Abstract. We introduce the $\omega $-tangential maximal function and the $\omega $-Littlewood--Paley function associated with Beurling's generalized distributions, which are extensions of Schwartz's distributions in terms of the weight function $\omega $. We compare the $\nu $-weighted $L^p$-norms of these maximal functions, and relations among several kinds of maximal functions. Also, we study the extension of Beurling's distributions on $R^n$ to $R^{n+1}_+$ by means of convolutions. As an application of these results we establish estimates between some of the above $\omega $-maximal functions associated to the extension.

AMS Subject Classification (1991): 42B25

Keyword(s): \omega, -tangential-maximal functions, \omega, -Littlewood--Paley functions

Received May 17, 2006, and in final form February 26, 2007. (Registered under 5960/2009.)