Divergence of General Localized Operators on the Sets of Measure Zero3

arXiv:0912.1453v1 [math.CA] 8 Dec 2009 n nteppr .L lynv[6 n .L ae[7.W aenot have w We operators [17]. linear Wade of L. sequences W. general and for [16] Ul’yanov common L. Fourier is P. divergent phenomena papers the f the concerning that in problems proved Lu m find other Walsh-Fejer A. [6] which About Goginava M. function set. bounded U. papers that a Recently the exists in there [13]. zero also Prochorenko measure considered I. is V. problems Walsh-Kachmaz and such and Walsh-Paley in ([2]), system Haar Haar for also same the proves lofrteVlni ytm.Te .M uaz 1 rvdta f that from proved taken [1] p be can Bugadze he theorem M. [10] mentioned V. paper the Then another in In function systems. system zero. Vilenkin measure the of for set also given a on diverges ec fFuirseries. Fourier of gence u elfnto hc ore eisdvre nta e.Tesa S from The systems. function ex orthonormal set. a there classical that constructed zero other on [9] measure the diverges of for series set also Fourier any investigated which for Kahane-Katzne function that developing real proved Essentially which ous [3] function zero. Buzdalin valued V. measure complex V. of continuous set a given of a existence the proved ore eisdvre on diverges series Fourier ea eoeCreo’ hoe.FrtS .Sehi 1]i 91p 1951 in of [14] investigation Stechkin the B. fact S. In First set any zero. theorem. for measure Carleson’s of before set began arbitrary given a ae from taken swehrteFuirsre fafnto from function a of series Fourier the whether is eeeyhr.O h te adacrigt alsnHn th Carleson-Hant to from according functions hand the other of the series Fourier On everywhere. be hc rgnmti ore eisdvre tsm on.I 93A 1923 In point. from function some a for at that diverges proved [11] series mogorov Fourier trigonometric which 1991 e od n phrases. and words Key n17 .D osRyod[]cntutda xml fcontinuo of example an constructed [5] Bois-Reymond Du P. 1876 In IEGNEO EEA OAIE PRTR ON OPERATORS LOCALIZED GENERAL OF DIVERGENCE ahmtc ujc Classification. Subject Mathematics Abstract. rpry ti rvdta o n set any for that proved is It property. faaoosterm nw o h ore usoeaosw operators sums Fourier the systems. orthogonal for different known theorems analogous of o which for E L p (0 ⊂ U , T n 2 ecnie eune flna operators linear of sequences consider We I π G fmauezr hr xssafunction a exists there zero measure of o n 1 any for ) H ESO ESR ZERO MEASURE OF SETS THE ( x iegsa ahpoint each at diverges ) eea prtr,lclzto rpry iegneo a on divergence property, localization operators, general E hni 93L .Tio 1]proved [15] Taikov V. L. 1963 in Then . .A KARAGULYAN A. G. 1. L ≤ p Introduction L (0 < p p , ( 1)(1 42A20. T ) 1 ∞ > p , E x n16 aaeadKtnlo [8] Katznelson and Kahane 1965 In . < p < fmauezr hr xssaset a exists there zero measure of ∈ L E 1 1 ( , hsrsl sageneralization a is result This . T ovreae.Antrlquestion natural A a.e.. converge L iegneo ore eiscan series Fourier of divergence ) p ∞ ( > p hc ore-as series Fourier-Walsh which ) U n f ( x )or 1) ihlocalization with ) L f t epc to respect ith ∞ ∈ C .V hldein Kheladze V. h. nfc Bugadze fact In . eisrae can reader series asdvre on diverges eans L a ieg on diverge may oe h same the roved t localization ith oe [] [7]) ([4], eorem eqeto is question me 2 ssacontinu- a ists snapproach lson (0 raystof set any or rteWalsh the or hsproblem this systems([1]). iegson diverges , sfunction us oe that roved 2 G .Kol- N. . f π ia[12] nina e,diver- set, cdthis iced which ) a be can 2 G. A. KARAGULYAN property. We consider sequences of linear operators b (1) Unf(x)= Kn(x, t)f(t)dt, n =1, 2,..., Za with

(2) |Kn(x, t)|≤ Mn. We say the sequence (1) has a localization property (L-property) if for any function f ∈ L1(a,b) with f(x)=1 as x ∈ I = (α, β)(⊂ [a,b]) we have

lim Unf(x)=1 as x ∈ I, n→∞ and the convergence is uniformly in each closed set A ⊂ I. We prove the following Theorem. If the sequence of operators (1) has a localization property, then for any set of measure zero E ⊂ [a,b] there exists a set G ⊂ [a,b] such that

lim inf UnIG(x) ≤ 0, lim sup UnIG(x) ≥ 1 for any x ∈ E, n→∞ n→∞ where IG(x) denotes the characteristic function of G. The result of the theorem can be applied to the Fourier partial sums operators with respect to all classical orthogonal systems(trigonometric, Walsh, Haar, Franklin and Vilenkin systems). Moreover instead of partial sum we can discuss also linear means of partial sums corresponding to an arbitrary regular method T = {aij}. All these operators have localization property. So the following corollary is an immediate consequence of the main result.

Corollary. Let Φ = {φn(x), n ∈ N}, x ∈ [a,b], be one of the above mentioned orthogonal systems and T is an arbitrary regular linear method. Then for any set E of measure zero there exists a set G ⊂ [a,b] such that the Fourier series of its characteristic function f(x)= IG(x) with respect to Φ diverges on E by T -method. Remark. The function f(x) in the corollary can not be continuous in general. There are variety of sequences of Fourier operators which uniformly converge while f(x) is continuous.

The following lemma gives a bound for the kernels of operators (1) if Un has L- property.

Lemma. If the sequence of operators Un has L-property, then there exists a positive decreasing function φ(u), u ∈ (0, +∞), such that if x ∈ [a,b] and n ∈ N then

(3) |Kn(x, t)|≤ φ(|x − t|) for almost all t ∈ [a,b]. Proof. We deﬁne

φ(u) = sup essup t:|t−x|≥u|Kn(x, t)|, n∈N,x∈[a,b] where essup t∈A|g(t)| denotes kgkL∞(A). It is clear φ(u) is decreasing and satisﬁes (3), provided φ(u) < ∞, u> 0. To prove φ(u) is ﬁnite, let us suppose the converse, that is φ(u0)= ∞ for some u0 > 0. It means for any γ > 0 there exist lγ ∈ N and cγ ∈ [a,b] such that

(4) |Klγ (cγ ,t)| >γ, t ∈ Eγ ⊂ [a,b] \ (cγ − u0,cγ + u0), |Eγ | > 0. DIVERGENCE OF GENERAL LOCALIZED OPERATORS ON THE SETS OF MEASURE ZERO3

Consider the sequences ck and lk corresponding to the numbers γk = k, k =1, 2,.... We can ﬁx an interval I with |I| = u0/3, where the sequence {ck} has inﬁnitely many terms. Using this it can be supposed cγ ∈ I in (4) and therefore we will have 2I ⊂ (cγ − u0,cγ + u0) . So we can write

(5) cγ ∈ I, Eγ ⊂ [a,b] \ 2I.

Thus we chose a sequence γk ր ∞ such that for corresponding sequences mk = lγk , xk = cγk and Ek = Eγk we have

(6) xk ⊂ I, Ek ⊂ (a,b) \ 2I, 3 (7) |Kmk (xk,t)|≥ k , t ∈ Ek, I (8) sup |Umk Ei (x)| < 1, x ∈ I, 1≤i

(9) |Ek| · max Mmi < 1, (k> 1). 1≤i

We do it by induction. Taking γ1 = 1 we will get m1 satisfying (7). This follows from (4). Now suppose we have already chosen the numbers γk and mk satisfying I (6)-(9) for k = 1, 2,...,p. According to L-property Un Ei (x) converge to 0 uniformly in I for any i =1, 2,...,p. On the other hand because of (2) and (4) from 3 γ → ∞ follows lγ → ∞. Hence we can chose a number γp+1 > (p + 1) such that the corresponding mp+1 satisﬁes the inequality I (10) |Ump+1 Ei (x)| < 1, x ∈ I, i =1, 2, . . . , p. 3 This gives (8) in the case k = p+1. According to (4) and the bound γp+1 > (p+1) we will have also (7). Finally, since Ep+1 may have enough small measure we can guarantee (9) for k = p + 1. So the construction of the sequence γk with (6)-(9) is complete. Now consider the function ∞ I (x) (11) g(x)= Ei . k2 Xi=1 1 We have g ∈ L and supp g ⊂ [a,b] \ (2I). Since xk ∈ I, using the relations (6)-(9), we obtain

|Umk g(xk)|≥ k−1 ∞ |U I (x )| |U I (x )| |U I (x )| ≥ mk Ek k − mk Ei k − mk Ei k k2 i2 i2 Xi=1 i=Xk+1 k−1 1 ∞ |E | ≥ k − − M i ≥ k − 2. i2 mk i2 Xi=1 i=Xk+1

This is a contradiction, because the convergence Ung(x) → 0 is uniformly on I according to L-property.

We say a family I of mutually disjoint semi-open intervals is a regular partition for an open set G ⊂ (a,b) if G = ∪I∈I I and each interval I ∈ I has two adjacent intervals I+, I− ∈ I with (12) 2I ⊂ I∗ = I ∪ I+ ∪ I−. It is clear any open set has a regular partition. 4 G. A. KARAGULYAN

Proof of Theorem. For a given set E of measure zero we will construct a definite sequence of open sets Gk, k = 1, 2,..., with regular partitions Ik, k = 1, 2,.... They will satisfy the conditions 1) if I ∈ Ik and I = [α, β) then α, β 6∈ E, k ∅ 2) if I,J ∈ ∪j=1Ij then J ∩ I ∈{ ,I,J}, 3) E ⊂ Gk ⊂ Gk−1, (G0 = [a,b]). In addition for any interval I ∈ I we fix a number ν(I) ∈ N such that k 4) if I,J ∈ ∪j=1Ij and I ⊂ J then ν(I) ≥ ν(J), I 2 5) supx∈I |Uν(I) Gl (x) − 1| < 1/k , if I ∈ Ik and l ≤ k, I 2 6) supx∈I |Uν(I) Gk (x)| < 1/k , if I ∈ Il and l < k. We define G1 and its partition I1 arbitrarily, just ensuring the condition 1). It may be done because |E| = 0 and so Ec is everywhere dense on [a,b]. Then using L-property for any interval I ∈ I1 we can find a number ν(I) ∈ N satisfying 5) for k = 1. Now suppose we have already chosen Gk and Ik with the conditions 1)-6) for all k ≤ p. Obviously we can chose an open set Gp+1, E ⊂ Gp+1 ⊂ Gp, satisfying 1), 2) and the bound p |Gp+1 ∩ I| <δ(I), I ∈ ∪k=1Ik, where 1 δ(I)= , φ(|I|/2) 2 − 6(p + 1) max Mν(I),Mν(I+),Mν(I ), |I| n o and the function φ(u) is taken from the lemma. Suppose I ∈ Il and l

I ∗ c |Uν(I) Gp+1∩(I ) (x)|

≤ φ(|x − t|)dt + − ZGp+1∩J J∈Il:JX6=I,I ,I |J| ≤ φ dt + − ZGp+1∩J 2 J∈Il:JX6=I,I ,I (14) |J| ≤ |Gp+1 ∩ J|φ + − 2 J∈Il:JX6=I,I ,I |J| ≤ δ(J)φ + − 2 J∈Il:JX6=I,I ,I 1 1 ≤ 2 |J| < 2 , x ∈ I. 6(p + 1) + − 6(p + 1) J∈Il:JX6=I,I ,I On the other hand we have

+ − 1 δ(I),δ(I ),δ(I ) ≤ 2 , 6 · (p + 1) Mν(I) DIVERGENCE OF GENERAL LOCALIZED OPERATORS ON THE SETS OF MEASURE ZERO5 and therefore

I ∗ ∗ |Uν(I) Gp+1∩I (x)|≤ Mν(I)|Gp+1 ∩ I | (15) 1 ≤ M (δ(I)+ δ(I+)+ δ(I−)) ≤ , x ∈ [a,b]. ν(I) 2(p + 1)2 Combining (13),(14) and (15) we get 6) in the case k = p + 1. Now we chose the partition Ip+1 satisfying just conditions 1) and 2). Using L property we may deﬁne numbers ν(I) for I ∈ Ip+1 satisfying the condition 5) with k = p + 1. Hence the construction of the sets Gk is complete. Now denote ∞ G = (G2i−1 \ G2i), i[=1 we have ∞ I k+1 I Un G(x)= (−1) Un Gk (x). Xk=1 For any x ∈ E there exists a unique sequence I1 ⊃ I2 ⊃ ... ⊃ Ik ⊃ ..., Ik ∈ Ik, such that x ∈ Ik, k =1, 2,.... According to 6) for l > k we have 1 |U I (x)|≤ , l > k. ν(Ik) Gl l2 From 5) it follows that 1 |U I (x) − 1|≤ , l ≤ k. ν(Ik) Gl k2 Thus we obtain k I l+1 |Uν(Ik) G(x) − (−1) | Xl=1 k ∞ I I ≤ |Uν(Ik) Gl (x) − 1| + |Uν(Ik) Gl (x)| Xl=1 l=Xk+1 1 ∞ 1 2 ≤ k · + < k2 l2 k l=Xk+1 k k+1 Since the sum i=1(−1) takes values 0 and 1 alternately we get P lim Uν(I )IG(x)=0, lim Uν(I )IG(x)=1 t→∞ 2t t→∞ 2t+1 for any x ∈ E. The proof of theorem is complate.

References [1] Bugadze, V. M. Divergence of Fourier-Walsh series of bounded functions on sets of measure zero. (Russian) Mat. Sb. 185 (1994), no. 7, 119–127 [2] Bugadze, V. M. On the divergence of Fourier-Haar series of bounded functions on sets of measure zero. (Russian) Mat. Zametki 51 (1992), no. 5, 20–26. [3] Buzdalin, V. V. Unboundedly diverging trigonometric Fourier series of continuous functions. (Russian) Mat. Zametki 7 1970 7–18. [4] Carleson L., On convergence and growth of partial sums of Fourier series, Acta Math., 1966, 116 , 135–157, [5] Du Bois-Reymond T. , Untersuchungen ¨uber die Convergenz und Divergenz der Fourierschen Darstellungsformen, Abh. Akad. Wiss. M¨unchen, X (1876), 1 103. 6 G. A. KARAGULYAN

[6] Goginava U., On devergence of Walsh-Fejer means of bounded functions on sets of measure zero. Acta Math. Hungarica 121 (3) (2008), 359-369 [7] Hunt R.A., On the convergence of Fourier series in orthogonal expansions and their continuous analogues, Southern Illinois University Press, 1968, 235–256, [8] Kahane J-P., Katznelson Y., Sur les ensembles de divergence des series trigonometriques, Studia math., XXVI (1966), 305306 [9] Kheladze Sh. V., On everywhere divergence of Fourier series by bounded type Vilenkin systems, Trudi Tbil. Math. Ins. Gruz. SSR, 1978, v. 58, 225-242. [10] Kheladze Sh. V., On everywhere divergence of Fourier-Walsh series, Soobsh. Geargian ANS, 1975, v.77, No 2, 305-307. [11] Kolmogoroﬀ A. N. , Une serie de Fourier-Lebesgue divergente partout, C. r. Acad, Sci., Paris, 183 (1926), 13271328. [12] Lunina M. A., On the sets of unbounded divergence of series with respect to Haar system, Vestnik MGU, math., 1976, No 4, 13-20. [13] Prochorenko V. I., Divergent series with respect to Haar system, Izvestia Vuzov, Math., 1971, No 1, 62-68. [14] Stechkin S. B., On the convergence and divergence of trigonometric series, Uspechi Math. Nauk, VI, vol. 2(42), 1951,148-149 (in Russian). [15] Taikov L. V., On the divergence of Fuorier series by rearanged trigonometric system, Uspechi Math. Nauk, XVIII, v. 5(113), 1963, 191-198 (in Russian). [16] Ul’yanov P. L., A. N. Kolmogorov and divergent Fourier series, Jour. Russ. Math. Surv. 1983, vol. 38, No.4, 57-100; [17] Wade W. R., Resent development in the theory of Walsh series, Internat. Journ. of Math. and Math, Sci., 1982, vol. 5, No 4, 625-673.

Institute of Mathematics of Armenian National Academy of Sciences, Baghramian Ave.- 24b, 375019, Yerevan, Armenia E-mail address: [email protected]