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Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/ Volume 5, Issue 3, Article 68, 2004 APPLICATION LEINDLER SPACES TO THE REAL INTERPOLATION METHOD VADIM KUKLIN D EPARTMENT OF M ATHEMATICS VORONEZH S TATE U NIVERSITY VORONEZH , 394693, RUSSIA . [email protected] Received 30 March, 2004; accepted 21 April, 2004 Communicated by H. Bor A BSTRACT. The paper is devoted to the important section the Fourier analysis in one variable (AMS subject classification 42A16). In this paper we introduce Leindler space of Fourier - Haar coefficients, so we generalize [2, Theorem 7.a.12] and application to the real method spaces. Key words and phrases: Leindler sequence space of Fourier - Haar coefficients, Lorentz space, Haar functions, real method spaces. 2000 Mathematics Subject Classification. 26D15, 40A05, 42A16, 40A99, 46E30, 47A30, 47A63. 1. I NTRODUCTION A Banach space E[0, 1] is said to be a rearrangement invariant space (r.i) provided f ∗ (t) ≤ g (t) for any t ∈ [0, 1] and g ∈ E implies that f ∈ E and kf kE ≤ kgkE , where g ∗ (t) is the rearrangement of |g(t)|. Denote by ϕE the fundamental function of (r.i) space E such that ϕE = kκe (t)k (see, [1, p. 137]). Given τ > 0, the dilation operator στ f (t) = f ( τt ), t ∈ [0, 1] and min(1, τ ) ≤ kστ kE→E ≤ max(1, τ ). Denote by ∗ ln kστ kE→E , τ →+0 ln τ αE = lim ln kστ kE→E τ →∞ ln τ βE = lim the Boyd indices of E. In general, 0 ≤ αE ≤ βE ≤ 1. R1 The associated space to E 0 is the space of all measurable functions f (t) such that 0 f (t)g(t)dt < ∞ for every g(t) ∈ E endowed with the norm Z 1 kf (t)kE 0 = sup f (t)g(t)dt. kg(t)kE ≤1 0 For every (r.i) space E space the embedding E ⊂ E 00 is isometric. If an (r.i) space E is separable, then (χkn ) is everywhere dense in E. ISSN (electronic): 1443-5756 c 2004 Victoria University. All rights reserved. 071-04 2 VADIM K UKLIN Denote by Ψ the set of increasing concave functions ψ(t) ≥ 0 on [0, 1] with ψ(0) = 0. Then each function ψ(t) ∈ Ψ generates the Lorentz space Λ(ψ) endowed with the norm Z 1 kg(t)kΛ(ψ) = g ∗ (t)dϕ(t) < ∞. 0 For every (r.i) space E space the embedding E ⊂ E 00 is isometric. Let be Ω the set of (n, k) such that 1 ≤ k ≤ 2n , n ∈ N∪ {0}. Put χ00 ≡ 1. If (n, k) ∈ Ω, k−1 < t < 2k−1 , 2n 2n+1 1, k 2k−1 k χn (t) = −1, 2n+1 < t < 2n , 0, for any t ∈ k−1 , 2kn . 2n The set of functions χkn is called the Haar functions, normalized in L∞ [0, 1] (see [2, p. k 15-18]). If an (r.i) space E is separable, then χn everywhere dense in E. Given f (t) ∈ L1 . The Fourier-Haar coefficients are given by Z 1 n f (t)χkn (t)dt. cn,k (f ) = 2 0 P k Put g(t) = cn,k χn for any g ∈ L1 [0, 1]. (n,k)∈Ω A Banach sequence space E is said to be a rearrangement invariant space (r.i) provided that k(an )kE ≤ k(a∗n )kE ,where a∗n the rearrangement of sequence (an )n∈N i.e. ( ) a∗n = inf sup |ai | : J ⊂ N, card(J) <n . i∈N\J It is maximal if the unit ball BE is closed in the poinwise convergence topology inducted by the space A of all real sequences. This condition is equivalent to E # = E 0 , where ) ( ∞ X E # = (bn )n∈N ⊂ A : |an bn | < ∞, (an )n∈N ⊂ E n=1 # is the Kother dual of E. Clearly, E is a maximal Banach space under the norm (∞ ) X k(bn )kE # = sup |an bn | < ∞ : k(an )kE ≤ 1 . n=1 (λn )∞ n=1 Denoting λ = be a sequence of positive numbers. We shall use the following notation (see [3, pp. 517-518]): ∞ ∞ X X (c) Λn = λk and Λn = λk Λ−c k , (Λ1 < ∞); k=n k=n furthermore, for c ≥ 0. By analogy with [3, pp. 517-518] we define Leindler sequence space of Fourier-Haar coefficients, for p > 0, c ≥ 0, with the norm: !p ! p1 ∞ 2n X X < ∞. k(cn,k )∞ λn Λ−c |cn,k | 2−n n=1 kλ(p,c) = n n=1 k=1 ∞ Why do we consider the sequence (cn,k )n=1 ? The answer to this n Theorem 7.a.3], i.e. g ∈ Λ(ψ) ⇔ sup 2− p cn,1 (g) < ∞. Here, as 0<t≤1 question follows from [2, usual, X ,→ Y stands for the continuous embedding, that is, kgkY ≤ C kgkX for some C > 0 and every g∈X. The sign ∼ = means that these spaces coincide to with within equivalence of norms. J. Inequal. Pure and Appl. Math., 5(3) Art. 68, 2004 http://jipam.vu.edu.au/ A PPLICATION L EINDLER S PACES 3 2. P ROBLEMS By [2, Theorem 7.a.12] for p = 2 we have ! 12 ∞ 2n X X X cn,k χkn 2−n c2n,k . = (n,k)∈Ω n=1 k=1 L2 If for (cn,k )∞ we put p = 2, c = 0, λn = 1, then n=1 λ(p,c) X ∞ k k(cn,k )n=1 kλ(2,0) ≤ M cn,k χn . (n,k)∈Ω L2 Denote by X T cn,k χkn = (cn,k )(n,k)∈Ω . (n,k)∈Ω Hence by [1, Chapter 2, §5, Theorem 5.5] we have the operator bounded from Λ(ψ) into λ(2, 0). In general we consider Problem 1. Let 0 < c < 1, 1 < p < ∞. Whether there exists a operator T bounded from Λ(ψ) into λ(p, c)? Let (E0 , E1 ) be a compatible pair of Banach spaces. We recall K(t, g) = K(t, g, E0 , E1 ) = inf g=g0 +g1 ,gi ∈Ei (i=0,1) kg0 kE0 + t kg1 kE1 . Here g ∈ E0 + E1 , 0 < t ≤ 1. If 0 < θ < 1, 1 ≤ p ≤ ∞, then the spaces (E0 , E1 )θ,p endowed with the norm p1 Z 1 −θ p dt < ∞, iff p < ∞ (K(t, g)t ) kgk(E0 ,E1 )θ,p = t 0 and kgk(E0 ,E1 )θ,p = sup K(t, g)t−θ < ∞, iff p = ∞ 0<t<1 are called real method spaces. Let 0 ≤ α0 < α1 < 1, ψ0 (t) = tα0 , ψ1, (t) = tα1 , 0 < θ < 1, ∼ 1 ≤ p ≤ ∞, ψ(t) = t . ψ(t) In [5, §2, p. 174] the problem was solved: when does the equivalence ∼ ∼ ∼ (Λ(ψ0 ), Λ(ψ1 ))θ,p = M (ψ0 ), M (ψ1 ) . θ,p holds? ∼ ∼ We consider the embedding (Λ(ψ0 ), Λ(ψ1 ))θ,p ,→ M (ψ0 ), M (ψ1 ) . Let 0 ≤ α0 = α1 < θ,p 1, ψ(t) = tα , 0 < θ < 1, 1 < p ≤ ∞. Problem 2. Whether there exists 0 < c < 1, 1 < p < ∞ such that T : (Λ(ψ), Λ(ψ))θ,p → (λ(p, c), λ(p, c))θ,p ? In this article we consider Leindler sequence space of Fourier-Haar coefficients λ(p, c). To prove our theorems we need the following Theorem 1 (see [4]). J. Inequal. Pure and Appl. Math., 5(3) Art. 68, 2004 http://jipam.vu.edu.au/ 4 VADIM K UKLIN Theorem 1. If p > 1, 0 ≤ c < 1, then ∞ X n X λn Λ−c n n=1 !p |ak | ≤ k=1 p 1−c p X ∞ p−c p λ1−p n Λn an . n=1 The constant is best possible. 3. L EMMAS AND T HEOREMS n Lemma 3.1. Let 1 < p < ∞, 0 ≤ c < 1 and sup 2− p cn,1 (g) < ∞. Then the operator T is 0<t≤1 bounded from Λ(ψ) into λ(p, c). Proof. By [2, Theorem 4.a.1] for 1 < p < ∞ we have p p p Z 1 X Z 1 X Z 1 X 2n ∞ X 2n k k p k dt, cn,k χn dt ≤ cn,k χn dt ≤ 2 c χ n,k n 0 0 0 k=1 n=l k=1 (n,k)∈Ω where n ≤ l ≤ ∞. On the other hand, p Z 1 Z 1 2n 2n 2n X X X p k −n −n cn,k χn dt = 2 |cn,k | dt = 2 |cn,k |p . 0 0 k=1 k=1 Lp Therefore, ! p1 n 2−n 2 X k=1 |cn,k |p ≤ 2 kgkLp . k=1 From the above and [1, Chapter 2, §5, Theorem 5.5] we get ! p1 ∞ X k(cn,k )∞ λn Λ−c kgkΛ(ψ) . n=1 kλ(p,c) ≤ 2 n n=1 Hence the operator T is bounded from Λ(ψ) into λ(p, c). This proves the assertion. Remark 3.2. In the Lemma 3.1 the condition 0 < c < 1, 1 < p < ∞ is necessary for the operator T. We shall formulate the sufficient condition of boundedness of the operator T from Λ(ψ) into λ(p, c). n Theorem 3.3. Let 0 ≤ c < 1, sup 2− p cn,1 (g) < ∞. For of boundedness the operator T 0<t≤1 bounded from Λ(ψ) into λ(p, c) is sufficient that 2 ≤ p < ∞. Proof. By Theorem 1 and Hölder’s inequality we have k(cn,k )∞ n=1 kλ(p,c) ≤ ∞ X p1 1 X −1 1− c p λnp Λn p |cn,k |p 2−n . 1 − c n=1 (n,k)∈Ω Now using [2, Theorem 7.a.12 (c. 2)] and [1, Chapter 2, §5, Theorem 5.5] we obtain that ∞ 1 c X X −1 1− p p p k (cn,k )∞ ≤ λ Λ c χ . n n n,k n n=1 λ(p,c) 1 − c n=1 (n,k)∈Ω Λ(ψ) This finishes the proof. J. Inequal. Pure and Appl. Math., 5(3) Art. 68, 2004 http://jipam.vu.edu.au/ A PPLICATION L EINDLER S PACES 5 Remark 3.4. If 1 ≤ p < 2, 0 < c < 1, then by [2, Theorem 7.a.12 (c. 1)] T : Λ(ψ) 9 λ(p, c). n Theorem 3.5. Let 0 ≤ c < 1, 2 ≤ p ≤ ∞, sup 2− p cn,1 (g) < ∞. Then 0<t≤1 T : (Λ(ψ), Λ(ψ))θ,p → (λ(p, c), λ(p, c))θ,p . Proof. Clearly, by Hölder’s inequality the estimate ∞ p X p1 −1 1− pc (cn,k )∞ (cn,k )∞ Λ ≤ λ n n n=1 `2 n=1 λ(p,c) 1 − c n=1 holds. It is known that the operator T is bounded from L2 into `2 . Then from the above and [1, Chapter 2, §5, Theorem 5.5] we obtain K(t, (cn,k )∞ n=1 , λ(p, c), λ(p, c)) ≤ K(t, g, Λ(ψ), Λ(ψ)). Hence T : (Λ(ψ), Λ(ψ))θ,p → (λ(p, c), λ(p, c))θ,p . This completes the proof. R EFERENCES [1] S.G. KREIN, YU. I. PETUNIN AND E.M. SEMENOV, Interpolation linear operators, Nauka, Moscow, 1978, in Russian; Math. Mono., Amer. Math. Soc., Providence, RI, 1982, English translation. [2] I. NOVIKOV AND E. SEMENOV, Haar Series and Linear Operators, Mathematics and Its Applications, Kluwer Acad. Publ., 1997. [3] L. LEINDLER, Hardy - Bennett - Type Theorems, Math. Ineq. and Appl., 4 (1998), 517–526. [4] L. LEINDLER, Two theorems of Hardy-Bennett - type, Acta Math. Hung., 79(4) (1998), 341–350. [5] E. SEMENOV, On the stability of the real interpolation method in the class of rearrangement invariant spaces, Israel Math. Proceedings, 13 (1999), 172–182. J. Inequal. Pure and Appl. Math., 5(3) Art. 68, 2004 http://jipam.vu.edu.au/