Some inequalities for Cesàro Means of double Vilenkin-Fourier Series
aa r X i v : . [ m a t h . C A ] D ec SOME INEQUALITIES FOR CESÀRO MEANS OF DOUBLEVILENKIN-FOURIER SERIES
T. TEPNADZE , L. E. PERSSON Abstract.
In this paper we state and prove some new inequalities related to the rate of L p approximation by Cesàro means of the quadratic partial sums of double Vilenkin-Fourierseries of functions from L p . [email protected] ; The Artic University of Norway, Campus Narvik, P.O. Box385, N-8505, Narvik, Norway. The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway.
Key words and phrases:
Inequalities, Approximation, Vilenkin system, Vilenkin-Fourierseries, Cesàro means, Convergence in norm.1.
Introduction
Let N + denote the set of positive integers, N := N + ∪ { } . Let m := ( m , m , ... ) denote asequence of positive integers not less then 2. Denote by Z m k := { , , ..., m k − } the additivegroup of integers modulo m k . Define the group G m as the complete direct product of thegroups Z m j , with the product of the discrete topologies of Z mj ’s.The direct product of the measures µ k ( { j } ) := 1 m k ( j ∈ Z m k ) is the Haar measure on G m with µ ( G m ) = 1 . If the sequence m is bounded, then G m is calleda bounded Vilenkin group. In this paper we will consider only bounded Vilenkin groups.The elements of G m can be represented by sequences x := ( x , x , ..., x j , ... ) , (cid:0) x j ∈ Z m j (cid:1) . The group operation + in G m is given by x + y = (( x + y ) mod m , ..., ( x k + y k ) mod m k , ... ) , where x := ( x , ..., x k , ... ) and y := ( y , ..., y k , ... ) ∈ G m . The inverse of + will be denoted by − . It is easy to give a base for the neighborhoods of G m : I ( x ) := G m ,I n ( x ) := { y ∈ G m | y = x , ..., y n − = x n − } for x ∈ G m , n ∈ N. Define I n := I n (0) for n ∈ N + . Set e n := (0 , ..., , , , ... ) ∈ G m the n th coordinate of which is and the rest are zeros ( n ∈ N ) . , L. E. PERSSON 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. Wealso use the following notation: | n | := max { k ∈ N : n k = 0 } (that is , M | n | ≤ n < M | n | +1 , n = 0 ). For every x ∈ G m we denote | x | := ∞ P j =0 x j M j +1 , (cid:0) x j ∈ Z m j (cid:1) .Next, we introduce on G m an orthonormal system, which is called Vilenkin system. At firstdefine the complex valued functions r k ( x ) : G m → C , the generalized Rademacher functions,in this way: r k ( x ) := exp 2 πix k m k (cid:0) i = − , x ∈ G m , k ∈ N (cid:1) . Now we define the Vilenkin system ψ := ( ψ n : n ∈ N ) on G m as follows: ψ n ( x ) := ∞ Y k =0 r n k k ( x ) , ( n ǫ N ) . In particular, we call the system the Walsh-Paley system if m = 2 . Each ψ n is a characterof G m and all characters of G m are of this norm. Moreover, ψ n ( − x ) = ¯ ψ n ( x ) .The Dirichlet kernels are defined by D n := n − X k =0 ψ k , ( n ∈ N + ) . Recall that (see [4] or [25])(1) D M n ( x ) = (cid:26) M n , if x ∈ I n , , if x / ∈ I n . The Vilenkin system is orthonormal and complete in L ( G m ) ( see [1]).Next, we introduce some notation with respect to the theory of two-demonsional Vilenkinsystem. Let ˜ m be a sequence like m . The relation between the sequences ( ˜ m n ) and (cid:16) ˜ M n (cid:17) is the same as between sequences ( m n ) and ( M n ) . The group G m × G ˜ m is called a two-dimensional Vilenkin group. The normalized Haar measure is denoted by µ as in the one-dimensional case. We also suppose that m = ˜ m and G m × G ˜ m = G m . The norm of the space L p ( G m ) is defined by k f k p := Z G m | f ( x, y ) | p dµ ( x, y ) /p , (1 ≤ p < ∞ ) . Denote by C ( G m ) the class of continuous functions on the group G m , endoved with thesupremum norm.For the sake of brevity in notation, we agree to write L ∞ ( G m ) instead of C ( G m ) . OME INEQUALITIES FOR CESÀRO MEANS 3
The two-dimensional Fourier coefficients, the rectangular partial sums of the Fourier series,the Dirichlet kernels with respect to the two-dimensional Vilenkin system are defined asfollows: b f ( n , n ) := Z G m f ( x, y ) ¯ ψ n ( x ) ¯ ψ n ( y ) dµ ( x, y ) ,S n ,n ( x, y, f ) := n − X k =0 n − X k =0 b f ( k , k ) ψ k ( x ) ψ k ( y ) ,D n ,n ( x, y ) := D n ( x ) D n ( y ) , Denote S (1) n ( x, y, f ) := n − X l =0 b f ( l, y ) ¯ ψ l ( x ) ,S (2) m ( x, y, f ) := m − X r =0 b f ( x, r ) ¯ ψ r ( y ) , where b f ( l, y ) = Z G m f ( x, y ) ψ l ( x ) dµ ( x ) and b f ( x, r ) = Z G m f ( x, y ) ψ r ( y ) dµ ( y ) . The ( C, − α ) means of double Vilenkin-Fourier series are defined as follows σ − αn ( f, x, y ) = 1 A − αn − n X j =1 A − α − n − j S j,j ( f, x, y ) , where A α = 1 , A αn = ( α + 1) ... ( α + n ) n ! . It is well known that (see [29])(2) A αn = n X k =0 A α − k . (3) A αn − A αn − = A α − n . and(4) c ( α ) n α ≤ A αn ≤ c ( α ) n α , where positive constants c and c are dependent on α . T. TEPNADZE , L. E. PERSSON The dyadic partial moduli of continuity of a function f ∈ L p ( G m ) in the L p -norm aredefined by ω (cid:18) f, M n (cid:19) p = sup u ∈ I n k f ( · + u, · ) − f ( · , · ) k p , and ω (cid:18) f, M n (cid:19) p = sup v ∈ I n k f ( · , · + v ) − f ( · , · ) k p , while the dyadic mixed modulus of continuity is defined as follows: ω , (cid:18) f, M n , M m (cid:19) p = sup ( u,v ) ∈ I n × I m k f ( · + u, · + v ) − f ( · + u, · ) − f ( · , · + v ) + f ( · , · ) k p . It is clear that ω , (cid:18) f, M n , M m (cid:19) p ≤ ω (cid:18) f, M n (cid:19) p + ω (cid:18) f, M m (cid:19) p . The dyadic total modulus of continuity is defined by ω (cid:18) f, M n (cid:19) p = sup ( u,v ) ∈ I n × I n k f ( · + u, · + v ) − f ( · , · ) k p . The problems of summability of partial sums and Cesàro means for Walsh-Fourier serieswere studied in [2], [15]-[24], [27].The convergence issue of Fejér (and Cesàro ) means on the Walsh and Vilenkin groups forunbouded case were studies in [5]-[11] .In his monography [28] L.V. Zhizhinashvili investigated the behavior of Cesàro ( C, α ) − meansfor double trigonometric Fourier series in detail. U.Goginava [20] studied the analogical ques-tion in case of the Walsh system. In particular, the following theorems were proved: Theorem A.
Let f belong to L p ( G ) for some p ∈ [1 , ∞ ] and α ∈ (0 , . Then, for any k ≤ n < k +1 , ( k, n ∈ N ) , the inequality (cid:13)(cid:13) σ − α k ( f ) − f (cid:13)(cid:13) p ≤ c ( α ) n kα ω (cid:0) f, / k − (cid:1) p + 2 kα ω (cid:0) f, / k − (cid:1) p ++ k − X r =0 r − k ω ( f, / r ) p + k − X s =0 s − k ω ( f, / s ) p ) holds. Theorem B.
Let f belong to L p ( G ) for some p ∈ [1 , ∞ ] and α ∈ (0 , . Then, for any k ≤ n < k +1 , ( k, n ∈ N ) , the inequality (cid:13)(cid:13) σ − αn ( f ) − f (cid:13)(cid:13) p ≤ c ( α ) n kα kω (cid:0) f, / k − (cid:1) p + 2 kα kω (cid:0) f, / k − (cid:1) p + OME INEQUALITIES FOR CESÀRO MEANS 5 + k − X r =0 r − k ω ( f, / r ) p + k − X s =0 s − k ω ( f, / s ) p ) holds. In this paper, we state and prove the analogous results in the case of double Vilenkin-Fourier series. Our main results read:
Theorem 1.
Let f belong to L p ( G m ) for some p ∈ [1 , ∞ ] and α ∈ (0 , . Then, for any M k ≤ n < M k +1 ( k, n ∈ N ) , the inequality (cid:13)(cid:13) σ − αM k ( f ) − f (cid:13)(cid:13) p ≤ c ( α ) (cid:16) ω ( f, /M k − ) p M αk + ω ( f, /M l − ) p M αk ++ k − X r =0 M r M k ω ( f, /M r ) p + k − X s =0 M s M k ω ( f, /M s ) p ! holds. Theorem 2.
Let f belong to L p ( G m ) for some p ∈ [1 , ∞ ] and α ∈ (0 , . Then, for any M k ≤ n < M k +1 ( k, n ∈ N ) , the inequality (cid:13)(cid:13) σ − αn ( f ) − f (cid:13)(cid:13) p ≤ c ( α ) (cid:16) ω ( f, /M k − ) p M αk log n + ω ( f, /M l − ) p M αk log n + k − X r =0 M r M k ω ( f, /M r ) p + k − X s =0 M s M k ω ( f, /M s ) p ! holds.In order to make the proofs of these Theorems more clear we formulate some auxiliaryLemmas in Section 2. Some of these Lemmas are new and of independent interest. Thedetailed proofs can be found in Section 3. T. TEPNADZE , L. E. PERSSON AUXILIARY LEMMAS
In order to prove Theorem 1 and Theorem 2 we need the following Lemmas (see [1], [3]and [12], respectively)
Lemma 1.
Let α , α , ..., α n be real numbers.Then n Z G (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 α k D k ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( x ) ≤ c √ n n X k =1 α k ! / . Lemma 2.
Let α , α , ..., α n be real numbers. Then n Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X k =1 α k D k ( x ) D k ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( x, y ) ≤ c √ n n X k =1 α k ! / . Lemma 3.
Let ≤ j < n s M s and ≤ n s < m s . Then D n s M s − j = D n s M s − ψ n s M s − ¯ D j . We also need the following new Lemmas of independent interest.
Lemma 4.
Let f belong to L p ( G m ) for some p ∈ [1 , ∞ ] . Then, for every α ∈ (0 , , thefollowing inequality holds I := 1 A − αn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m M k − X i =1 A − α − n − i D i ( u ) D i ( v ) [ f ( · − u, ·− ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p ≤ k − X r =0 M r M k ω ( f, /M r ) p + k − X s =0 M s M k ω ( f, /M s ) p , where M k ≤ n < M k +1 . Lemma 5.
Let α ∈ (0 , and p = M k , M k + 1 , .... Then II := Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k X i =1 A − α − p − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) < ∞ , k = 1 , ... Lemma 6.
The inequality
III := Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 A − α − n − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) log n holds. OME INEQUALITIES FOR CESÀRO MEANS 7 The detailed proofs
Proof of Lemma 3.
Applying Abel’s transformation, from (2) we get that(5) I ≤ A − αn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m M k − − X i =1 A − α − n − i i X l =1 D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αn (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m A − α − n − M k − M k − X i =1 D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p := I + I , where the first and the second terms on the right side of inequality (5) should be denotedby I and I respectively.For I we can estimate as follows:(6) I ≤ A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m A − α − n − M k − k − X r =1 M r +1 − X i = M r D i ( u ) D i ( v ) × [ f ( · − u, · − v ) − f ( · , · )] (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v ) ≤ A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m A − α − n − M k − k − X r =1 M r +1 − X i = M r D i ( u ) D i ( v ) × [ f ( · − u, · − v ) − S M r ,M r ( · − u, · − v, f )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m A − α − n − M k − k − X r =1 M r +1 − X i = M r D i ( u ) D i ( v ) × [ S M r ,M r ( · − u, · − v, f ) − S M r ,M r ( · , · , f )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m A − α − n − M k − k − X r =1 M r +1 − X i = M r D i ( u ) D i ( v ) × [ S M r ,M r ( · , · , f ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p := I + I + I , where the first, the second and the third terms on the right side of inequality (6) should bedenoted by I , I and I respectively. T. TEPNADZE , L. E. PERSSON It is evident that Z G m M r +1 − X i = M r D i ( u ) D i ( v ) [ S M r ,M r ( · − u, · − v, f ) − S M r ,M r ( · , · , f )] dµ ( u, v )= M r +1 − X i = M r Z G m D i ( u ) D i ( v ) S M r ,M r ( · − u, · − v, f ) dµ ( u, v ) − S M r ,M r ( · , · , f ) = M r +1 − X i = M r ( S i ( · , · , S M r ,M r ( f )) − S M r ,M r ( · , · , f ))= M r +1 − X i = M r ( S M r ,M r ( · , · , f ) − S M r ,M r ( · , · , f )) = 0 . Hence,(7) I = 0 . Moreover, according to the generalized Minkowski inequality, Lemma 2 and by (1) and (4)we obtain that(8) I ≤ A − αn (cid:12)(cid:12)(cid:12) A − α − n − M k − (cid:12)(cid:12)(cid:12) k − X r =1 Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M r +1 − X i = M r D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:13)(cid:13)(cid:13)(cid:13) f ( · − u, · − v ) − S M r ,M r ( · − u, · − v, f ) (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v ) ≤ c ( α ) M k k − X r =1 (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) × Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M r +1 − X i = M r D i ( x ) D i ( y ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) k − X r =1 M r M k (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) . The estimation of I is analogous to the estimation of I and we get that(9) I ≤ c ( α ) k − X r =1 M r M k (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) . OME INEQUALITIES FOR CESÀRO MEANS 9
Analogously, we can estimate I in the following way(10) I ≤ A − αn k − X r =1 (cid:13)(cid:13)(cid:13)(cid:13)Z G m M r +1 − X i = M r A − α − n − i i X l =1 D l ( u ) D l ( v ) × [ f ( · − u, · − v ) − S M r ,M r ( · − u, · − v, f )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αn k − X r =1 (cid:13)(cid:13)(cid:13)(cid:13)Z G m M r +1 − X i = M r A − α − n − i i X l =1 D l ( u ) D l ( v ) × [ S M r ,M r ( · − u, · − v, f ) − S M r ,M r ( · , · , f )] (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v )+ 1 A − αn k − X r =1 (cid:13)(cid:13)(cid:13)(cid:13)Z G m M r +1 − X i = M r A − α − n − i i X l =1 D l ( u ) D l ( v ) × [ S M r ,M r ( · , · , f ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ A − αn k − X r =1 Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M r +1 − X i = M r A − α − n − i i X l =1 D l ( u ) D l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:13)(cid:13)(cid:13)(cid:13) f ( · − u, · − v ) − S M r ,M r ( · − u, · − v, f ) (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v )+ 1 A − αn k − X r =1 Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M r +1 − X i = M r A − α − n − i i X l =1 D l ( u ) D l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:13)(cid:13)(cid:13)(cid:13) S M r ,M r ( · , · , f ) − f ( · , · ) (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v ) ≤ c ( α ) M αk k − X r =1 M r +1 − X i = M r ( n − i ) − α − i (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) ≤ c ( α ) M αk k − X r =1 M r +1 − X i = M r ( n − M r +1 − − α − i (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) ≤ c ( α ) k − X r =0 M r M k (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) . By combining (7)-(9) with (10) for I we find that , L. E. PERSSON (11) I ≤ c ( α ) k − X r =0 M r M k (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) . The proof of Lemma 3 is complete. (cid:3)
Proof of Lemma 4.
It is evident that(12) II ≤ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i D M k − i ( u ) D M k − i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ (cid:12)(cid:12) A − α − p − M k (cid:12)(cid:12) Z G m D M k ( u ) D M k ( v ) dµ ( u, v ) := II + II , where the first and the second terms on the right side of inequality (12) should be denotedby II and II respectively.From (1) by (cid:12)(cid:12) A − α − p − M k (cid:12)(cid:12) ≤ we get that(13) II ≤ . Moreover, by Lemma 3 we have that(14) II ≤ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i ¯ D i ( u ) ¯ D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D M k ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i ¯ D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i ¯ D i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z G m D M k ( u ) D M k ( v ) dµ ( u, v ):= II + II + II + II , where the first, the second, the third and the fourth terms on the right side of inequality(14) should be denoted by II , II , II and II respectively.From (1) and (4) it follows that(15) II ≤ c ( α ) ∞ X v =1 v − α − < ∞ . By Applying Abel’s transformation, in view of Lemma 2 we have that(16) II ≤ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i i X l =1 ¯ D l ( u ) ¯ D l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) OME INEQUALITIES FOR CESÀRO MEANS 11 + Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A − α − p − M k − X i =1 ¯ D i ( u ) ¯ D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) ( M k − X v =1 ( p − M k + i ) − α − i + ( p − − α − M k ) ≤ c ( α ) ( ∞ X i =1 i − α − + M − αk ) < ∞ . The estimation of II and II are analogous to the estimation of II . By ApplyingAbel’s transformation, in view of Lemma 1 we find that(17) II ≤ Z G m D M k ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i i X l =1 ¯ D l ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D M k ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A − α − p − M k − X i =1 ¯ D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) ( M k − X v =1 ( p − M k + i ) − α − i + ( p − − α − M k ) ≤ c ( α ) ( ∞ X i =1 i − α − + M − αk ) < ∞ . and(18) III ≤ Z G m D M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k − X i =1 A − α − p − M k + i i X l =1 ¯ D l ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) A − α − p − M k − X i =1 ¯ D i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) ( M k − X v =1 ( p − M k + i ) − α − i + ( p − − α − M k ) ≤ c ( α ) ( ∞ X i =1 i − α − + M − αk ) < ∞ . The proof is complete by combining (12)-(18). (cid:3) , L. E. PERSSON Proof of Lemma 5.
Let n = n k M k + ... + n k s M k s , k > ... > k s ≥ . Denote n ( i ) = n k i M k i + ... + n k s M k s , i = 1 , , ...s. Since ( see [4])(19) D j + n A M A = D n A M A + ψ n A M A D j , we find that(20) III ≤ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n k M k X i =1 A − α − n − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (2) X i =1 A − α − n (2) − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D n k M k ( u ) D n k M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (2) X i =1 A − α − n (2) − i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D n k M k ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (2) X i =1 A − α − n (2) − i D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D n k M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (2) X i =1 A − α − n (2) − i D i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ):= III + III + III + III + III , where the first, the second, the third, the fourth and the fifth terms on the right side ofinequality (20) should be denoted by III , III , III , III and III respectively.By (1) we have that(21) III ≤ c ( α ) . Moreover, since (see [26] )(22) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 A − α − n − i D i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:0) | u | α − (cid:1) . for III we get that OME INEQUALITIES FOR CESÀRO MEANS 13 (23)
III ≤ Z G m D n k M k ( u ) | v | α − dµ ( u, v ) ≤ Z G m | v | α − dµ ( v ) = 1 α < ∞ . Analogously, we find that(24)
III ≤ Z G m D n k M k ( v ) | u | α − dµ ( u, v ) ≤ Z G m | u | α − dµ ( v ) = 1 α < ∞ . For r ∈ { , ...m A − } , ≤ j < M A , ( see [4]), it yields that D j + rM A = r − X q =0 ψ qM A ! D M A + ψ rM A D j . Thus, we have that Z G m n k M k − X i =1 A − α − n − i D i ( u ) D i ( v ) dµ ( u, v ) ≤ Z G m n k − X r =0 M k − X i =0 A − α − n − i − rM k D i + rM k ( u ) D i + rM k ( v ) dµ ( u, v ) ≤ Z G m n k − X r =0 M k − X i =0 A − α − n − i − rM k r − X q =0 ψ qM k ! D M k ( u ) × r − X q =0 ψ qM k ! D M k ( v ) dµ ( u, v )+ Z G m n k − X r =0 M k − X i =0 A − α − n − i − rM k r − X q =0 ψ qM k ! D M k ( u ) ψ rM A D i ( v ) dµ ( u, v )+ Z G m n k − X r =0 M k − X i =0 A − α − n − i − rM k ψ rM A D i ( u ) r − X q =0 ψ qM k ! D M k ( v ) dµ ( u, v ) , L. E. PERSSON + Z G m n k − X r =0 M k − X i =0 A − α − n − i − rM k ψ rM A D i ( u ) ψ rM A D i ( v ) dµ ( u, v ) . On the other hand, by (1) and (4) we obtain that Z G m A − α − n − n k M k D n k M k ( u ) D n k M k ( v ) dµ ( u, v ) ≤ c ( α ) . Consequently, for
III we have the estimate(25) III ≤ Z G m D M k ( u ) D M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n k − X r =0 M k X i =1 A − α − n − i − rM k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D M k ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n k − X r =0 M k X i =1 A − α − n − i − rM k D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m D M k ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n k − X r =0 M k X i =1 A − α − n − i − rM k D i ( u ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v )+ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n k − X r =0 M k X i =1 A − α − n − i − rM k D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) + c ( α ):= III + III + III + III + c ( α ) . where the first, the second, the third and the fourth terms on the right side of inequality(25) should be denoted by III , III , III and III respectively.From Lemma 4 we have that(26) III ≤ c ( α ) . The estimation of
III is analogous to the estimation of III and we find that(27) III ≤ c ( α ) . The estimation of
III and III is analogous to the estimation of III and we obtainthat(28) III < ∞ , and(29) III < ∞ . OME INEQUALITIES FOR CESÀRO MEANS 15
After substituting (21) and (23)- (29) into (20) we conclude that Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 A − α − n − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (2) X i =1 A − α − n (2) − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) + c ( α ) ≤ ... ≤ Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n ( s ) X i =1 A − α − n ( s ) − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) + c ( α ) s ≤ c ( α ) + c ( α ) s ≤ c ( α ) log n. The proof is complete. (cid:3)
Now we are ready to prove the main results
Proof of Theorem 1.
It is evident that(30) (cid:13)(cid:13) σ − αM k ( f ) − f (cid:13)(cid:13) p ≤ A − αM k − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m M k − X i =1 A − α − M k − i D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αM k − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m M k X i = M k − +1 A − α − M k − i D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p := I + II.
From Lemma 5 it follows that(31) I ≤ c ( α ) k − X r =0 M r M k (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) . Moreover, for II we have the estimate(32) II ≤ A − αM k − (cid:13)(cid:13)(cid:13)(cid:13)Z G m M k X i = M k − +1 A − α − M k − i D i ( u ) D i ( v ) × h f ( · − u, · − v ) − S (1) M k − ( · − u, · − v, f ) i dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p , L. E. PERSSON + 1 A − αM k − (cid:13)(cid:13)(cid:13)(cid:13)Z G m M k X i = M k − +1 A − α − M k − i D i ( u ) D i ( v ) × h S (1) M k − ( · − u, · − v, f ) − f ( · , · ) i dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p := II + II , where the first and the second terms on the right side of inequality (32) should be denotedby II and II respectively.In view of generalized Minkowski inequality, by (4) and using Lemma 5 we get that(33) II ≤ A − αM k − Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M k X i = M k − +1 A − α − M k − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:13)(cid:13)(cid:13) f ( · − u, · − v ) − S (1) M k − ( · − u, · − v, f ) (cid:13)(cid:13)(cid:13) p dµ ( u, v ) ≤ c ( α ) M αk ω ( f, /M k − ) p . The estimation of II is analogous to the estimation of II and we find that(34) II ≤ c ( α ) M αk ω ( f, /M k − ) p . Combining (30)- (34) we receive the proof of Theorem 1. (cid:3)
Proof of Theorem 2.
It is evident that(35) (cid:13)(cid:13) σ − αn ( f ) − f (cid:13)(cid:13) p ≤ A − αn − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m M k − X i =1 A − α − n − i D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αn − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m M k X i = M k − +1 A − α − n − i D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p + 1 A − αn − (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z G m n X i = M k +1 A − α − n − i D i ( u ) D i ( v ) [ f ( · − u, · − v ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p := I + II + III, where the first, the second and the third terms on the right side of inequality (35) should bedenoted by I , II and III respectively.
OME INEQUALITIES FOR CESÀRO MEANS 17
From Lemma 4 it follows that(36) I ≤ c ( α ) k − X r =0 M r M k (cid:16) ω ( f, /M r ) p + ω ( f, /M r ) p (cid:17) . Next, we repeat the arguments just in the same way as in the proof of Theorem 1 and findthat(37) II ≤ c ( α ) M αk (cid:16) ω ( f, /M k − ) p + ω ( f, /M k − ) p (cid:17) . On the other hand, for III we have that(38)
III ≤ A − αn − (cid:13)(cid:13)(cid:13)(cid:13)Z G m n X i = M k +1 A − α − n − i D i ( u ) D i ( v ) × [ f ( · − u, · − v ) − f ( · , · )] (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v ) ≤ A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m n X i = M k +1 A − α − n − i D i ( u ) D i ( v ) ×× [ f ( · − u, · − v ) − S M k ,M k ( · − u, · − v, f )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m n X i = M k +1 A − α − n − i D i ( u ) D i ( v ) ×× [ S M k ,M k ( · − u, · − v, f ) − S M k ,M k ( · , · , f )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p ≤ A − αn (cid:13)(cid:13)(cid:13)(cid:13)Z G m n X i = M k +1 A − α − n − i D i ( u ) D i ( v ) ×× [ S M k ,M k ( · , · , f ) − f ( · , · )] dµ ( u, v ) (cid:13)(cid:13)(cid:13)(cid:13) p := III + III + III , where the first, the second and the third terms on the right side of inequality (38) should bedenoted by III , III and III respectively.It is easy to show that(39) III = 0 . , L. E. PERSSON According to the generalized Minkowski inequality and by using Lemma 5 for
III we obtainthat(40) III ≤ A − αn Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i = M k +1 A − α − n − i D i ( u ) D i ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:13)(cid:13)(cid:13)(cid:13) f ( · − u, · − v ) − S M r ,M r ( · − u, · − v, f ) (cid:13)(cid:13)(cid:13)(cid:13) p dµ ( u, v ) ≤ c ( α ) M αk (cid:16) ω ( f, /M k − ) p + ω ( f, /M k − ) p (cid:17) × Z G m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X v = M k +1 A − α − n − v D v ( u ) D v ( v ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dµ ( u, v ) ≤ c ( α ) M αk log n (cid:16) ω ( f, /M k − ) p + ω ( f, /M k − ) p (cid:17) . The estimation of
III is analogous to the estimation of III and we find that(41) III ≤ c ( α ) M αk log n (cid:16) ω ( f, /M k − ) p + ω ( f, /M k − ) p (cid:17) . After substituting (36)- (37), (41) into (35), we receive the proof of Theorem 2. (cid:3)
Author details The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway. The Artic University of Norway, Campus Narvik, P.O. Box 385, N-8505, Narvik, Norway.
Authors’ contributions
The authors contributed equally to the writing of this paper. Both authors approved thefinal version of the manuscript.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
The authors would like to thank the referees for helpful suggestions.
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