INTEGRABILITY CONDITIONS PERTAINING TO ORLICZ SPACE

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1 INTEGRABILITY CONDITIONS PERTAINING TO ORLICZ SPACE L. LEINDLER University of Szeged, Bolyai Institute Aradi vértanúk tere 1, 6720 Szeged, Hungary Received: 04 September, 2006 Accepted: 01 June, 2007 Communicated by: S.S. Dragomir 2000 AMS Sub. Class.: 42A32, 46E30. Key words: Trigonometric series, Integrability, Orlicz space. Page 1 of 14 Abstract: Acknowledgements: Recently S. Tikhonov proved two theorems on the integrability of sine and cosine series with coefficients from the R + 0 BV S class. These results are extended such that the R + 0 BV S class is replaced by the MRBV S class. The author was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T

2 1 Introduction 3 2 New Result 5 3 Notions and Notations 6 4 Lemmas 8 5 Proof of Theorem Page 2 of 14

3 1. Introduction There are many known and classical theorems pertaining to the integrability of formal sine and cosine series 1.1 gx := λ n sin nx, and 1.2 fx := λ n cos nx. We do not recall such theorems because a nice short survey of these results with references can be found in a recent paper of S. Tikhonov [3], where he proved two theorems providing sufficient conditions of belonging of fx and gx to Orlicz spaces. In his theorems the sequence of the coefficients λ n belongs to the class of sequences of rest bounded variation. For notions and notations, please, consult the third section. In the present paper we shall verify analogous results assuming only that the sequence λ := {λ n } is a sequence of mean rest bounded variation. We emphasize that the latter sequences may have many zero terms, while the previous ones have no zero term. Tikhonov s theorems read as follows: Theorem 1.1. Let Φx p, 0 0 p. If {λ n } R + 0 BV S, and the sequence {γ n } is such that {γ n n 1+ε } is almost decreasing for some ε > 0, then 1.3 γ n n 2 Φn λ n < ψx LΦ, γ, Page 3 of 14

4 where a function ψx is either a sine or cosine series. Theorem 1.2. Let Φx p, q 0 q p. If {λ n } R + 0 BV S, and the sequence {γ n } is such that {γ n n 1+q+ε } is almost decreasing for some ε > 0, then 1.4 γ n n 2+q Φn2 λ n < gx LΦ, γ. Page 4 of 14

5 2. New Result Now, we formulate our result in a terse form. Theorem 2.1. Theorems 1.1 and 1.2 can be improved when the condition {λ n } R 0 + BV S is replaced by the assumption {λ n } MRBV S. Furthermore the conditions of 1.3 and 1.4 may be modified as follows: 2.1 and 2.2 respectively. γ n n 2 Φ γ n n 2+q Φ 2n 1 ν=n n 2n 1 ν=n λ ν < ψx LΦ, γ, λ ν < gx LΦ, γ, Remark 1. It is easy to see that if {λ n } R + 0 BV S also holds, then 2n 1 ν=n λ ν n λ n, that is, our assumptions are not worse than 1.3 and 1.4. Page 5 of 14

6 3. Notions and Notations A null-sequence c := {c n } c n 0 of positive numbers satisfying the inequalities c n Kcc m, c n := c n c n+1, m = 1, 2,... n=m with a constant Kc > 0 is said to be a sequence of rest bounded variation, in brief, c R + 0 BV S. A null-sequence c of nonnegative numbers possessing the property n=2m 2m 1 c n Kcm 1 is called a sequence of mean rest bounded variation, in symbols, c MRBV S. It is clear that the class MRBV S includes the class R + 0 BV S. The author is grateful to the referee for calling his attention to an inaccurancy in the previous definition of the class MRBV S and to some typos. A sequence γ of positive terms will be called almost increasing decreasing if ν=m c ν Kγγ n γ m γ n Kγγ m holds for any n m. Denote by p, q 0 q p the set of all nonnegative functions Φx defined on [0, such that Φ0 = 0 and Φx/x p is nonincreasing and Φx/x q is nondecreasing. In this paper a sequence γ := {γ n } is associated to a function γx being defined in the following way: γ π n := γn, n N and K 1 γγ n+1 γx K 2 γγ n holds for all x π, π n+1 n. Page 6 of 14

7 A locally integrable almost everywhere positive function γx : [0, π] [0, is said to be a weight function. Let Φt be a nondecreasing continuous function defined on [0, such that Φ0 = 0 and lim Φt = +. For a weight function γx the weighted Orlicz t space LΦ, γ is defined by LΦ, γ := { h : π 0 } γxφε hx dx < for some ε > 0. Later on D k x and D k x shall denote the Dirichlet and the conjugate Dirichlet kernels. It is known that, if x > 0, D k x = Ox 1 and D k x = Ox 1 hold. We shall also use the notation L R if there exists a positive constant K such that L KR. Page 7 of 14

8 4. Lemmas Lemma 4.1 [1]. If a n 0, ρ n > 0, and if p 1, then ρ n ν=1 a ν p ρ 1 p n a p n p ρ ν. Lemma 4.2 [2]. Let Φ p, q 0 q p and t j 0, j = 1, 2,..., n, n N. Then ν=n 1. Q p Φt ΦQt Q q Φt, 0 Q 1, t 0, p 2. Φ t j Φ 1/p t j, p := max1, p. j=1 j=1 Lemma 4.3. Let Φ p, q 0 q p. If ρ n > 0, a n 0, and if 4.1 holds for all m N, then k ρ k Φ a ν ν=1 where p := max1, p. 2 m+1 1 ν=2 m a ν 2k 1 Φ 2 m 1 ν=1 a ν a ν ρ k 1 kρ k p ρ ν, Page 8 of 14

9 Proof. Denote by A n := n 1 2n 1 ν=n a ν. Let ξ be an integer such that 2 ξ k < 2 ξ+1. Then k ξ 2 m+1 1 ξ 4.2 a ν a ν = 2 m A 2 m. ν=1 m=0 ν=2 m m=0 Utilizing the properties of Φ, furthermore 4.1, 4.2 and Lemma 4.2, we obtain that k ξ Φ ν=1 a ν Φ Φ m=0 ξ 1 m=0 ξ 1 m=0 Hence, by Lemma 4.1, we have k ρ k Φ a ν ν=1 2 m A 2 m 2 m A 2 m Φ 1/p 2 m A 2 m p k ν 1 Φ 1/p ν A ν ν=1 ν 1 Φ 1/p ν A ν p k ρ k ν=1 p k 1 Φ 1/p k A k p ρ ν Herewith the proof is complete. ρ 1 p k p ρ k Φk A k k ρ k 1 ρ ν. p. Page 9 of 14

10 Lemma 4.4. If λ := {λ n } MRBV S and Λ n := n 1 2n 1 ν=n λ ν, then holds for all k 2l. Proof. It is clear that if m 2l, then Λ k Λ l whence obviously follows. 2l 1 l 1 ν=l λ ν λ ν ν=2l 2l 1 Λ l = l 1 ν=l λ ν λ m, ν=m 2k 1 λ ν k 1 m=k λ m = Λ k Page 10 of 14

11 5. Proof of Theorem 2.1 Proof of Theorem 2.1. Let x π, π n+1 n. Using Abel s rearrangement, the known estimate of D k x and the fact that λ MRBV S, we obtain that fx λ k + λ k cos kx k=n+1 λ k + λ k D k x + λ n D n x λ k + k=n k n/2 Hence, λ MRBV S, and we obtain that also holds. A similar argument yields thus we have fx gx 5.1 ψx λ k + n λ n. λ k λ k, λ k, Page 11 of 14

12 where ψx is either fx or gx. By Lemma 4.4, the condition 4.1 with λ ν in place of a ν is satisfied, thus we can apply Lemma 4.3, consequently 5.1 and some elementary calculations give that π π/n γxφ ψx dx Φ λ k γxdx π/n+1 γ n n 2 Φ λ k 2k 1 Φ λ ν γ k k 2 k γ 1 k Since the sequence {γ n n 1+ε } is almost decreasing, then k γ 1 k γ ν ν 2 1, therefore 5.2 proves 2.1. To prove 2.2 we follow a similar procedure as above. Then gx kxλ k + λ k sin kx k=n+1 x kλ k + λ k Dk x + λ n D n x 5.3 n 1 k=n kλ k + k n/2 λ k + n λ n n 1 p γ ν ν 2. kλ k. Page 12 of 14

13 Using Lemmas 4.2, 4.3, 4.4 and the estimate 5.3, we obtain that π π/n γxφ gx dx Φ n 1 kλ k γxdx γ n n 2 q Φ kλ k Φ k By the assumption on {γ n }, and thus 5.4 yields that π 0 k 1+q γ 1 k 2k 1 λ ν π/n+1 γ k k 2 q γ ν ν 2 q 1, γxφ gx dx γ k k 2 q Φ k holds, which proves 2.2. Herewith the proof of Theorem 2.1 is complete. k 1+q γ 1 k 2k 1 p γ ν ν 2 q. λ ν Page 13 of 14

14 References [1] L. LEINDLER, Generalization of inequalities of Hardy and Littlewood, Acta Sci. Math. Szeged, , [2] M. MATELJEVIC AND PAVLOVIC, L p -behavior of power series with positive coefficients and Hardy spaces, Proc. Amer. Math. Soc., , [3] S. TIKHONOV, On belonging of trigonometric series of Orlicz space, J. Inequal. Pure and Appl. Math., , Art. 22. [ONLINE: edu.au/article.php?sid=395]. Page 14 of 14