz interpolation theorem, discovered by , is a result bounding the norms of non-linear operators acting on Lp spaces.

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

Preliminaries

Let f be a measurable function with real or complex values, defined on a measure space (X,F,ω). The distribution function of f is defined by

Then f is called 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 norm Similarly the space is usually denoted by L1,w or L1,∞.

, which has norm 4 not 2.)

function belongs to L1,w and in addition one has the inequality

This is nothing but Markov's inequality ( Chebyshev's Inequality). The converse is not true. For example, the function 1/x belongs to L1,w but not to L1.

Similarly, one may define the L^p space as the space of all functions f belong to L1,w, and the norm using

More directly, the Lp,w norm is defined as the best constant C in the inequality

for all t>0.

Formulation

Informally, Marcinkiewicz's theorem is

1. Theorem. Let T be a bounded linear operator. Then T for any r between p and q.

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. See Riesz-Thorin theorem for these details.

norm of T but this bound increases to infinity as r converges to either p or q. Specifically , 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:

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

The latter formulation follows from the former through an application of Hölder's inequality and a duality argument.

Applications and examples

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 for any 1 < p < 2. 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 follows immediately from the weak (1,1) estimate and interpolation. The weak (1,1) estimate can be obtained from the Vitali covering lemma.

History

The theorem was first announced by , 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 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 A. Hunt and Guido Weiss published a new proof of the Marcinkiewicz interpolation theorem.

See also

• Interpolation space

References

• .

• .

• .