Sharp \left( H_{p},L_{p}\right) type inequalities of maximal operators of T means with respect to Vilenkin systems with monotone coefficients
aa r X i v : . [ m a t h . C A ] J a n SHARP ( H p , L p ) TYPE INEQUALITIES OF MAXIMAL OPERATORS OF T MEANS WITH RESPECT TO VILENKIN SYSTEMS WITHMONOTONE COEFFICIENTS
G. TUTBERIDZE
Abstract.
In this paper we prove and discuss some new ( H p , L p ) type inequalities of max-imal operators of T means with respect to the Vilenkin systems with monotone coefficients.We also apply these inequalities to prove strong convergence theorems of such T means. Wealso show that these results are the best possible in a special sense. As applications, bothsome well-known and new results are pointed out. Key words and phrases:
Vilenkin groups, Vilenkin systems, partial sums of Vilenkin-Fourier series, T means, Vilenkin-Nörlund means, Fejér mean, Riesz means, martingaleHardy spaces, L p spaces, weak − L p spaces, maximal operator, strong convergence, inequal-ities. 1. Introduction
The definitions and notations used in this introduction can be found in our next Section.It is well-known that Vilenkin systems do not form bases in the space L . Moreover, thereis a function in the Hardy space H p , such that the partial sums of f are not bounded in L p -norm, for < p ≤ . Approximation properties of Vilenkin-Fourier series with respect toone- and two-dimensional case can be found in [17] and [32]. Simon [24] proved that thereexists an absolute constant c p , depending only on p, such that the inequality [ p ] n n X k =1 k S k f k pp k − p ≤ c p k f k pH p (0 < p ≤ holds for all f ∈ H p and n ∈ N + , where [ p ] denotes the integer part of p. For p = 1 analogousresults with respect to more general systems were proved in Blahota [2] and Gát [4] and for < p < simpler proof was given in Tephnadze [31]. Some new strong convergence resultfor partial sums with respect to Vilenkin system was considered in Tutberidze [33].In the one-dimensional case the weak (1,1)-type inequality for the maximal operator ofFejér means σ ∗ f := sup n ∈ N | σ n f | can be found in Schipp [21] for Walsh series and in Pál,Simon [15] for bounded Vilenkin series. Fujji [8] and Simon [23] verified that σ ∗ is boundedfrom H to L . Weisz [38] generalized this result and proved boundedness of σ ∗ from themartingale space H p to the space L p , for p > / . Simon [22] gave a counterexample, whichshows that boundedness does not hold for < p < / . A counterexample for p = 1 / was given by Goginava [6] (see also Tephnadze [25]). Moreover, Weisz [40] proved that the The research was supported by Shota Rustaveli National Science Foundation grant no.FR-19-676. maximal operator of the Fejér means σ ∗ is bounded from the Hardy space H / to the space weak − L / . In [26] and [27] the following result was proved: Theorem T1:
Let < p ≤ / . Then the following weighted maximal operator of Fejérmeans e σ ∗ p f := sup n ∈ N + | σ n f | ( n + 1) /p − log / p ] ( n + 1) is bounded from the martingale Hardy space H p to the Lebesgue space L p .Moreover, the rate of the weights n / ( n + 1) /p − log p +1 / ( n + 1) o ∞ n =1 in n -th Fejér meanwas given exactly.Similar results with respect to Walsh-Kachmarz systems were considered in [7] for p = 1 / and in [28] for < p < / . Approximation properties of Fejér means with respect toVilenkin and Kaczmarz systems can be found in Tephnadze [29], Tutberidze [34], Persson,Tephnadze and Tutberidze [19].In [3] it was proved that there exists an absolute constant c p , depending only on p , suchthat the inequality(1) [1 / p ] n n X k =1 k σ k f k pp k − p ≤ c p k f k pH p (0 < p ≤ / , n = 2 , , . . . ) . holds. Some new strong convergence result for Vilenkin-Fejér means was considered [20].Móricz and Siddiqi [11] investigated the approximation properties of some special Nör-lund means of Walsh-Fourier series of L p function in norm. In the two-dimensional caseapproximation properties of Nörlund means was considered by Nagy [12, 13, 14]. In [16] itwas proved that the maximal operators of Nörlund means t ∗ f := sup n ∈ N | t n f | either withnon-decreasing coefficients, or non-increasing coefficients, satisfying condition(2) Q n = O (cid:18) n (cid:19) , as n → ∞ are bounded from the Hardy space H / to the space weak − L / and are not bounded fromthe Hardy space H p to the space L p , when < p ≤ / . In [18] it was proved that for < p < / , f ∈ H p and non-decreasing sequence { q k : k ≥ } there exists an absolute constant c p , depending only on p, such that the inequality holds ∞ X k =1 k t k f k pp k − p ≤ c p k f k pH p Moreover, if f ∈ H / and { q k : k ≥ } be a sequence of non-decreasing numbers, satisfyingthe condition(3) q n − Q n = O (cid:18) n (cid:19) , as n → ∞ , then there exists an absolute constant c, such that the inequality holds n n X k =1 k t k f k / / k ≤ c k f k / H / In [35] was proved that the maximal operators T ∗ f := sup n ∈ N | T n f | of T means either withnon-increasing coefficients, or non-decreasing sequence satisfying condition (3) are bounded MEANS 3 from the Hardy space H / to the space weak − L / . Moreover, there exists a martingaleand such T means for which boundedness from the Hardy space H p to the space L p do nothold when < p ≤ / . One of the most well-known mean of T means is Riesz summability. In [30] it was provedthat the maximal operator R ∗ of Riesz means is bounded from the Hardy space H / to thespace weak − L / and is not bounded from H p to the space L p , for < p ≤ / . There alsowas proved that Riesz summability has better properties than Fejér means. In particular,the following weighted maximal operators log n | R n f | ( n + 1) /p − log / p ] ( n + 1) are bounded from H p to the space L p , for < p ≤ / and the rate of weights are sharp.Moreover, in [9] was also proved that if < p < / and f ∈ H p ( G m ) , then there exists anabsolute constant c p , depending only on p, such that the inequality holds:(4) ∞ X n =1 log p n k R n f k pH p n − p ≤ c p k f k pH p If we compare strong convergence results, given by (1) and (4), we obtain that Riesz meanshas better properties then Fejér means, for < p < / , but in the case p = 1 / is was notpossible to show even similar result for Riesz means as it is proved for Fejér means given byinequality (1).In this paper we prove and discuss some new ( H p , L p ) type inequalities of maximal operatorsof T means with respect to the Vilenkin systems with monotone coefficients. Moreover, weapply these inequalities to prove strong convergence theorems of such T means. In particular,we also study strong convergence theorems of T means with non-increasing sequences in thecase p = 1 / , but under the condition (2). For example, this condition is fulfilled for Fejérmeans but does not hold for Riesz means. We also show that these inequalities are thebest possible in a special sense. As applications, both some well-known and new results arepointed out.This paper is organized as follows: In order not to disturb our discussions later on somedefinitions and notations are presented in Section 2. The main results and some of itsconsequences can be found in Section 3. For the proofs of the main results we need someauxiliary Lemmas, some of them are new and of independent interest. These results arepresented in Section 4. The detailed proofs are given in Section 5.2. Definitions and Notation
Denote by N + the set of the positive integers, N := N + ∪ { } . Let m := ( m , m , ... ) be asequence of the positive integers not less than 2. Denote by Z m k := { , , ..., m k − } the additive group of integers modulo m k .Define the group G m as the complete direct product of the groups Z m i with the productof the discrete topologies of Z m j ‘ s.The direct product µ of the measures µ k ( { j } ) := 1 /m k ( j ∈ Z m k ) is the Haar measureon G m with µ ( G m ) = 1 . G. TUTBERIDZE
In this paper we discuss bounded Vilenkin groups, i.e. the case when sup n m n < ∞ . The elements of G m are represented by sequences x := ( x , x , ..., x j , ... ) , (cid:0) x j ∈ Z m j (cid:1) . Set e n := (0 , ..., , , , ... ) ∈ G, the n − th coordinate of which is 1 and the rest are zeros ( n ∈ N ) . It is easy to give a basis for the neighborhoods of G m : I ( x ) := G m , I n ( x ) := { y ∈ G m | y = x , ..., y n − = x n − } , where x ∈ G m , n ∈ N . If we define I n := I n (0) , for n ∈ N and I n := G m \ I n , then(5) I N = N − [ k =0 N − [ l = k +1 I k,lN ! [ N − [ k =1 I k,NN ! , where I k,lN := (cid:26) I N (0 , ..., , x k = 0 , , ..., , x l = 0 , x l +1 , ..., x N − , ... ) , for k < l < N,I N (0 , ..., , x k = 0 , , ..., , x N − = 0 , x N , ... ) , for l = N. If we define the so-called generalized number system based on m in the following way : M := 1 , M k +1 := m k M k ( k ∈ N ) , then every n ∈ N can be uniquely expressed as n = P ∞ j =0 n j M j , where n j ∈ Z m j ( j ∈ N + ) and only a finite number of n j ‘ s differ from zero.We introduce on G m an orthonormal system which is called the Vilenkin system. Atfirst, we define the complex-valued function r k ( x ) : G m → C , the generalized Rademacherfunctions, by r k ( x ) := exp (2 πix k /m k ) , (cid:0) i = − , x ∈ G m , k ∈ N (cid:1) . Next, we define the Vilenkin system ψ := ( ψ n : n ∈ N ) on G m by: ψ n ( x ) := ∞ Y k =0 r n k k ( x ) , ( n ∈ N ) . Specifically, we call this system the Walsh-Paley system when m ≡ . The norms (or quasi-norms) of the spaces L p ( G m ) and weak − L p ( G m ) (0 < p < ∞ ) arerespectively defined by k f k pp := Z G m | f | p dµ, k f k pweak − L p := sup λ> λ p µ ( f > λ ) < + ∞ . The Vilenkin system is orthonormal and complete in L ( G m ) (see [36]).Now, we introduce analogues of the usual definitions in Fourier-analysis. If f ∈ L ( G m ) we can define Fourier coefficients, partial sums and Dirichlet kernels with respect to theVilenkin system in the usual manner: b f ( n ) := Z G m f ψ n dµ, S n f := n − X k =0 b f ( k ) ψ k , D n := n − X k =0 ψ k , ( n ∈ N + ) . MEANS 5
It is well known that if n ∈ N , then(6) D M n ( x ) = (cid:26) M n , x ∈ I n , , x / ∈ I n . Moreover, if n = P ∞ i =0 n i M i , and ≤ s n ≤ m n − , then we have the following identity:(7) D n = ψ n ∞ X j =0 D M j m j − X k = m j − n j r kj , The σ -algebra generated by the intervals { I n ( x ) : x ∈ G m } will be denoted by ̥ n ( n ∈ N ) . Denote by f = (cid:0) f ( n ) , n ∈ N (cid:1) a martingale with respect to ̥ n ( n ∈ N ) . (for details see e.g.[37]). The maximal function of a martingale f is defined by f ∗ = sup n ∈ N (cid:12)(cid:12) f ( n ) (cid:12)(cid:12) . For
. Then the summability method (8) generated by { q k : k ≥ } is regular if and only if lim n →∞ Q n = ∞ . If we invoke Abel transformation we get the following identities, which are very importantfor the investigations of T summability: Q n := n − X j =0 q j = n − X j =0 ( q j − q j +1 ) j + q n − ( n − , (9)(10) F n = 1 Q n n − X j =0 ( q j − q j +1 ) jK j + q n − ( n − K n − ! . G. TUTBERIDZE and(11) T n f = 1 Q n n − X j =0 ( q j − q j +1 ) jσ j f + q n − ( n − σ n − f ! . Let { q k : k ≥ } be a sequence of nonnegative numbers. The n -th Nörlund mean t n for aFourier series of f is defined by(12) t n f = 1 Q n n X k =1 q n − k S k f, where Q n := P n − k =0 q k . If q k ≡ , we respectively define the Fejér means σ n and Kernels K n as follows: σ n f := 1 n n X k =1 S k f , K n := 1 n n X k =1 D k . It is well-known that (for details see [1])(13) n | K n | ≤ c | n | X l =0 M l | K M l | and(14) k K n k ≤ c < ∞ . The well-known example of Nörlund summability is the so-called ( C, α ) -mean (Cesàromeans) for < α < , which are defined by σ αn f := 1 A αn n X k =1 A α − n − k S k f, where A α := 0 , A αn := ( α + 1) ... ( α + n ) n ! . We also consider the "inverse" ( C, α ) -means, which is an example of a T -means: U αn f := 1 A αn n − X k =0 A α − k S k f, < α < . Let V αn denote the T mean, where { q = 0 , q k = k α − : k ∈ N + } , that is V αn f := 1 Q n n − X k =1 k α − S k f, < α < . The n -th Riesz logarithmic mean R n and the Nörlund logarithmic mean L n are defined by R n f := 1 l n n − X k =1 S k fk and L n f := 1 l n n − X k =1 S k fn − k , respectively, where l n := P n − k =1 /k. MEANS 7
Up to now we have considered T means in the case when the sequence { q k : k ∈ N } isbounded but now we consider T summabilities with unbounded sequence { q k : k ∈ N } .Let α ∈ R + , β ∈ N + and log ( β ) x := β − times z }| { log ... log x. If we define the sequence { q k : k ∈ N } by n q = 0 , q k = log ( β ) k α : k ∈ N + o , then we get the class of T means with non-decreasingcoefficients: B α,βn f := 1 Q n n − X k =1 log ( β ) k α S k f. We note that B α,βn are well-defined for every n ∈ N B α,βn f = n − X k =1 log ( β ) k α Q n S k f. It is obvious that n log ( β ) n α α ≤ Q n ≤ n log ( β ) n α . It follows that q n − Q n ≤ c log ( β ) n α n log ( β ) n α = O (cid:18) n (cid:19) → , as n → ∞ . (15)We also define the maximal operator of T and Nörlund means by T ∗ f := sup n ∈ N | T n f | , t ∗ f := sup n ∈ N | t n f | . Some well-known examples of maximal operators of T means are the maximal operator ofFejér σ ∗ and Riesz R ∗ logarithmic means, which are defined by: σ ∗ f := sup n ∈ N | σ n f | , R ∗ f := sup n ∈ N | R n f | . The Main Results and Applications
Our first main result reads:
Theorem 1.
Let < p ≤ / , f ∈ H p and { q k : k ≥ } be a sequence of non-increasingnumbers. Then the maximal operator (16) e T ∗ p f := sup n ∈ N + | T n f | ( n + 1) /p − log / p ] ( n + 1) is bounded from the Hardy space H p to the space L p . Corollary 1.
Let < p ≤ / and f ∈ H p . Then the maximal operator e R ∗ p f := sup n ∈ N + | R n f | ( n + 1) /p − log / p ] ( n + 1) is bounded from the Hardy space H p to the space L p . Corollary 2.
Let < p ≤ / and f ∈ H p . Then the maximal operator e U α, ∗ p f := sup n ∈ N + | U αn f | ( n + 1) /p − log / p ] ( n + 1) is bounded from the Hardy space H p to the space L p . G. TUTBERIDZE
Corollary 3.
Let < p ≤ / and f ∈ H p . Then the maximal operator e V α, ∗ p f := sup n ∈ N + | V αn f | ( n + 1) /p − log / p ] ( n + 1) is bounded from the Hardy space H p to the space L p . Next, we consider maximal operators of T means with non-decreasing sequence: Theorem 2.
Let < p ≤ / , f ∈ H p and { q k : k ≥ } be a sequence of non-decreasingnumbers, satisfying the condition (17) q n − Q n = O (cid:18) n (cid:19) , as n → ∞ . Then the maximal operator (18) e T ∗ p f := sup n ∈ N + | T n f | ( n + 1) /p − log / p ] ( n + 1) is bounded from the martingale Hardy space H p to the space L p . Corollary 4.
Let < p ≤ / , f ∈ H p and { q k : k ≥ } be a sequence of non-decreasingnumbers, such that (19) sup n ∈ N q n < c < ∞ . Then q n − Q n ≤ cQ n ≤ cq n = c n = O (cid:18) n (cid:19) , as n → , and weighted maximal operators of such T means, given by (18) are bounded from the Hardyspace H p to the space L p . Corollary 5.
Let < p ≤ / and f ∈ H p . Then the maximal operator e T ∗ p f := sup n ∈ N + (cid:12)(cid:12) B α,βn f (cid:12)(cid:12) ( n + 1) /p − log / p ] ( n + 1) is bounded from the martingale Hardy space H p to the space L p . Remark 1.
According to Theorem T1 we obtain that weights in (16) and (18) are sharp.
Theorem 3. a) Let < p < / , f ∈ H p and { q k : k ≥ } be a sequence of non-increasingnumbers. Then there exists an absolute constant c p , depending only on p, such that theinequality holds: ∞ X k =1 k T k f k pp k − p ≤ c p k f k pH p b)Let f ∈ H / and { q k : k ≥ } be a sequence of non-increasing numbers, satisfying thecondition (20) Q n = O (cid:18) n (cid:19) , as n → ∞ . Then there exists an absolute constant c, such that the inequality holds: (21) n n X k =1 k T k f k / / k ≤ c k f k / H / MEANS 9
Corollary 6.
Let < p ≤ / and f ∈ H p . Then there exists absolute constant c p , dependingonly on p, such that the following inequality holds: [1 / p ] n n X k =1 k σ k f k pp k − p ≤ c p k f k pH p . Corollary 7.
Let < p ≤ / and f ∈ H p . Then there exists an absolute constant c p , depending only on p, such that the following inequalities hold: ∞ X k =1 k U αk f k pp k − p ≤ c p k f k pH p , ∞ X k =1 k V αk f k pp k − p ≤ c p k f k pH p , ∞ X k =1 k R k f k pp k − p ≤ c p k f k pH p . Theorem 4. a) Let < p < / , f ∈ H p and { q k : k ≥ } be a sequence of non-decreasingnumbers. Then there exists an absolute constant c p , depending only on p, such that theinequality holds: ∞ X k =1 k T k f k pp k − p ≤ c p k f k pH p b)Let f ∈ H / and { q k : k ≥ } be a sequence of non-increasing numbers, satisfying thecondition (17) . Then there exists an absolute constant c, such that the inequality holds: (22) n n X k =1 k T k f k / / k ≤ c k f k / H / Corollary 8.
Let < p ≤ / , f ∈ H p and { q k : k ≥ } be a sequence of non-decreasingnumbers, such that sup n ∈ N q n < c < ∞ . Then condition (17) is satisfied and for all such T means there exists an absolute constant c, such that the inequality (22) holds. We have already considered the case when the sequence { q k : k ≥ } is bounded. Now, weconsider some Nörlund means, which are generated by a unbounded sequence { q k : k ≥ } . Corollary 9.
Let < p ≤ / and f ∈ H p . Then there exists an absolute constant c p , depending only on p, such that the following inequality holds: [1 / p ] n n X k =1 (cid:13)(cid:13)(cid:13) B α,βk f (cid:13)(cid:13)(cid:13) pp k − p ≤ c p k f k pH p . Auxiliary lemmas
We need the following auxiliary Lemmas:
Lemma 1 (see e.g. [39]) . A martingale f = (cid:0) f ( n ) , n ∈ N (cid:1) is in H p (0 < p ≤ if and onlyif there exists a sequence ( a k , k ∈ N ) of p-atoms and a sequence ( µ k , k ∈ N ) of real numberssuch that, for every n ∈ N , (23) ∞ X k =0 µ k S M n a k = f ( n ) , a.e., where ∞ X k =0 | µ k | p < ∞ . Moreover, k f k H p ∽ inf ∞ X k =0 | µ k | p ! /p where the infimum is taken over all decompositions of f of the form (23). Lemma 2 (see e.g. [39]) . Suppose that an operator T is σ -sublinear and for some < p ≤ Z − I | T a | p dµ ≤ c p < ∞ , for every p -atom a , where I denotes the support of the atom. If T is bounded from L ∞ to L ∞ , then k T f k p ≤ c p k f k H p , < p ≤ . Lemma 3 (see [5]) . Let n > t, t, n ∈ N . Then K M n ( x ) = M t − r t ( x ) , x ∈ I t \ I t +1 , x − x t e t ∈ I n , M n − , x ∈ I n , , otherwise. For the proof of our main results we also need the following new Lemmas of independentinterest:
Lemma 4.
Let n ∈ N and { q k : k ∈ N } be a sequence either of non-increasing numbers, ornon-decreasing numbers satisfying condition (17) . Then (24) k F n k < c < ∞ . Proof:
Let n ∈ N and { q k : k ∈ N } be a sequence of non-increasing numbers. Bycombining (9) and (11) with (14) we can conclude that k T n k ≤ Q n n − X j =0 | q j − q j +1 | j k σ j k + q n − ( n − k σ n − k ! ≤ cQ n n − X j =0 ( q j − q j +1 ) j + q n − ( n − ! ≤ c < ∞ . Let n ∈ N and { q k : k ∈ N } be a sequence non-decreasing sequence satisfying condition(17). Then By using again (9) and (11) with (14) we find that k T n k ≤ Q n n − X j =0 | q j − q j +1 | j k σ j k + q n − ( n − k σ n − k ! ≤ cQ n n − X j =0 ( q j +1 − q j ) j + q n − ( n − ! = cQ n q n − ( n − − n − X j =0 ( q j − q j +1 ) j + q n − ( n − !! = cQ n (2 q n − ( n − − Q n ) ≤ c < ∞ . The proof is complete.
MEANS 11
Lemma 5.
Let { q k : k ∈ N } be a sequence of non-increasing numbers and n > M N . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cM N | n | X j =0 M j (cid:12)(cid:12) K M j (cid:12)(cid:12) , Proof.
Since sequence is non-increasing number we get that Q n q M N + n − X j = M N | q j − q j +1 | + q n − ! ≤ Q n q M N + n − X j = M N ( q j − q j +1 ) + q n − ! ≤ q M N Q n ≤ q M N Q M N +1 ≤ cM N . If we apply Abel transformation and (13) we immediately get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 1 Q n q M n K M n − + n − X j = M N ( q j − q j +1 ) K j + q n − K n − ! ≤ Q n q M n + n − X j = M N | q j − q j +1 | + q n − ! | n | X i =0 M i | K M i |≤ cM N | n | X i =0 M i | K M i | . The proof is complete. (cid:3)
Lemma 6.
Let x ∈ I k,lN , k = 0 , . . . , N − , l = k + 1 , . . . , N and { q k : k ∈ N } be a sequenceof non-increasing numbers. Then there exists an absolute constant, such that Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) ≤ cM l M k M N . Proof : Let x ∈ I k,lN , for ≤ k < l ≤ N − and t ∈ I N . First, we observe that x − t ∈ I k,lN . Next, we apply Lemmas 3 and 5 to obtain that Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) (25) ≤ cM N | n | X i =0 M i Z I N | K M i ( x − t ) | dµ ( t ) ≤ cM N Z I N l X i =0 M i M k dµ ( t ) ≤ cM k M l M N and the first estimate is proved.Now, let x ∈ I k,NN . Since x − t ∈ I k,NN for t ∈ I N , by combining (6) and (7) we have that | D i ( x − t ) | ≤ M k and Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) (26) ≤ cQ n | n | X i =0 q i Z I N | D i ( x − t ) | dµ ( t ) ≤ cQ n | n |− X i =0 q i Z I N M k dµ ( t ) ≤ cM k M N . According to (25) and (26) the proof is complete.
Lemma 7.
Let n > M N and { q k : k ∈ N } be a sequence of non-increasing numbers, satisfyingcondition (20) . Then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ cn | n | X j =0 M j (cid:12)(cid:12) K M j (cid:12)(cid:12) , where c is an absolute constant.Proof. Since sequence is non-increasing number satisfying condition (20), we get that Q n q M n + n − X j = M N | q j − q j +1 | + q n − ! ≤ Q n q M n + n − X j = M N ( q j − q j +1 ) + q n − ! ≤ q M N Q n ≤ cQ n ≤ cn . If we apply (13) we immediately get that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Q n q M n + n − X j = M N +1 | q j − q j +1 | + q n − !! | n | X i =0 M i | K M i |≤ cn | n | X i =0 M i | K M i | . The proof is complete. (cid:3)
MEANS 13
Lemma 8.
Let x ∈ I k,lN , k = 0 , . . . , N − , l = k + 1 , . . . , N − and { q k : k ∈ N } be asequence of non-increasing numbers, satisfying condition (20) . Then Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) ≤ cM l M k nM N . Let x ∈ I k,NN , k = 0 , . . . , N − . Then Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) ≤ cM k M N . Here c is an absolute constant. Proof : Let x ∈ I k,lN , for ≤ k < l ≤ N − and t ∈ I N . First, we observe that x − t ∈ I k,lN . Next, we apply Lemmas 3 and 7 to obtain that Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) (27) ≤ cn | n | X i =0 M i Z I N | K M i ( x − t ) | dµ ( t ) ≤ cn Z I N l X i =0 M i M k dµ ( t ) ≤ cM k M l nM N and the first estimate is proved.Now, let x ∈ I k,NN . Since x − t ∈ I k,NN for t ∈ I N , by combining again Lemmas 3 and 7 wehave that Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) (28) ≤ cn | n | X i =0 M i Z I N | K M i ( x − t ) | dµ ( t ) ≤ cn | n |− X i =0 M i Z I N M k dµ ( t ) ≤ cM k M N . By combining (27) and (28) we complete the proof.
Lemma 9.
Let n ≥ M N , x ∈ I k,lN , k = 0 , . . . , N − , l = k + 1 , . . . , N and { q k : k ∈ N } be a sequence of non-increasing sequence, satisfying condition (20) . Then Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) ≤ cM l M k M N , where c is an absolute constant. Proof : Since n ≥ M N if we apply Lemma 8 we immediately get the proof. Lemma 10.
Let { q k : k ∈ N } be a sequence of non-decreasing numbers satisfying (17). Then | F n | ≤ cn | n | X j =0 M j (cid:12)(cid:12) K M j (cid:12)(cid:12) , where c is an absolute constant.Proof. Since sequence { q k : k ∈ N } be non-decreasing if we apply condition (17) we canconclude that Q n n − X j =0 | q j − q j +1 | + q n − ! ≤ Q n n − X j =0 ( q j +1 − q j ) + q n − ! ≤ q n − − q Q n ≤ q n − Q n ≤ cn . If we apply (10) and (13) we immediately get that | F n | ≤ Q n n − X j =1 | q j − q j +1 | + q !! | n | X i =0 M i | K M i | = Q n n − X j =1 ( q j − q j +1 ) + q !! | n | X i =0 M i | K M i |≤ q n − Q n | n | X i =0 M i | K M i | ≤ cn | n | X i =0 M i | K M i | . The proof is complete. (cid:3)
Lemma 11.
Let x ∈ I k,lN , k = 0 , . . . , N − , l = k + 1 , . . . , N − and { q k : k ∈ N } be asequence of non-decreasing numbers, satisfying condition (17). Then Z I N | F n ( x − t ) | dµ ( t ) ≤ cM l M k nM N . Let x ∈ I k,NN , k = 0 , . . . , N − . Then Z I N | F n ( x − t ) | dµ ( t ) ≤ cM k M N . Here c is an absolute constant. Proof : The proof is quite analogously to Lemma 8. So we leave out the details.
Lemma 12.
Let n ≥ M N , x ∈ I k,lN , k = 0 , . . . , N − , l = k + 1 , . . . , N and { q k : k ∈ N } be a sequence of non-decreasing sequence, satisfying condition (17). Then Z I N | F n ( x − t ) | dµ ( t ) ≤ cM l M k M N . Proof : Since n ≥ M N if we apply Lemma 11 we immediately get the proof. MEANS 15 Proofs of the Theorems
Proof of Theorem 1.
Let < p ≤ / and sequence { q k : k ≥ } be non-increasing. Bycombining (9) and (11) we get that e T ∗ p f := | T n f | ( n + 1) /p − log / p ] ( n + 1) ≤ n + 1) /p − log / p ] ( n + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j =1 q j S j f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n + 1) /p − log / p ] ( n + 1) 1 Q n n − X j =1 | q j − q j +1 | j | σ j f | + q n − ( n − | σ n f | ! ≤ Q n n − X j =1 | q j − q j +1 | j | σ j f | ( j + 1) /p − log / p ] ( j + 1) + q n − ( n − | σ n f | ( n + 1) /p − log / p ] ( n + 1) ! ≤ Q n n − X j =1 ( q j − q j +1 ) j + q n − ( n − ! sup n ∈ N + | σ n f | ( n + 1) /p − log / p ] ( n + 1) ≤ sup n ∈ N + | σ n f | ( n + 1) /p − log / p ] ( n + 1) := e σ ∗ p f, so that e T ∗ p f ≤ e σ ∗ p f. Hence, if we apply Theorem T1 we can conclude that the maximaloperators e T ∗ p of T means with non-increasing sequence { q k : k ≥ } are bounded from theHardy space H p to the space L p for < p ≤ / . The proof is complete. (cid:3) Proof of Theorem 2.
Let < p ≤ / and sequence { q k : k ≥ } be non-decreasing satisfyingthe condition (17). By combining (9) and (11) we find that e T ∗ p f := | T n f | ( n + 1) /p − log / p ] ( n + 1) ≤ n + 1) /p − log / p ] ( n + 1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n − X j =1 q j S j f (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n + 1) /p − log / p ] ( n + 1) 1 Q n n − X j =1 | q j − q j +1 | j | σ j f | + q n − ( n − | σ n f | ! ≤ Q n n − X j =1 | q j − q j +1 | j | σ j f | ( j + 1) /p − log / p ] ( j + 1) + q n − ( n − | σ n f | ( n + 1) /p − log / p ] ( n + 1) ! ≤ Q n n − X j =1 ( q j +1 − q j ) j + q n − ( n − ! sup n ∈ N + | σ n f | ( n + 1) /p − log / p ] ( n + 1) ≤ q n − ( n − − Q n Q n sup n ∈ N + | σ n f | ( n + 1) /p − log / p ] ( n + 1) ≤ sup n ∈ N + | σ n f | ( n + 1) /p − log / p ] ( n + 1) = e σ ∗ p f. so that(29) e T ∗ p f ≤ e σ ∗ p f If we apply (29) and Theorem T1 we can conclude that the maximal operators e T ∗ p of T means with non-decreasing sequence { q k : k ≥ } , are bounded from the Hardy space H p tothe space L p for < p ≤ / . The proof is complete. (cid:3)
Proof of Theorem 3.
Let < p < / and sequence { q k : k ≥ } be non-increasing. ByLemma 1, the proof of part a) will be complete, if we show that ∞ X m =1 k T m a k pH p m − p ≤ c p , for every p -atom a, with support I , µ ( I ) = M − N . We may assume that I = I N . It is easy tosee that S n ( a ) = T n ( a ) = 0 , when n ≤ M N . Therefore, we can suppose that n > M N .Let x ∈ I N . Since T n is bounded from L ∞ to L ∞ (boundedness follows Lemma 4) and k a k ∞ ≤ M /pN we obtain that Z I N | T m a | p dµ ≤ k a k p ∞ M N ≤ c < ∞ . Hence,(30) ∞ X m =1 R I N | T m a | p dµm − p ≤ ∞ X k =1 m − p ≤ c < ∞ , < p < / . It is easy to see that | T m a ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z I N a ( t ) F n ( x − t ) dµ ( t ) (cid:12)(cid:12)(cid:12)(cid:12) (31) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z I N a ( t ) 1 Q n n X j = M N q j D j ( x − t ) dµ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k a k ∞ Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) ≤ M /pN Z I N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q n n X j = M N q j D j ( x − t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( t ) Let T n be T means, with non-decreasing coefficients { q k : k ≥ } and x ∈ I k,lN , ≤ k < l ≤ N. Then, in the view of Lemma 6 we get that(32) | T m a ( x ) | ≤ cM l M k M /p − N , for < p < / . MEANS 17
Let < p < / . By using (5), (31) and (32) we find that Z I N | T m a | p dµ = N − X k =0 N − X l = k +1 m j − X x j =0 , j ∈{ l +1 ,...,N − } Z I k,lN | T m a | p dµ + N − X k =0 Z I k,NN | T m a | p dµ (33) ≤ c N − X k =0 N − X l = k +1 m l +1 · · · m N − M N ( M l M k ) p M − pN + N − X k =0 M N M pk M − pN ≤ cM − pN N − X k =0 N − X l = k +1 ( M l M k ) p M l + N − X k =0 M pk M pN ≤ cM − pN . Moreover, according to (33) we get that(34) ∞ X m = M N +1 R I N | T m a | p dµm − p ≤ ∞ X m = M N +1 cM − pN m − p < c < ∞ , (0 < p < / . The proof of part a) is complete by just combining (30) and (34).Let p = 1 / and T n be T means, with non-increasing coefficients { q k : k ≥ } , satisfyingcondition (20). By Lemma 1, the proof of part b) will be complete, if we show that n n X m =1 k T m a k / H / m ≤ c, for every / -atom a, with support I , µ ( I ) = M − N . We may assume that I = I N . It is easyto see that S n ( a ) = T n ( a ) = 0 , when n ≤ M N . Therefore, we can suppose that n > M N .Let x ∈ I N . Since T n is bounded from L ∞ to L ∞ (boundedness follows from Lemma 4) and k a k ∞ ≤ M N we obtain that Z I N | T m a | / dµ ≤ k a k / ∞ M N ≤ c < ∞ . Hence,(35) n n X m =1 R I N | T m a | / dµm ≤ n n X k =1 m ≤ c < ∞ . Analogously to (31) we find that | T m a ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z I N a ( t ) 1 Q n n X j = M N q j D j ( x − t ) dµ ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (36) ≤ k a k ∞ Z I N | F m ( x − t ) | dµ ( t ) ≤ M N Z I N | F m ( x − t ) | dµ ( t ) . Let x ∈ I k,lN , ≤ k < l < N. Then, in the view of Lemma 8 we get that(37) | T m a ( x ) | ≤ cM l M k M N m . Let x ∈ I k,NN . Then, according to Lemma 8 we obtain that(38) | T m a ( x ) | ≤ cM k M N . By combining (5), (36), (37) and (38) we obtain that Z I N | T m a ( x ) | / dµ ( x ) ≤ c N − X k =0 N − X l = k +1 m l +1 · · · m N − M N ( M l M k ) / M / N m / + N − X k =0 M N M / k M / N ≤ M / N N − X k =0 N − X l = k +1 ( M l M k ) / m / M l + N − X k =0 M / k M / N ≤ cM / N Nm / + c. It follows that n n X m = M N +1 R I N | T m a ( x ) | / dµ ( x ) m ≤ n n X m = M N +1 cM / N Nm / + cm ! < c < ∞ . (39)The proof of part b) is completed by just combining (35) and (39). (cid:3) Proof of Theorem 4.
If we use Lemmas 11 and 12 and follows analogical steps of proof ofTheorem 3 we immediately get the proof of Theorem 4. So, we leave out the details. (cid:3)
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