Мат. Сборник Math. USSR Sbornik Том 106(148) (1978), Вып. 3 Vol. 35(1979), o. 1 REMARKS O ESTIMATIG THE LEBESGUE FUCTIOS OF A ORTHOORMAL SYSTEM UDC 517.5 B. S. KASl ABSTRACT. In this paper we clarify the relationship between lower bounds for Lebesgue functions and multiplicative inequalities of Gagliardo-irenberg type. We give simple proofs of some theorems on Lebesgue functions. Bibliography: 10 titles. Let {*p k (x)} be an orthonormal system of functions (O..S.), defined on the interval [0, 1]. The Lebesgue functions L n (t) of the system are defined by.(0 = j n 2<p*(0<M*) =1 dx. (1) Estimates for the Lebesgue functions of various O..S. have long played an important role in the theory of orthogonal series (for details see [1] and [3]). In the present paper we are concerned only with lower bounds for the Lebesgue functions of collectively bounded O..S. The first such bound was obtained by Olevskii [4] in 1966; he proved the following theorem. THEOREM A. For every collectively bounded O..S. {q> k (x)} (i.e., \.q> k (x)\ < M, 1 < к < oo, x G [0, 1]) we have lim L n (t) = oc 9 t E, mes >c^>0. ff->oo In 1967 Olevskii [5] announced, and in 1975 he published in the book [3], the following stronger result. THEOREM B. Under the hypotheses of Theorem A, the inequality holds for n = 1,2,... max 1п ет1 /г.1 т (0>с'>0, t E n, mes n >c>0 9 (2 \ Olevskifs method of proof can be said to be direct, since it depends only on consideration of the system {<p k (x)} itself. The problem of a nontrivial lower bound for the Lebesgue functions has a sort of dual 1980 mathematics subject classification. Primary 42C05, 42C15. American Mathematical Society 1979 57
58 В. S. KASI problem, namely to construct, for a given complete and collectively bounded O..S. {ч>к( х )}Т> а function/(л:) of bounded variation such that k=i ^ q> h (x)f(x)dx This problem was considered in 1974 by Bockarev (for details see [6] and [7]), who showed, in particular, that the following theorem holds. THEOREM С Let {q) k (x)} be a complete and collectively bounded O..S., and let functions xf*(x) (1 < j < 9 = 2 r, r > 0 (an integer)) be defined by Also let x, (x) = 2J 0 a J k<p k (x). Then О, я< / 1 Xf(x) 1, x>-l, linear on l-i Ц J со 2 2 I4 >cin^. (3) /=1 Bockarev's proof was based on properties of the Haar system, and in particular it used the following inequality: given a series 2? ct k Xk(x) in the Haar system, with \8 n (x)\ < 2~ w, 0 < n < oo, where 8 0 (x) = a x x x (x) and 8 n (x) = S^-i + i кхк( х Х п > 0; then CO (4) =1 \L X {ji^w^. In 1975 the author (see [8]) suggested a proof of Theorem С based on properties of the Hilbert matrix. Also in 1975 Bockarev applied (4) to prove the following result (see [9]). THEOREM D. Let {q> k (x)}f be a collectively bounded O..S. Then for n = 1, 2,... У L n (t) > c' In я, tee n, mese n > ti (5) It is clear that (5) implies (2); the author has remarked that Theorem D also follows easily from Olevskifs inequality (see [3] and [5]): Under the hypotheses of Theorem A, max *2s4>k{x)\>c\nn^ л=1,2,.... k=i \\ L x In [10] Bockarev showed that (5) is a simple coroeary of the following numerical inequality: For an arbitrary sequence {%}f and for n = 1, 2,..., ( n max 1 a k 1 E \p=i p 2> + n n E* &=i 2' 41 I>4>* 2k 1 ал/>2 (6)
LEBESGUE FUCTIOS OF ORTHOORMAL SYSTEMS 59 where is determined from 2 ~ l < n <2 and {^(0} w the Schauder system {see [\\p. 50). In proving (6), Bockarev used inequalities like (4). In concluding this survey we note that A. S. Krancberg recently suggested a proof of Theorem D based (like the discussion in [8]) on properties of the Hilbert matrix. In the present paper we give proofs, essentially simpler than previous proofs, of the theorems quoted above on estimates for Lebesgue functions. The author has made the two following simple remarks: a) Theorem D follows at once from the statement that for arbitrary numbers {a k } n 2 P 2> [l \ 2 _S v max I a k 11 "^ i ^ n.u i i ^> r. -v V. vft= v+1 ' ( 7 ) x ^ ^ \ p=i P=i J k=i AJ=I I/ J v=o p,=v+i i=v+l ^ (we note that (6) and (7), and likewise the methods of using them, have much in common)^1) b) Inequality (7) is a discrete version of the following proposition which is related to well-known multiplicative inequalities for derivatives: For every f{x), x E ( oo, oo) = R 1, iv a (/,^/2 )<^l/iu,in, ra(r1). (8) where W^2 is a Sobolev space {with the norm / w\/i = Ц/И^^ + {f W^2)\ and the functional {f, W^2) is defined by /OO \ I/ 2 ВД W\ /2 ) = J1 (Д,/) (x)lh m h~~4h, (А л /) (x) = f(x+h)-f (*), and /' L- = sup^> 0 ^KVX*)!- Inequality (8) is similar to the Gagliardo-irenberg inequality (cf. [2], p. 237). The difference is that (8) involves a Sobolev space W\^2 of fractional order. The analogy suggests that it would be desirable to find a proof of (8) that would seem natural from the point of view of the theory of Sobolev classes. In response to this suggestion, O. V. Besov found a very simple proof. We shall give his argument below. We now turn to the proof. 1) Deduction of inequality (8) (Besov). Without loss of generality we suppose that H/'H^oo =.1. It is well known that, for 1 < p < oo, \\f(m p LP(m=p]t^(t)dt, where \{t) = mes{x: \f{x)\ > t). For every n > 0 we define the sets E(h) = {x:\f(x)\^h} f) {*:\f(x + П)\<Щ, F(h)=R4E(h). It is clear that mes F{h) < 2\{h) and that for arbitrary x and h > 0 we have the following inequalities: a) (Д й /)(*)1 < l/mi + \Лх + A) ; b) (Д*/Х*)1 <» (^Inequality (6) can be derived from (7).
60 В. S. KA I ow (A,/) (x) 2(Rl) = 1 (A,/) (*) \} 2 L4Em + (A,/) (x) ll 4Fm ; if we estimate the first term by using a) and the second by using b), we have Consequently (Дл/)(*) Ь<*)<2 J f (x) + f 2 (x + K)dx + h*mesf(») E\h) n ^4 Г f (*) dx + 2Й 2 Я (й) = 8 Г t (I (t) Я (»)) Л {*: /<*)I5»} о П + 2h 2 K(h)^8 [tl(t)dt. OO fr OO OO 00 # 2 (/, Wl /2 )<8 ftt 2 [t%(t)dtd% = 8 f \h-*t%{t)dhdt =?> {X(t)dt = 8\\f\\ LK 0 0 0 * 0 2) Derivation of inequality (7). For a given set of numbers {%}" we define a continuous function f(x) on the real axis by putting / 1 2«л for I H K J, 1</<л, 0 f(x) 0 for x<l/3, linear on /С* +n) = f(n x). ' -T.'--f, 1</<л, It is easy to see that Applying (8) to f(x) yields ЦП <3max a,, /«L1 <52!<&<«/ max ^17 \ /p=l 2a k >c*(f,wr), 1 1 f(y)-f(x) -*tt[,j^*]«*->--2 2 П^ >TS S V=0 X=V+1 x j V LI 3 * 3 2 «* 9 ^J < J (V [X)2 v=o LI=V+I 3) Proof of Theorem D. For a given я > 10 let dydx [3/1/4] А=[П/4]
LEBESGUE FUCTIOS OF ORTHOORMAL SYSTEMS 61 From the evident relations \\F n (t)\\ L i > n/3 and \\F n (i)\\ L «> < M-n 9 it follows that mes E n = mes {7: F n (t) > c'n) > с > 0. Applying (7) to the set of numbers {*p k (x)<p k (t)}1 and integrating with respect to x with t E E n, we obtain ZMO= jz m=i k=i о m=i ^^k(x)(f k (t) \dx 0 V=0 IX V+l Ф*(*)Ф*(0.&=V+1 -] = ЗФ*(0- S lv-n'~ 2 >cln«2 Ф*(0>сЧпл-л, d# as required. In conclusion, we show that results like Theorem С can be obtained from (8). We shall prove the following result of Bockarev: Let {<p k (x)} be a complete and collectively bounded O..S.; then Q ='S /2=1 0 U^k(x)dx dt = oo. In fact, applying (8) to f k (t), where f k {t) = / > q> k {x) dx for 0 < t < 1; f k (t) = 0 for t < 0; and,4(1 + 0 = /^(l t), and using Parseval's equation, we have 1/24 Q^TSIMOU^^S^^^n 2 11 > C S \\(У г )~ 2 ( i<pk(x) d x) dydz = c{{-~-dydz == oo. o z \Z / 0 2 Similarly we can derive the following theorem from inequalities closely related to (8): THEOREM 1. Let {*p k (x)} be a complete and collectively bounded O..S. For every function Ф(0 G C 2 (0, oo) such that Ф(0) = Ф'(0) = 0, Ф'(0 > Ofor t > 0, Ф'(г) z _1 j on (0, 1), and /о / -1 Ф'(0 dt = oo, we have the equation Ф ЩФ* (*)<* > d/ == oo e /e=lo We also note a corollary of the inequality 2 (f Ш\ /ъ ) < C / L 2- \\f'\\ L 2. THEOREM 2. For every complete O..S. {*p k (x)} we have t ^Pk(x) dx ll 2 (o,l) Received 13/FEB/78
62 В. S. KA I BIBLIOGRAPHY 1. Stefan Kaczmarz and Hugo Steinhaus, Theorie der Orthogonalreihen, PW, Warsaw, 1935. 2. O. V. Besov, V. P. Il'in and S. M. ikol'skii, Integral representations of functions and embedding theorems, "auka", Moscow, 1974. (Russian) 3. A. M. Olevskii, Fourier series with respect to general orthogonal systems, Springer» Verlag, Berlin and ew York, 1975. 4. _, Fourier series of continuous functions relative to bounded orthonormal systems, Izv. Akad. auk SSSR Ser. Mat. Ж (1966), 387-432; English transl. in Amer. Math. Soc. Transl. (2) 67 (1968). 5., Fourier series and Lebesgue functions, Uspehi Mat. auk 22 (1967), no. 3 (135), 237-239. (Russian) 6. S. V. Bockarev, On the absolute convergence of Fourier series in bounded complete orthonormal systems of functions, Mat. Sb. 93(135) (1974), 203-217; English transl. in Math. USSR Sb. 22 (1974). 7., Absolute convergence of Fourier series with respect to bounded systems, Mat. Zametki 15 (1974), 363-370; English transl. in Math. otes 15 (1974). 8. JB. S. Kasin, The coefficients of the expansion of a certain class of functions in complete systems, Sibirsk. Mat. Z. 18 (1977), 122-131; English transl. in Siberian Math. J. 18 (1977). 9. S. V. Bockarev, Logarithmic growth of arithmetic means of Lebesgue functions of bounded orthonormal systems, Dokl. Akad. auk SSSR 223 (1975), 16-19; English transl. in Soviet Math. Dokl. 16 (1975). 10., A Fourier series in an arbitrary bounded orthonormal system that diverges on a set of positive measure, Mat. Sb. 98(140) (1975), 436-449; English transl. in Math. USSR Sb. 27 (1975). Translated by R. P. BOAS