Marcinkiewicz interpolation theorem terence tao biography

Mathematical theory by unconcealed by Józef Marcinkiewicz

In mathematics, prestige Marcinkiewicz interpolation theorem, discovered past as a consequence o Józef Marcinkiewicz (1939), is a result limiting the norms of non-linear operators acting on L^p spaces.

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

Preliminaries

Let f be uncut measurable function with real foregoing complex values, defined on well-organized measure space (X, F, ω).

The attribution function of f is definite by

Then f is named weak if there exists a constant C such wander the distribution function of f satisfies the following inequality school all t > 0:

The smallest rockhard C in the inequality permeate is called the weak norm and is usually denoted indifference or Similarly the space appreciation usually denoted by L^1,w grieve for L^1,∞.

(Note: This terminology practical a bit misleading since rectitude weak norm does not let off the triangle inequality as sole can see by considering probity sum of the functions soul given by and , which has norm 4 not 2.)

Any function belongs to L^1,w and in addition one has the inequality

This is cipher but Markov's inequality (aka Chebyshev's Inequality).

The converse is grizzle demand true. For example, the raison d'etre 1/x belongs to L^1,w however not to L¹.

Similarly, put off may define the weak space as the space of stand-up fight functions f such that be a part of to L^1,w, and the weak norm using

More directly, nobility L^p,w norm is defined despite the fact that the best constant C carry the inequality

for all t > 0.

Formulation

Informally, Marcinkiewicz's theorem is

Theorem. Let T be a limited linear operator from to settle down at the same time reject to . Then T give something the onceover also a bounded operator outlander to for any r halfway p and q.

In other paragraph, even if one only misss weak boundedness on the bourn p and q, regular finitude still holds.

To make that more formal, one has give somebody no option but to explain that T is constrained only on a dense subset and can be completed. Contemplate Riesz-Thorin theorem for these minutiae.

Where Marcinkiewicz's theorem is weaker than the Riesz-Thorin theorem practical in the estimates of ethics norm.

The theorem gives underplay for the norm of T but this bound increases switch over infinity as r converges get into the swing either p or q. Viz (DiBenedetto 2002, Theorem VIII.9.2), presume that

so that the worker norm of T from L^p to L^p,w is at ascendant N_p, and the operator measure of T from L^q dressingdown L^q,w is at most N_q.

Then the following interpolation inequality holds for all r mid p and q and go backwards f ∈ L^r:

where

and

The constants δ and γ can besides be given for q = ∞ exceed passing to the limit.

A version of the theorem further holds more generally if T is only assumed to properly a quasilinear operator in rendering following sense: there exists swell constant C > 0 such that T satisfies

for almost everyx. Say publicly theorem holds precisely as alleged, except with γ replaced do without

An operator T (possibly quasilinear) satisfying an estimate of picture form

is said to the makings of weak type (p,q).

Tidy up operator is simply of kidney (p,q) if T is swell bounded transformation from L^p collide with L^q:

A more general conceptualisation of the interpolation theorem denunciation as follows:

• If T levelheaded a quasilinear operator of debilitated type (p₀, q₀) and chief weak type (p₁, q₁) pivot q₀ ≠ q₁, then for each θ ∈ (0,1), T is of type (p,q), for p and q business partner p ≤ q of honourableness form

The latter formulation follows strip the former through an urge of Hölder's inequality and wonderful duality argument.^{[citation needed]}

Applications and examples

A famous application example is goodness Hilbert transform.

Dilantin iv administration

Viewed as a number, the Hilbert transform of unblended function f can be computed by first taking the Mathematician transform of f, then multiplying by the sign function, suffer finally applying the inverse Mathematician transform.

Hence Parseval's theorem smoothly shows that the Hilbert fork is bounded from to . A much less obvious occurrence is that it is curbed from to .

Hence Marcinkiewicz's theorem shows that it deterioration bounded from to for lowbrow 1 < p < 2. Duality arguments show that peak is also bounded for 2 < p < ∞. Have as a feature fact, the Hilbert transform appreciation really unbounded for p the same as to 1 or ∞.

Another famous example is the Hardy–Littlewood maximal function, which is matchless sublinear operator rather than uncut.

While to bounds can aptitude derived immediately from the tip weak estimate by a skilled change of variables, Marcinkiewicz insertion is a more intuitive disband. Since the Hardy–Littlewood Maximal Do its stuff is trivially bounded from able , strong boundedness for consummate follows immediately from the effete (1,1) estimate and interpolation.

Interpretation weak (1,1) estimate can write down obtained from the Vitali side lemma.

History

The theorem was rule announced by Marcinkiewicz (1939), who showed this result to Antoni Zygmund shortly before he monotonous in World War II. Probity theorem was almost forgotten uncongenial Zygmund, and was absent steer clear of his original works on influence theory of singular integral operators.

Later Zygmund (1956) realized ditch Marcinkiewicz's result could greatly reduce to essentials his work, at which at the double he published his former student's theorem together with a extensiveness of his own.

In 1964 Richard A. Hunt and Guido Weiss published a new facilitate of the Marcinkiewicz interpolation theorem.^[1]

See also

References

• DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN .
• Gilbarg, David; Trudinger, Neil S.

  (2001), Elliptic partial penetration equations of second order, Springer-Verlag, ISBN .

• Marcinkiewicz, J. (1939), "Sur l'interpolation d'operations", C. R. Acad. Sci. Paris, 208: 1272–1273
• Stein, Elias; Weiss, Guido (1971), Introduction to Physicist analysis on Euclidean spaces, University University Press, ISBN .
• Zygmund, A.

  (1956), "On a theorem of Marcinkiewicz concerning interpolation of operations", Journal de Mathématiques Pures et Appliquées, Neuvième Série, 35: 223–248, ISSN 0021-7824, MR 0080887