Marcinkiewicz interpolation theorem

In mathematics, 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. 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 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 norm and is usually denoted by or Similarly the space is usually denoted by L1,w or L1,∞. (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 given by and , which has norm 4 not 2.) Any function belongs to L1,w and in addition one has the inequality 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 space as the space of all functions f such that belong to L1,w, and the weak norm using More directly, the Lp,w norm is defined as the best constant C in the inequality for all t > 0. Informally, Marcinkiewicz's theorem is Theorem. Let T be a bounded linear operator from to and at the same time from to . Then T is also a bounded operator from to 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. Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem is in the estimates of the norm. The theorem gives bounds for the norm of T but this bound increases to infinity as r converges to either p or q.