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 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. History The theorem was first announced by