# Riesz–Thorin theorem

Riesz–Thorin theorem For more theorems that are called Riesz's theorem, see Riesz theorem.

In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin.

This theorem bounds the norms of linear maps acting between Lp spaces. Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L2 which is a Hilbert space, or to L1 and L∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps.

Contents 1 Motivation 2 Statement of the theorem 3 Proof 3.1 Simple Functions 3.1.1 Proof of (3) 3.2 Extension to All Measurable Functions in '"UNIQ--postMath-0000005E-QINU"' 4 Interpolation of analytic families of operators 5 Applications 5.1 Hausdorff–Young inequality 5.2 Convolution operators 5.3 The Hilbert transform 6 Comparison with the real interpolation method 7 Mityagin's theorem 8 See also 9 Notes 10 References 11 External links Motivation First we need the following definition: Definition. Let p0, p1 be two numbers such that 0 < p0 < p1 ≤ ∞. Then for 0 < θ < 1 define pθ by:   1 / pθ = 1 − θ / p0 + θ / p1 . By splitting up the function  f  in Lpθ as the product | f | = | f |1−θ | f |θ and applying Hölder's inequality to its pθ power, we obtain the following result, foundational in the study of Lp-spaces: Proposition (log-convexity of Lp-norms) — Each  f  ∈ Lp0 ∩ Lp1 satisfies: {displaystyle |f|_{p_{theta }}leq |f|_{p_{0}}^{1-theta }|f|_{p_{1}}^{theta }.}         (1) This result, whose name derives from the convexity of the map 1⁄p ↦ log || f ||p on [0, ∞], implies that Lp0 ∩ Lp1 ⊂ Lpθ. On the other hand, if we take the layer-cake decomposition  f  =  f 1{| f |>1} +  f 1{| f |≤1}, then we see that  f 1{| f |>1} ∈ Lp0 and  f 1{| f |≤1} ∈ Lp1, whence we obtain the following result: Proposition — Each  f  in Lpθ can be written as a sum:  f  = g + h, where g ∈ Lp0 and h ∈ Lp1.

In particular, the above result implies that Lpθ is included in Lp0 + Lp1, the sumset of Lp0 and Lp1 in the space of all measurable functions. Therefore, we have the following chain of inclusions: Corollary — Lp0 ∩ Lp1 ⊂ Lpθ ⊂ Lp0 + Lp1.

In practice, we often encounter operators defined on the sumset Lp0 + Lp1. For example, the Riemann–Lebesgue lemma shows that the Fourier transform maps L1(Rd) boundedly into L∞(Rd), and Plancherel's theorem shows that the Fourier transform maps L2(Rd) boundedly into itself, hence the Fourier transform {displaystyle {mathcal {F}}} extends to (L1 + L2) (Rd) by setting {displaystyle {mathcal {F}}(f_{1}+f_{2})={mathcal {F}}_{L^{1}}(f_{1})+{mathcal {F}}_{L^{2}}(f_{2})} for all  f1  ∈ L1(Rd) and  f2  ∈ L2(Rd). It is therefore natural to investigate the behavior of such operators on the intermediate subspaces Lpθ.

To this end, we go back to our example and note that the Fourier transform on the sumset L1 + L2 was obtained by taking the sum of two instantiations of the same operator, namely {displaystyle {mathcal {F}}_{L^{1}}:L^{1}(mathbf {R} ^{d})to L^{infty }(mathbf {R} ^{d}),} {displaystyle {mathcal {F}}_{L^{2}}:L^{2}(mathbf {R} ^{d})to L^{2}(mathbf {R} ^{d}).} These really are the same operator, in the sense that they agree on the subspace (L1 ∩ L2) (Rd). Since the intersection contains simple functions, it is dense in both L1(Rd) and L2(Rd). Densely defined continuous operators admit unique extensions, and so we are justified in considering {displaystyle {mathcal {F}}_{L^{1}}} and {displaystyle {mathcal {F}}_{L^{2}}} to be the same.

Therefore, the problem of studying operators on the sumset Lp0 + Lp1 essentially reduces to the study of operators that map two natural domain spaces, Lp0 and Lp1, boundedly to two target spaces: Lq0 and Lq1, respectively. Since such operators map the sumset space Lp0 + Lp1 to Lq0 + Lq1, it is natural to expect that these operators map the intermediate space Lpθ to the corresponding intermediate space Lqθ.

Statement of the theorem There are several ways to state the Riesz–Thorin interpolation theorem;[1] to be consistent with the notations in the previous section, we shall use the sumset formulation.

Riesz–Thorin interpolation theorem — Let (Ω1, Σ1, μ1) and (Ω2, Σ2, μ2) be σ-finite measure spaces. Suppose 1 ≤ p0 , q0 , p1 , q1 ≤ ∞, and let T : Lp0(μ1) + Lp1(μ1) → Lq0(μ2) + Lq1(μ2) be a linear operator that boundedly maps Lp0(μ1) into Lq0(μ2) and Lp1(μ1) into Lq1(μ2). For 0 < θ < 1, let pθ, qθ be defined as above. Then T boundedly maps Lpθ(μ1) into Lqθ(μ2) and satisfies the operator norm estimate {displaystyle |T|_{L^{p_{theta }}to L^{q_{theta }}}leq |T|_{L^{p_{0}}to L^{q_{0}}}^{1-theta }|T|_{L^{p_{1}}to L^{q_{1}}}^{theta }.}         (2) In other words, if T is simultaneously of type (p0, q0) and of type (p1, q1), then T is of type (pθ, qθ) for all 0 < θ < 1. In this manner, the interpolation theorem lends itself to a pictorial description. Indeed, the Riesz diagram of T is the collection of all points ( 1 / p , 1 / q ) in the unit square [0, 1] × [0, 1] such that T is of type (p, q). The interpolation theorem states that the Riesz diagram of T is a convex set: given two points in the Riesz diagram, the line segment that connects them will also be in the diagram. The interpolation theorem was originally stated and proved by Marcel Riesz in 1927.[2] The 1927 paper establishes the theorem only for the lower triangle of the Riesz diagram, viz., with the restriction that p0 ≤ q0 and p1 ≤ q1. Olof Thorin extended the interpolation theorem to the entire square, removing the lower-triangle restriction. The proof of Thorin was originally published in 1938 and was subsequently expanded upon in his 1948 thesis.[3] Proof We will first prove the result for simple functions and eventually show how the argument can be extended by density to all measurable functions. Simple Functions By symmetry, let us assume {textstyle p_{0}1}} and define {textstyle g=fmathbf {1} _{E}} , {textstyle g_{n}=f_{n}mathbf {1} _{E}} , {textstyle h=f-g=fmathbf {1} _{E^{mathrm {c} }}} and {textstyle h_{n}=f_{n}-g_{n}} . Note that, since we are assuming {textstyle p_{0}leq p_{theta }leq p_{1}} , {displaystyle {begin{aligned}lVert frVert _{p_{theta }}^{p_{theta }}&=int _{Omega _{1}}leftvert frightvert ^{p_{theta }},mathrm {d} mu _{1}geq int _{Omega _{1}}leftvert frightvert ^{p_{theta }}mathbf {1} _{E},mathrm {d} mu _{1}geq int _{Omega _{1}}leftvert fmathbf {1} _{E}rightvert ^{p_{0}},mathrm {d} mu _{1}=int _{Omega _{1}}leftvert grightvert ^{p_{0}},mathrm {d} mu _{1}=lVert grVert _{p_{0}}^{p_{0}}\lVert frVert _{p_{theta }}^{p_{theta }}&=int _{Omega _{1}}leftvert frightvert ^{p_{theta }},mathrm {d} mu _{1}geq int _{Omega _{1}}leftvert frightvert ^{p_{theta }}mathbf {1} _{E^{mathrm {c} }},mathrm {d} mu _{1}geq int _{Omega _{1}}leftvert fmathbf {1} _{E^{mathrm {c} }}rightvert ^{p_{1}},mathrm {d} mu _{1}=int _{Omega _{1}}leftvert hrightvert ^{p_{1}},mathrm {d} mu _{1}=lVert hrVert _{p_{1}}^{p_{1}}end{aligned}}} and, equivalently, {textstyle gin L^{p_{0}}(Omega _{1})} and {textstyle hin L^{p_{1}}(Omega _{1})} .

Let us see what happens in the limit for {textstyle nto infty } . Since {textstyle leftvert f_{n}rightvert leq leftvert frightvert } , {textstyle leftvert g_{n}rightvert leq leftvert grightvert } and {textstyle leftvert h_{n}rightvert leq leftvert hrightvert } , by the dominated convergence theorem one readily has {displaystyle {begin{aligned}lVert f_{n}rVert _{p_{theta }}&to lVert frVert _{p_{theta }}&lVert g_{n}rVert _{p_{0}}&to lVert grVert _{p_{0}}&lVert h_{n}rVert _{p_{1}}&to lVert hrVert _{p_{1}}.end{aligned}}} Similarly, {textstyle leftvert f-f_{n}rightvert leq 2leftvert frightvert } , {textstyle leftvert g-g_{n}rightvert leq 2leftvert grightvert } and {textstyle leftvert h-h_{n}rightvert leq 2leftvert hrightvert } imply {displaystyle {begin{aligned}lVert f-f_{n}rVert _{p_{theta }}&to 0&lVert g-g_{n}rVert _{p_{0}}&to 0&lVert h-h_{n}rVert _{p_{1}}&to 0end{aligned}}} and, by the linearity of {textstyle T} as an operator of types {textstyle (p_{0},q_{0})} and {textstyle (p_{1},q_{1})} (we have not proven yet that it is of type {textstyle (p_{theta },q_{theta })} for a generic {textstyle f} ) {displaystyle {begin{aligned}lVert Tg-Tg_{n}rVert _{p_{0}}&leq |T|_{L^{p_{0}}to L^{q_{0}}}lVert g-g_{n}rVert _{p_{0}}to 0&lVert Th-Th_{n}rVert _{p_{1}}&leq |T|_{L^{p_{1}}to L^{q_{1}}}lVert h-h_{n}rVert _{p_{1}}to 0.end{aligned}}} It is now easy to prove that {textstyle Tg_{n}to Tg} and {textstyle Th_{n}to Th} in measure: For any {textstyle epsilon >0} , Chebyshev’s inequality yields {displaystyle mu _{2}(yin Omega _{2}:leftvert Tg-Tg_{n}rightvert >epsilon )leq {frac {lVert Tg-Tg_{n}rVert _{q_{0}}^{q_{0}}}{epsilon ^{q_{0}}}}} and similarly for {textstyle Th-Th_{n}} . Then, {textstyle Tg_{n}to Tg} and {textstyle Th_{n}to Th} a.e. for some subsequence and, in turn, {textstyle Tf_{n}to Tf} a.e. Then, by Fatou’s lemma and recalling that (4) holds true for simple functions, {displaystyle lVert TfrVert _{q_{theta }}leq liminf _{nto infty }lVert Tf_{n}rVert _{q_{theta }}leq |T|_{L^{p_{theta }}to L^{q_{theta }}}liminf _{nto infty }lVert f_{n}rVert _{p_{theta }}=|T|_{L^{p_{theta }}to L^{q_{theta }}}lVert frVert _{p_{theta }}.} Interpolation of analytic families of operators The proof outline presented in the above section readily generalizes to the case in which the operator T is allowed to vary analytically. In fact, an analogous proof can be carried out to establish a bound on the entire function {displaystyle varphi (z)=int (T_{z}f_{z})g_{z},dmu _{2},} from which we obtain the following theorem of Elias Stein, published in his 1956 thesis:[5] Stein interpolation theorem — Let (Ω1, Σ1, μ1) and (Ω2, Σ2, μ2) be σ-finite measure spaces. Suppose 1 ≤ p0 , p1 ≤ ∞, 1 ≤ q0 , q1 ≤ ∞, and define: S = {z ∈ C : 0 < Re(z) < 1}, S = {z ∈ C : 0 ≤ Re(z) ≤ 1}. We take a collection of linear operators {Tz : z ∈ S} on the space of simple functions in L1(μ1) into the space of all μ2-measurable functions on Ω2. We assume the following further properties on this collection of linear operators: The mapping {displaystyle zmapsto int (T_{z}f)g,dmu _{2}} is continuous on S and holomorphic on S for all simple functions  f  and g. For some constant k < π, the operators satisfy the uniform bound: {displaystyle sup _{zin S}e^{-k|{text{Im}}(z)|}log left|int (T_{z}f)g,mu _{2}right|alpha }right)leq left({frac {C_{p,q}|f|_{p}}{alpha }}right)^{q},} real interpolation theorems such as the Marcinkiewicz interpolation theorem are better-suited for them. Furthermore, a good number of important operators, such as the Hardy-Littlewood maximal operator, are only sublinear. This is not a hindrance to applying real interpolation methods, but complex interpolation methods are ill-equipped to handle non-linear operators. On the other hand, real interpolation methods, compared to complex interpolation methods, tend to produce worse estimates on the intermediate operator norms and do not behave as well off the diagonal in the Riesz diagram. The off-diagonal versions of the Marcinkiewicz interpolation theorem require the formalism of Lorentz spaces and do not necessarily produce norm estimates on the Lp-spaces.

Mityagin's theorem B. Mityagin extended the Riesz–Thorin theorem; this extension is formulated here in the special case of spaces of sequences with unconditional bases (cf. below).

Assume: {displaystyle |A|_{ell _{1}to ell _{1}},|A|_{ell _{infty }to ell _{infty }}leq M.} Then {displaystyle |A|_{Xto X}leq M} for any unconditional Banach space of sequences X, that is, for any {displaystyle (x_{i})in X} and any {displaystyle (varepsilon _{i})in {-1,1}^{infty }} , {displaystyle |(varepsilon _{i}x_{i})|_{X}=|(x_{i})|_{X}} .

The proof is based on the Krein–Milman theorem.

See also Marcinkiewicz interpolation theorem Interpolation space Notes ^ Stein and Weiss (1971) and Grafakos (2010) use operators on simple functions, and Muscalu and Schlag (2013) uses operators on generic dense subsets of the intersection Lp0 ∩ Lp1. In contrast, Duoanddikoetxea (2001), Tao (2010), and Stein and Shakarchi (2011) use the sumset formulation, which we adopt in this section. ^ Riesz (1927). The proof makes use of convexity results in the theory of bilinear forms. For this reason, many classical references such as Stein and Weiss (1971) refer to the Riesz–Thorin interpolation theorem as the Riesz convexity theorem. ^ Thorin (1948) ^ Bernard, Calista. "Interpolation theorems and applications" (PDF). ^ Stein (1956). As Charles Fefferman points out in his essay in Fefferman, Fefferman, Wainger (1995), the proof of Stein interpolation theorem is essentially that of the Riesz–Thorin theorem with the letter z added to the operator. To compensate for this, a stronger version of the Hadamard three-lines theorem, due to Isidore Isaac Hirschman, Jr., is used to establish the desired bounds. See Stein and Weiss (1971) for a detailed proof, and a blog post of Tao for a high-level exposition of the theorem. ^ Fefferman and Stein (1972) ^ Elias Stein is quoted for saying that interesting operators in harmonic analysis are rarely bounded on L1 and L∞. References Dunford, N.; Schwartz, J.T. (1958), Linear operators, Parts I and II, Wiley-Interscience. Fefferman, Charles; Stein, Elias M. (1972), " {displaystyle H^{p}} Spaces of Several variables", Acta Mathematica, 129: 137–193, doi:10.1007/bf02392215 Glazman, I.M.; Lyubich, Yu.I. (1974), Finite-dimensional linear analysis: a systematic presentation in problem form, Cambridge, Mass.: The M.I.T. Press. Translated from the Russian and edited by G. P. Barker and G. Kuerti. Hörmander, L. (1983), The analysis of linear partial differential operators I, Grundl. Math. Wissenschaft., vol. 256, Springer, doi:10.1007/978-3-642-96750-4, ISBN 3-540-12104-8, MR 0717035. Mitjagin [Mityagin], B.S. (1965), "An interpolation theorem for modular spaces (Russian)", Mat. Sb., New Series, 66 (108): 473–482. Thorin, G. O. (1948), "Convexity theorems generalizing those of M. Riesz and Hadamard with some applications", Comm. Sem. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.], 9: 1–58, MR 0025529 Riesz, Marcel (1927), "Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires", Acta Mathematica, 49 (3–4): 465–497, doi:10.1007/bf02564121 Stein, Elias M. (1956), "Interpolation of Linear Operators", Trans. Amer. Math. Soc., 83 (2): 482–492, doi:10.1090/s0002-9947-1956-0082586-0 Stein, Elias M.; Shakarchi, Rami (2011), Functional Analysis: Introduction to Further Topics in Analysis, Princeton University Press Stein, Elias M.; Weiss, Guido (1971), Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press External links "Riesz convexity theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994] 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: Theorems involving convexityTheorems in harmonic analysisTheorems in Fourier analysisTheorems in functional analysisBanach spacesOperator theory

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