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.
Contents
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 weak
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
Any
This is nothing but Markov's inequality (aka 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 weak
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
Theorem: Let T be a bounded linear operator from
In other words, even if you only require weak boundedness on the extremes p and q, you still get regular boundedness inside. 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.
Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the
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. That is, 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.
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 easily shows that the Hilbert transform is bounded from
Another famous example is the Hardy–Littlewood maximal function, which is only sublinear operator rather than linear. While
History
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.