Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)
GÁT, G., On a theorem of type Hardy-Littlewood with respect to the Vilenkin-like systems
84 György Gát for x G G m, n G P: = N \ {0}. Let 0 - (0,i G N) G G m denote the nullelement of G m, I n '•= -fn(O) (n G N). Let I : = {I n(x) : x G G m, TL G N}. The elements of 1 are called intervals on G m . Furthermore, let L p(G m ) (1 < p < oo) denote the usual Lebesgue spaces (I . jp the corresponding norms) on G m, A n the a algebra generated by the sets I n(x) ( x G Cm) and E n the conditional expectation operator with respect to A n (n G N) (/ G L l.) Let Mo : = 1, M n +i : = m nM n (n G N) be the generalized powers. Then each natural number n can be uniquely expressed as oo n = ^ n lM l (rii G {0,1,. . ., m l - 1}, i G N), i=0 where only a finite number of n t's differ from zero. The generalized Rademacher functions are defined as r n(x) exp(27n-~) (x G G m, n G N, i:=y/^l). TTl n Then oo j= 0 the nth Vilenkin function. The system ip := (ip n: n G N) is called a Vilenkin system. Each ip n is a character of G m and all the characters of G m are of this form. Define the m-adic addition as oo k © n := kj + rij (mod mj))Mj (fc, n G N). i=o Then, 1p k@ n = Ipk^n, 1pn{x + y) = 1pn(-x) = ÍVi(z), = 1 (k,n £ N, x,y G G m). Let functions a n,a^:G m —» C (n,j,k G N) satisfy: (i) Q^ is measurable with respect to Aj (j,k G N), (ii) |4 fc )| = a^(0) = a« f c> = t t«°» = l(;,fc eN), (iii) o n := nr=o n «> := n,M, (n 6 N). Let Xn := ^nön (ft G N). The system {xn : n € N} is called a Vilenkin-hke (or ilia) system ([2]-[4]).
