# Marcinkiewicz interpolation theorem

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

Contents 1 Preliminaries 2 Formulation 3 Applications and examples 4 History 5 See also 6 References 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 {displaystyle lambda _{f}(t)=omega left{xin Xmid |f(x)|>tright}.} Then f is called weak {displaystyle L^{1}} if there exists a constant C such that the distribution function of f satisfies the following inequality for all t > 0: {displaystyle lambda _{f}(t)leq {frac {C}{t}}.} The smallest constant C in the inequality above is called the weak {displaystyle L^{1}} norm and is usually denoted by {displaystyle |f|_{1,w}} or {displaystyle |f|_{1,infty }.} 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 {displaystyle (0,1)} given by {displaystyle 1/x} and {displaystyle 1/(1-x)} , which has norm 4 not 2.) Any {displaystyle L^{1}} function belongs to L1,w and in addition one has the inequality {displaystyle |f|_{1,w}leq |f|_{1}.} 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 {displaystyle L^{p}} space as the space of all functions f such that {displaystyle |f|^{p}} belong to L1,w, and the weak {displaystyle L^{p}} norm using {displaystyle |f|_{p,w}=left||f|^{p}right|_{1,w}^{frac {1}{p}}.} More directly, the Lp,w norm is defined as the best constant C in the inequality {displaystyle lambda _{f}(t)leq {frac {C^{p}}{t^{p}}}} for all t > 0.

Formulation Informally, Marcinkiewicz's theorem is Theorem. Let T be a bounded linear operator from {displaystyle L^{p}} to {displaystyle L^{p,w}} and at the same time from {displaystyle L^{q}} to {displaystyle L^{q,w}} . Then T is also a bounded operator from {displaystyle L^{r}} to {displaystyle L^{r}} for any r between p and q.

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 {displaystyle L^{r}} norm of T but this bound increases to infinity as r converges to either p or q. Specifically (DiBenedetto 2002, Theorem VIII.9.2), suppose that {displaystyle |Tf|_{p,w}leq N_{p}|f|_{p},} {displaystyle |Tf|_{q,w}leq N_{q}|f|_{q},} 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: {displaystyle |Tf|_{r}leq gamma N_{p}^{delta }N_{q}^{1-delta }|f|_{r}} where {displaystyle delta ={frac {p(q-r)}{r(q-p)}}} and {displaystyle gamma =2left({frac {r(q-p)}{(r-p)(q-r)}}right)^{1/r}.} 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 {displaystyle |T(f+g)(x)|leq C(|Tf(x)|+|Tg(x)|)} for almost every x. The theorem holds precisely as stated, except with γ replaced by {displaystyle gamma =2Cleft({frac {r(q-p)}{(r-p)(q-r)}}right)^{1/r}.} An operator T (possibly quasilinear) satisfying an estimate of the form {displaystyle |Tf|_{q,w}leq C|f|_{p}} 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: {displaystyle |Tf|_{q}leq C|f|_{p}.} 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 {displaystyle {frac {1}{p}}={frac {1-theta }{p_{0}}}+{frac {theta }{p_{1}}},quad {frac {1}{q}}={frac {1-theta }{q_{0}}}+{frac {theta }{q_{1}}}.} The latter formulation follows from the former through an application of Hölder's inequality and a duality argument.[citation needed] 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 {displaystyle L^{2}} to {displaystyle L^{2}} . A much less obvious fact is that it is bounded from {displaystyle L^{1}} to {displaystyle L^{1,w}} . Hence Marcinkiewicz's theorem shows that it is bounded from {displaystyle L^{p}} to {displaystyle L^{p}} 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 {displaystyle L^{p}} to {displaystyle L^{p}} bounds can be derived immediately from the {displaystyle L^{1}} to weak {displaystyle L^{1}} estimate by a clever change of variables, Marcinkiewicz interpolation is a more intuitive approach. Since the Hardy–Littlewood Maximal Function is trivially bounded from {displaystyle L^{infty }} to {displaystyle L^{infty }} , strong boundedness for all {displaystyle p>1} 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 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.

In 1964 Richard A. Hunt and Guido Weiss published a new proof of the Marcinkiewicz interpolation theorem.[1] See also Interpolation space References ^ Hunt, Richard A.; Weiss, Guido (1964). "The Marcinkiewicz interpolation theorem". Proceedings of the American Mathematical Society. 15 (6): 996–998. doi:10.1090/S0002-9939-1964-0169038-4. ISSN 0002-9939. DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN 3-7643-4231-5. Gilbarg, David; Trudinger, Neil S. (2001), Elliptic partial differential equations of second order, Springer-Verlag, ISBN 3-540-41160-7. Marcinkiewicz, J. (1939), "Sur l'interpolation d'operations", C. R. Acad. Sci. Paris, 208: 1272–1273 Stein, Elias; Weiss, Guido (1971), Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, ISBN 0-691-08078-X. 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 hide vte Functional analysis (topics – glossary) Spaces BanachBesovFréchetHilbertHölderNuclearOrliczSchwartzSobolevtopological vector Properties barrelledcompletedual (algebraic/topological)locally convexreflexiveseparable Theorems Hahn–BanachRiesz representationclosed graphuniform boundedness principleKakutani fixed-pointKrein–Milmanmin–maxGelfand–NaimarkBanach–Alaoglu Operators adjointboundedcompactHilbert–Schmidtnormalnucleartrace classtransposeunboundedunitary Algebras Banach algebraC*-algebraspectrum of a C*-algebraoperator algebragroup algebra of a locally compact groupvon Neumann algebra Open problems invariant subspace problemMahler's conjecture Applications Hardy spacespectral theory of ordinary differential equationsheat kernelindex theoremcalculus of variationsfunctional calculusintegral operatorJones polynomialtopological quantum field theorynoncommutative geometryRiemann hypothesisdistribution (or generalized functions) Advanced topics approximation propertybalanced setChoquet theoryweak topologyBanach–Mazur distanceTomita–Takesaki theory Categories: Fourier analysisTheorems in functional analysis

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