EXTREMAL SEQUENCES FOR THE BELLMAN FUNCTION OF THE DYADIC MAXIMAL OPERATOR 3 1 ELEFTHERIOS N. NIKOLIDAKIS 0 2 n Abstract: We give a characterization of the extremal sequences for the Bellman func- a J tion of the dyadic maximal operator. In fact we prove that they behave approximately 0 like eigenfunctions of this operator for a specific eigenvalue. 2 ] 1. Introduction A F The dyadic maximal operator on Rn is defined by . h 1 at (1.1) Mφ(x) = sup |φ(u)|du :x ∈ Q, Q ⊆ Rn is a dyadic cube m (cid:26)|Q| ZQ (cid:27) [ for every φ ∈ L1 (Rn) where |·| is the Lesbesgue measure on Rn and the dyadic cubes loc 2 are those formed by the grids 2−NZn, N = 0,1,2,... . v 8 As it is well known it satisfies the following weak type (1,1) inequality: 9 1 8 (1.2) |{x ∈ Rn : M φ(x) > λ}| ≤ |φ(u)|du, d 2 λ . Z{Mdφ>λ} 1 for every φ ∈ L1(Rn) and λ > 0. 0 3 From (1.2) it is easy to prove the following Lp-inequality 1 : p v (1.3) kM φk ≤ kϕk , d p p p−1 i X for every p > 1 and φ∈ Lp(Rn). r a It is easy to see that (1.2) is best possible, while (1.3) is sharp as it can be seen in [10]. (See [1] and [2] for general martingales). One way of studying the dyadic maximal operator is to find certain refinements of inequalities satisfied by it such as (1.2) and (1.3). Certain refinements of (1.2) have been done in [5], [6] and [7]. Refinements of (1.3) have been given in [3] or even more general in [4]. In studying (1.3) an interesting function has been introduced which is the following 1 (1.4) BQ(f,F) = sup (M φ)p :φ ≥ 0, Av (φ) = f, Av (φp)= F p |Q| d Q Q (cid:26) ZQ (cid:27) where Q is a fixed dyadic cube in Rn, and φ∈ Lp(Q) 1 Av (h) = |h(y)|du. Q |Q| ZQ ThisistheBellmanfunctionoftwovariablesassociatedtothedyadicmaximaloperator. 1 2 ELEFTHERIOSN.NIKOLIDAKIS The function given in (1.4) has been explicitly computed. Actually this is done in a much more general setting of a non-atomic probability measure space (X,µ) where the dyadic sets are now given in a family of sets T (called tree), which satisfies conditions similar to those that are satisfied by the dyadic cubes on [0,1]n. Then the associated dyadic maximal operator M is defined by T 1 (1.5) M φ(x) = sup |φ|dµ : x ∈ I ∈ T , T µ(I) (cid:26) ZI (cid:27) for φ ∈ L1(X,µ). Then the Bellman function of two variables for p > 1 associated to M is given by T (1.6) B (f,F) = sup (M φ)pdµ : φ≥ 0, φdµ = f, φpdµ = F , p T (cid:26)ZX ZX ZX (cid:27) where 0 < fp ≤ F. In[3](1.6)hasbeenfoundtobeequaltoB (f,F) = Fω (fp/F)p whereω : [0,1] −→ p p p p p 1, is the inverse function H−1 of H defined for z ∈ 1, by H (z) = p−1 p p p−1 p h−(p−1)zip +pzp−1. h i Recently L. Slavin and A. Stokolos [9], linked the Bellman function computation to solving certain PDEs of the Monge-Amp`ere type, and in this way they obtained an alternative proof of the results in [3]. In this paper we study those sequences of functions (φ ) that are extremal for the n n Bellman function (1.6). That is φ : (X,µ) → R+ must satisfy n φ dµ = f, φpdµ = F and lim (M φ )pdµ = Fω (fp/F)p. n n T n p n ZX ZX ZX In[8]itisproved thatevery suchextremalsequencemustsatisfy aself similarproperty, namely that for every I ∈ T we have that 1 1 (1.7) lim φ dµ = f and lim φpdµ = F, n µ(I) n µ(I) n ZI ZI This gives as an immediate result that extremal functions do not exist for the Bellman function. The core of this paper is the following: We prove a characterization of these extremal sequences of functions. In fact we prove p Theorem:Let(φ ) be such that φ dµ = f and φ dµ = F Then it is extremal n n X n X n for (1.6), if and only if lim |MRTφn−cφn|pdµ = 0R, for c= ωp(fp/F). (cid:3) n ZX We end this section with the following question. We ask if there is essentially unique extremal sequence for (1.6). By this we mean the following: Does lim |φ −g |pdµ = 0 whenever (φ ) and (g ) are extremal n n n n n n n ZX sequences for (1.6)?We hope to answer to this question in the near future. EXTREMAL SEQUENCES FOR THE BELLMAN FUNCTION OF THE DYADIC MAXIMAL OPERATOR3 2. Preliminaries Let (X,µ) be a non-atomic probability measure space. We give the following from [3]. Definition 2.1. A set T of measurable subsets of X will be called a tree if the following are satisfied i) X ∈T and for every I ∈ T, µ(I) > 0. ii) For every I ∈ T there corresponds a finite or countable subset C(I) of T con- taining at least two elements such that a) the elements of C(I) are pairwise disjoint subsets of I b) I = ∪C(I). iii) T = T , where T = {X} and T = C(I). (m) (0) (m+1) I∈T(m) m≥0 iv) The foSllowing holds S lim sup µ(I) = 0. m→∞I∈T(m) (cid:3) We state now the following lemma given in [3]. Lemma 2.1. For every I ∈ T and every a ∈ (0,1) there exists a subfamily F(I) ⊆ T consisting of pairwise disjoint subsets of I such that µ J = µ(J) = (1−a)µ(I). (cid:18)J∈F(I) (cid:19) J∈F(I) [ X (cid:3) Now given a tree T we define the maximal operator associated to it as follows 1 M φ(x) = sup |φ|dµ : x ∈ I ∈T , for every φ∈ L1(X,µ). T µ(I) (cid:26) ZI (cid:27) From [3] we recall the following Theorem 2.1. The following holds: sup (M φ)pdµ : φ ≥ 0, φdµ = f, φpdµ = F = Fω (fp/F)p, T p (cid:26) ZX ZX (cid:27) for every f,F such that 0< fp ≤F. (cid:3) At last we give the following Definition 2.2. Let (φ ) be a sequence of µ-measurable nonnegative functions defined n n on X, p > 1 and 0 < fp ≤ F. Then (φ ) is called (p,f,F) extremal or simply extremal n n if the following hold: φ dµ = f, φpdµ = F and lim (M φ )pdµ = Fω (fp/F)p. n n T n p n ZX ZX ZX (cid:3) 4 ELEFTHERIOSN.NIKOLIDAKIS 3. Characterization of the extremal sequences For the proof of the Theorem 2.1 an effective linearization for the operator M was T introduced valid for certain functions φ. We describe it. 1 For φ∈ L1(X,µ) nonnegative function and I ∈ T we define Av (φ) = φdµ. I µ(I) I We will say that φ is T-good if the set R A = {x ∈ X : M φ(x) > Av (φ) for all I ∈ T such that x ∈ I} φ T I has µ-measure zero. Let now φ be T-good and x ∈ X rA . φ We define I (x) to be the largest in the nonempty set φ {I ∈ T : x ∈ I and M φ(x) = Av (φ)}. T I Now given I ∈ T let A(φ,I) = {x ∈ X rA : I (x) =I} ⊆ I and φ φ S = {I ∈T : µ(A(φ,I)) > 0}∪{X}. φ Obviously, M φ = Av (φ)J , µ-a.e. where J is the characteristic function T I A(φ,I) E I∈Sφ of E. P We define also the following correspondence I → I∗ by: I∗ is the smallest element of {J ∈ S : I ( J}. It is defined for every I ∈ S except X. It is obvious that the φ φ A(φ,I)’sarepairwisedisjointandthatµ (A(φ,I)) = 0,sothat A(φ,I) ≈ X, (cid:16)I∈/Sφ (cid:17) I∈Sφ where by A ≈ B we mean that µ(ArB)=Sµ(B rA) = 0. S Now the following is true, obtained by [3]. Lemma 3.1. Let φ be T-good i) If I, J ∈ S then either A(φ,J)∩I = ∅ or J ⊆ I. φ ii) If I ∈ S then there exists J ∈ C(I) such that J ∈/ S . φ φ iii) For every I ∈ S we have that φ I ≈ A(φ,J). J[∈Sφ J⊆I iv) For every I ∈ S we have that φ A(φ,I) = I r J, so that J[∈Sφ J∗∈I µ(A(φ,I)) = µ(I)− µ(J). JX∈Sφ (cid:3) J∗=I EXTREMAL SEQUENCES FOR THE BELLMAN FUNCTION OF THE DYADIC MAXIMAL OPERATOR5 From the above we see that 1 Av (φ) = φdµ =:y I I µ(I) JX∈Sφ ZA(φ,J) J⊆I I ∈ S , we define also φ −1+1 χ = a p φdµ, forI ∈ S , where a =µ(A(φ,I)). I I φ I ZA(φ,I) We prove now the following Theorem 3.1. Let φ be T-good function such that φdµ = f. Let also B = {I } be a j X family of pairwise disjoint elements of S , which isRmaximal on S under ⊆ relation. φ φ That is if I ∈ S then I ∩(∪I ) 6=∅. (For example B = T satisfies this property). φ j (1) Then the following inequality holds: fp− µ(I )yp φp ≥ j j Ij + (p−1)β (M φ)pdµ (βP+1)p−1 (β +1)p T ZXrSIj ZXrSIj j j for every β > 0, where y = Av (φ). Ij Ij Proof. We follow [3]. We have that (3.1) φp = φpdµ, ZXr∪Ij I)pXiece(B) ZA(φ,I) I∈Sφ where by writing I ) piece(B) we mean that I ) I for some j. Of course (3.1) is true j since X r I ≈ A(φ,I) in view of the maximality of B and Lemma 3.1. j j J∈Sφ S I)piSece(B) Now from (3.1) we have by Holder’s inequality that p φdµ (3.2) φpdµ ≥ xp = (cid:16)A(φR,I) (cid:17) . I p−1 ZXrSj Ij IX∈Sφ IX∈Sφ aI I)piece(B) I)piece(B) Now, it is true that µ(I)y = µ(J)y + φdµ, for every I ∈ S . I J φ JX∈Sφ ZA(φ,I) J∗=I So by using Holder’s inequality in the form (λ +···+λ )p λp λp λp 1 m ≤ 1 + 2 +···+ n , we have (σ1+···+σm)p−1 σ1p−1 σ2p−1 σmp−1 6 ELEFTHERIOSN.NIKOLIDAKIS p µ(I)y − µ(J)y I J φp ≥ (cid:16) J∗=I (cid:17) P p−1 ZXr∪Ij IX∈Sφ µ(I)− µ(J) I)piece(B) J∗=I (cid:16) (cid:17) (µ(I)y )Pp (µ(J)y )p I J (3.3) ≥ − , (τ µ(I))p−1 ((β +1)µ(J))p−1 IX∈Sφ (cid:26) I JX∗=I (cid:27) I)piece(B) a I where τ = (β +1)−βρ , ρ = , β > 0. I I I µ(I) So by (3.3) we have because of the maximality of B that: p p µ(I)y µ(I)y (3.4) φp ≥ I − I , ZXrSj Ij IX∈Sφ τIp−1 X(∗) (β +1)p−1 I)piece(B) where the summation in (∗) is extended to: (a) I ∈ S : I ) piece(B) with I 6= X, or (b)I ∈ S is a piece of B (I = I , for some φ φ j j). So we can write: p y 1 1 φpdµ ≥ x + ZXr∪Ij τxp−1 IX∈Sφ ρI(cid:18)τIp−1 I 6=X I )piece(B) 1 1 p p (3.5) − a y − µ(I )y . (β +1)p−1 I I (β +1)p−1 j Ij (cid:19) j X It is true now that 1 1 (p−1)βx (3.6) − ≥ , (β +1−βx)p−1 (β +1)p−1 (β+1)p for any x ∈ [0,1]. Then (3.5) becomes p y (p−1)β 1 φpdµ ≥ x + a yp− µ(I )yp ZXr∪Ij τxp−1 (β +1)p IX6=X I I (β +1)p−1 Xj j Ij I ∈S φ I )piece(B) 1 (p−1)βρ (p−1)β = − x fp+ a yp ((β +1)−βρ )p−1 (β+1)p (β +1)p I I (cid:20) x (cid:21) IX∈Sφ I)piece(B) 1 p (3.7) − µ(I )y , (β +1)p−1 j Ij j X EXTREMAL SEQUENCES FOR THE BELLMAN FUNCTION OF THE DYADIC MAXIMAL OPERATOR7 But a yp = (M φ)pdµ, so in view of (3.6) we must have that I I T I∈Sφ Xr∪Ij I)pPiece(B) P φp ≥ fp− µ(Ij)yIpj + (p−1)β (M φ)pdµ, (β +1)p−1 β +1)p T ZXr∪Ij P ZXr∪Ij for every β > 0, and the proof of the theorem is now complete. (cid:3) In the same lines as above we can prove: Theorem 3.2. Let φ be T-good and A= {I } be a pairwise disjoint family of elements j of S . Then for every β > 0 we have that: φ p φpdµ ≥ µ(Ij)yIj + (p−1)β (M φ)pdµ. (β +1)p−1 (β +1)p T ZSIj P ZSIj j j (cid:3) We have now the following: Corollary 3.1. φ be a T-good and A = {I } be a pairwise disjoint family of elements j of S . Then for every β > 0 φ fp− µ(I )yp φpdµ ≥ j j Ij + (p−1)β (M φ)pdµ, (βP+1)p−1 (β +1)p T ZXrSIj ZXrSIj j j where f = φdµ. X R Proof. Obvious, since there exist families B,Γ of pairwise disjoint elements of S φ with B as in the statement of Theorem 3.1, and B = I′, Γ = J with I′ = j i j j i j S S S I ∪ J with the additional property that I disjoint to J for every j,i. j i j i j i (cid:16)SApp(cid:17)lyin(cid:16)gSThe(cid:17)orem 3.1 for B and 3.2 for Γ we obtain Corollary 3.1. (cid:3) We have now the following Theorem 3.3. Let (φ ) an extremal sequence consisting of T-good functions. Con- n n sider for every n ∈ N a pairwise disjoint family A = {In} of elements of S such n j φn that the following limit exists p lim µ(I)y , where y = Av (ϕ ), I ∈ A . I,n I,n I n n n IX∈An Then lim (Mφ )pdµ = ω (fp/F)plim φpdµ n p n n n Z∪An Z∪An meaning that if one of the limits on the above relation exists then the other also does and we have the stated equality. 8 ELEFTHERIOSN.NIKOLIDAKIS Proof. In view of Theorem 3.2 and Corollary 3.1 we have that for every n ∈ N fp− µ(I)yp I,n (p−1)β (3.8) φpdµ ≥ I∈An + (M φ )pdµ, and n (βP+1)p−1 (β +1)p T n ZXr∪An ZXr∪An p µ(I)y (p−1)β (3.9) φpdµ ≥ I∈An I,n + (M φ )pdµ, n (β+1)p−1 (β+1)p T n Z∪An P Z∪An or every β > 0 and n∈ N. Summing relations (3.8) and (3.9) for every n∈ N we obtain fp (p−1)β (3.10) F = φpdµ ≥ + (M φ )pdµ, n (β +1)p−1 (β +1)p T n ZX ZX Since (φ ) is extremal we have equality in the limit in (3.10) for β = ω (fp/F)−1 n n p (see [3]). So we must have equality on (3.8) and (3.9) in the limit for this value of β. p Suppose now that h = µ(I)y and that h → h (3.9) now can be written in n I,n n I∈An the form P (β +1)p−1 φp −h n n 1 (3.11) (M φ )pdµ ≤ 1+ ∪An , T n R β p−1 Z∪An (cid:18) (cid:19) (see [3], relations (4.24) and (4.25)), for every β > 0 The right hand side of (3.11), n ∈ N, is minimized for β = β = ω h φp −1. n p n n (cid:16) ∪An (cid:17) Since, we have equality in the limit in(cid:14)(3R.11) we must have that h fp n (3.12) lim = , p n φndµ F ∪An R Thus (3.12) and (3.11) now give lim (M φ )pdµ = ω (fp/F)plim φp T n p n n n Z∪An Z∪An in the sense stated above. (cid:3) Thetheorem justproved is thecore forprovingthecharacterization weneed foreach extremal sequence. We just need some Lemmas that we are going to state and prove below. We give now some notation. Let φ be T-good. For each I ∈ S we consider the set A = A(φ,I) is a union φ I of elements of T, because of the definition of tree T and Lemma 3.1 iv). Using now Lemma 2.1 we construct for each a ∈ (0,1) a pairwise disjoint family AI of elements φ of T and subset (3.13) µ(J) = aµ(A ). I JX∈AIφ EXTREMAL SEQUENCES FOR THE BELLMAN FUNCTION OF THE DYADIC MAXIMAL OPERATOR9 We define the following function g : X → R+ in the following way. For each I ∈ S φ φ we define: g := cφ, on ∪AI (3.14) φ I φ := 0, on A r∪AI I φ such that φ φ g dµ =c γ = φdµ and AI φ I I (3.15) AI , whereγφ = µ(∪AI) = aµ(A ). R gpdµ = (cφ)pγφ =R φpdµ,  I φ I φ I I   AI AI R R It is easy to see that such chases of γIφ and cφI, for every I ∈ Sp are possible. In fact (3.15) give p 1/(p−1) φdµ γIφ = (cid:16)ARIφpdµ(cid:17)  ≤ µ(AI), by Holder’s inequality  AI   R    so we just need to set p 1/(p−1) φdµ a = 1 (cid:16)ARI (cid:17)  . µ(A ) φpdµ I  AI   R  Then, if AI is such that (3.13) is satisfied for this a, by setting γφ = aµ(A ) and φ I I φdµ cφ = AI , we have that (3.15) are valid. I R φ γ I Since A ≈ X, g is well defined on X. I φ I∈Sφ It is obSvious that φdµ = f and gpdµ = F It is easy also to see that for every φ X X I ∈ S it holds R R φ µ({g = 0}∩A ) ≥ µ({φ = 0}∩A ) φ I I Therefore by 3.1, µ({φ = 0}) ≤ µ({g = 0}). φ Let now (φ ) be an extremal sequence consisting of T-good functions and let g = n n n g . We prove now the following φn Lemma3.2. Foranextremal(φ ) sequenceofT-goodfunctionswehavethatlim ({φ = n n n n 0}) = 0. Proof. Fix n ∈ N and let φ = φ and g = g and S = S the respective subtree of n φ φn φ φ. We consider two cases: i) p ≥ 2 φpdµ We set P = AI , for every I ∈ S . I R φ a I 10 ELEFTHERIOSN.NIKOLIDAKIS We obviously have a P = F, we consider now the sum Σ = γ P , where I I φ I I I∈Sφ I∈Sφ φ P P γ = γ as above. Then I I φp γ ·cp cp γ2ap−2cp Σ = γ AI = γ I I = γ2 I = I I I φ I RaI I aI IaI ap−1 IX∈Sφ IX∈Sφ IX∈Sφ IX∈Sφ I p φ p≥≥2 (γIcI)p = (cid:16)AI (cid:17) . p−1 Rp−1 a a IX∈Sφ I IX∈Sφ I From the first inequality in (4.20) in [3], and since φ is extremal we have that the last n sum in the last inequality tends to F, as φ moves along (φ ) . We then write n n (3.16) γ P ≈F I I IX∈Sp since Σ ≤ F. Consider now for every R > 0 and every φ the following set φ S = ∪{A = A(φ,I) : I ∈ S , P < R}. φ,R I φ I For every I ∈ S such that P < R we have that φp < Ra . Summing for all such I φ I I AI we obtain R (3.17) φp <Rµ(S ). φ,R ZSφ,R Additionally we have that (3.18) (cid:12) a P −F(cid:12) = φpdµ, and I I (cid:12) (cid:12) (cid:12)(cid:12)IX∈Sφ (cid:12)(cid:12) ZSφ,R (cid:12)PI≥R (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (3.19) γ P ≤ a P ≤ φpdµ. I I I I IX∈Sφ IX∈Sφ ZSφ,R PI<R PI<R From (3.15) and (3.19) we have that (3.20) limsup(cid:12) γ P −F(cid:12) ≤ lim φp, I I φ (cid:12)(cid:12)(cid:12)IX∈Sφ (cid:12)(cid:12)(cid:12) φ ZSφ,R (cid:12)PI≥R (cid:12) (cid:12) (cid:12) where we have supposed tha(cid:12)t the last limit(cid:12)exists (we just pass to a subsequence of (cid:12) (cid:12) (φ ) . From (3.18) and (3.20) we conclude that n n (3.21) limsup (a −γ )P ≤ 2lim φp. I I I φ IX∈Sφ φ ZSφ,R PI≥R