Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

GÁT, G., On a norm convergence theorem with respect to the Vilenkin system in the Hardy spaces

102 György Gát for x G G

m, n G P: = N \ {0}. Let 0 = (0G N) G G

m dénoté the nullelement of G

m,I

n: = I

n{ 0) (n G N). Furthermore, let L

p(G

m) (1 < p < oo) dénoté the usual Lebesgue spaces (||.||

p the corresponding norms) on G

m, A

n the a algebra generated by the sets I

n(x) (x G G

m) and E

n the conditional expectation operator with respect to A

n (n G N) (/ G X

1.) The concept of the maximal Hardy space ([Sch, Sim]) H

l(G

m) is defined by the maximal function /* : = sup

n |E

nf \ (/ G L

l{G

m)), saying that / belongs to the Hardy space H

1(G

m) if /* G L

l(G

m). H

1(G

m) is a Banach space with the norm Il/liThe so-called atomic Hardy space H(G

m

) is defined for bounded Vilenkin groups as follows [Sch, Sim]. A function a G L°°(G

m) is called an atom, if either a — 1 or a has the following properties: suppa Ç I

a

i || a ||oo< f[ a = 0, where I

a G J:= {I

n{x) x G G

m,n G N}. The elements of X are called intervais on G

m. We say that the function / belongs to H(G

m

), if / can be represented as / = X^o where ai 's are atoms and for the coefficients A; (z G N) YJÎLQ l-M < oo is true. It is known that H(G

m

) is a Banach space with respect to the norm h :=m

î 11, 1=0 where the infinum is taken ail over décompositions / = Y^Xiüi G H(G

m). i= 0 If the sequence m is not bounded, then we define the set of intervais in a différent way ([5]), that is we have "more" intervais than in the bounded case. A set / C G m is called an interval if for some x G G

m and n G N, J is of the form I — [J

ke

U I

n(x, k

) where U is one of the following sets Ui = 0,..., m

7 -1 },U

2 = TÏLR 1 /73- 0, [mj2] - 1 -1U« = [m

n/2] - 1 1. m

r -1 etc., and I

n(x,k):={y G G

m : yj = xj(j < n),y

n = k}, (x G G

m,k G Zm

n -,

n £ N"). The rest of the définition of the atomic Hardy space H is the same as in the bounded case.