Марцинкевич интерполяция теоремасы және оның қолданылуы

Mathematical theory by discovered by Józef Marcinkiewicz
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.

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 to A much less obvious fact is that it is bounded from to Hence Marcinkiewicz's theorem shows that it is bounded from to 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 rather than linear. While to bounds can be derived immediately from the to weak estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the Hardy–Littlewood Maximal Function is trivially bounded from to , strong boundedness for all follows immediately from the weak (1,1) estimate and interpolation. The weak (1,1) estimate can be obtained from the Vitali covering lemma.

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.