K. NAGY Now, introduce an orthonormal system on G m called Vilenkin-like system (see [G]). The complex valued functions rk n G m! C are called general

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1 A HARDY{LITTLEWOOD{LIKE INEQUALITY ON TWO{DIMENSIONAL COMPACT TOTALLY DISCONNECTED SPACES K. Nagy Abstract. We prove a Hardy-Littlewood type ineuality with respect to a system called Vilenkin-like system (which is a common generalisation of several well-known systems ) in the two-dimensional case.. Introduction Let P denote the set of positiv integers, N = P [f0g. For any set E let E the cartesian product E E. Thus N is the set of integral lattice points in the rst uadrant. Let m = (m 0 ;m ; ; m k ; ) ( m k N;k N) be a seuence of natural numbers and denote by G mk a set of which the number of elements is m k A measure on G mk is given in the way that k (fjg) = m k (j G mk ;k N) Let the topology be the discrete topology on the set G mk Let G m be the complete direct product of the compact spaces G mk (k N) with the product of the topologies and measures (). The elements of G m are of the form x = (x 0 ;x ; ; x k ; ) with x k G mk (k N) G m is called a Vilenkin space. The Vilenkin space G m is said to be bounded Vilenkin space if the generating seuence m is a bounded one. In this paper the boundedness of G m is supposed. A base of the neighborhoods can be given in the following way I 0 (x) =G m ; I n (x) =fy G m y =(x 0 ; ; x n, ;y n ; )g (x G m ;n P); I n = I n (0) for n N Denote by L p (G m ) the usual Lebesgue spaces (kk p the coresponding norms ) ( p ), A n the -algebra generated by the sets I n (x) (x G m ) and E n the conditional expectation operator with respect to A n (n N) If we dene the seuence ( k N) by M 0 = and = m 0 m m k, (k P) then each n N P has a unie representation of the for = n k ; where 0 n k <m P k (n k N) Let jnj = maxfk N n k 6=0g (that is M jnj n<m jnj+ ) and n (k) = j=k n k. 980 Mathematics Subject Classication (985 Revision). primary 4C0, secondary 4C5, 43A75, 40G05. Research supported by the Hungarian National Foundation for Sientic Research (OTKA), grant no. F05765, F00334 the foundation "Pro Regione" grant no 538-I/3/997, by the Hungarian "M}uvel}odesi es Kozoktatasi Miniszterium ", grant no. PFP 5306/997 and FKFP 070/997.. Typeset by AMS-TE

2 K. NAGY Now, introduce an orthonormal system on G m called Vilenkin-like system (see [G]). The complex valued functions rk n G m! C are called generalised Rademacher functions if the following properties holds i. rk n is A k+ measurable, rk 0 =for all k; n N ii. If is a divisor of n; l and n k+ = l k+ (n; lk N), then if E k (rk n rl k )= nk = l k 0 if n k 6= l k ; where z is the complex conjugate of z. iii. If is a divisor of n (that is n = n k + n k n jnj M jnj ), then m k, n k =0 jr n k (x)j = m k for all x G m iv. There exists a > for which kr n k k p m k Now we dene the Vilenkin-like system = f n n Ng by n = Y (r k ) n(k) (n N) The notation of Vilenkin-like systems is due to Gat [G]. We remark that is an orthonormal system and some well-known systems are Vilenkin-like systems (e.g. the Vilenkin system, the Walsh system and the UDMD product system) (see [G], [SWS], [V], [SW], [GT] ). Suppose that the seuences m = (m 0 ;m ; ; m k ; ) and ~m = (~m 0 ; ~m ; ; ~m k ; ) are bounded. The Kronecker product f n;m n; m Ng of two Vilenkin-like systems f n n Ng and f ~ n n Ng is said to be the two-dimensional Vilenkin-like system. Thus n;m(x; y) = n (x) ~ m(y); where x G m ; y G ~m For a function f in L (G m ) the Fourier coecients, the partial sums of the Fourier series, the Diriclet kernels are dened as follows. ^f(n) = G m f n d; S n f = D n (y; x) = n, n, ^f(k) k ; (n P; S 0 f = 0) k(y) k (x) (n P; D 0 = 0)

3 It is well known that A HARDY{LITTLEWOOD INEQUALITY ON... 3 S n f(y) = G m f(x)d n (y; x)d(x) (x G m ; n N) If f L (G m G ~m ) then the (n; k)-th Fourier coecients, the (n; k)-th partial sum of double Fourier series are the following. ^f(n; k) = G m G ~m f n;k ; S n;k f = It is simple to show that, in case f L (G m G ~m ); S n;k f(x; y)= Gm n, k, j=0 l=0 ^f(j; l) j;l ; G ~m f(t; u)d n (x; t) ~ D k (y; u)d(t)d(u) Let ~ In (x) (x G ~m ) denote the n-th intervals generated by ~m; that is ~I 0 (x) =G ~m ; ~ In (x) =fy G ~m y =(x 0 ; ; x n, ;y n ; )g (x G ~m ; n P) Dene ~n = ~n(n) = min(l N M n Ml ~ ) Then there exists a constant c for which M n M~n ~ <cm n for all n N (c does not depend on n, but do depends oax jn m j and max nn ~m j ). The atomic decomposition is a useful characterisation of Hardy spaces, to show this let us introduce the concept of an atom. A function a L (G m G ~m ) is said to be an atom if there exit a rectangle I k (x ) ~ I~k (x ) (x = (x ;x ) G m G ~m ; k N) such that (i.) supp a I k (x ) ~ I~k (x ) (ii.) kak (iii.) R I k (x ) ~ I ~k (x ) a =0 Wesay that f L (G m G ~m ) is an element of the Hardy space H(G m G ~m ) (or in brief H ), if there P P exists j C (j P) constants and a j (j P) atoms that j= P j jj < and f = j= ja j Moreover, H is a Banach space with the norm kfk H = inf( j=0 j jj) where the ium is taken over all decompositions of f. The Main result and the proof Theorem. Let > be a constant, for all f H(G m G ~m ) there exists a C > 0 constant that n= m= j ^f(n; m)j Ckfk H

4 4 K. NAGY Proof of Theorem. Throughout this paper C will denote a constant which may vary at dierent occurances and may depend only on ;sup and sup ~ Since f P P H(G m G ~m ), f can be written in the form f = j= ja j, where a j are atoms and j= j jj < n= m= j ^f(n; m)j = n= m= j P j= j^a j (n; m)j j= j j j n= m= j^a j (n; m)j Because of this the only thing we need to prove is that for an arbitrary atom a one has n= m= C Let I k (x ) ~ I~k (x ) be an interval for which (i), (ii) and (iii) hold. Thus ^a(n; m) = a(x; y) n;m (x; y) I k (x ) I ~ ~k (x ) If 0 n < and 0 m < M~k ~ then n;m (x; y) = n (x) ~ m(y) is constant on the set I k (x ) I~k ~ (x ) Conseuently, ^a(n; m) =0and n= m= Mk, + n=, n= = 0 + ~M ~k, m= ~M ~k, n= m= By the Cauchy-Buniakovski-Schwarz ineuality and Bessel's ineuality, u t u t Using the properties of the atoms we have kak = s I k (x ) ~ I ~k (x ) jaj () kak u t n= M k ~ M ~ k (I k(x ) ~ I ~k (x )) = ()

5 Notice that P m k=n k P Discuss. A HARDY{LITTLEWOOD INEQUALITY ON... 5 n, m From this s C P M If < ~ ~k then M =0 Conseuently we have ~ ~k From Cauchy-Buniakovski- Schwarz ineuality and Bessel's ineuality we have, n= The denition of ~ k implies P P 3 v uuuut, M~k ~ Mk, () C [ ~M ~k ] [ ] n= () C () [ ]M k l=[ ~M ~k ] ~ M ~k C can be proved in the similar way as we have done in case P This completes the proof. Thanks to Professor Gyorgy Gat for his professional advice. References l C ~ M ~k [SWS]. F. Schipp, W. R. Wade, P. Simon and J. Pal, Walsh series an introduction to dyadic harmonic analysis, Akademiai Kiado, Budapest, and Adam Hilger, Bristol and New York, 990. [G]. G. Gat, On (C; ) summability of integrable functions on compact totally disconnected spaces, (to appear). [FS]. S. Fridli, P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system, Acta Mathemetika Hungarica 45 (-) (985), [SW]. F. Schipp, W. R. Wade, Norm convergence and summability of Fourier series with respect to certain product systems in Pure and AAppl. Math. Approx. Theory, Marcel Dekker, New York-Basel-Hong Kong. [GT]. G. Gat, R. Toledo, L p -norm convergence of series in compact totally disconnected groups, Analysis Math. (996), 3-4. [HR]. E. Hewitt, K. Ross, Abstract Harmonic Analysis, Springer-Verlag, Heidelberg, 963. [W]. Weisz, F., The boundedness of the two-parameter Sunouchi operators on Hardy spaces, Acta Math. Hung., (to appear). [V]. N. Ja. Vilenkin, On a class of complete orthonormal systems (in Russian ), Izv. Akad. Nauk. SSSr,Ser. Math. (947), Bessenyei College, Dept. of Math., Nyregyhaza, P.O.Box 66., H-4400, Hungary nagyk@ ny.bgytf.hu