Main resultsTo be brief, the purpose of this paper is as follows. First, we introduce function spaces C;k(B) for arbitrary open subset B of IR". Our space C;'(B) is a modification of the space C",B) of DE VORE and SHARPLEY [DS]. The space C>'l(B) consists of Lebesgue measurable functions on B but not necessarily of distributions on B; if lip > 1 + u/n, then Cb,.'(B)contains a function which is not locally integrable. If l/p < 1 + a/n and if B satisfies a certain condition (which assures the extension of functions to R"), then C;k(B) for certain k can be identified with a Sobolev space or a Hardy-Sobolev space or a Triebel-Lizorkin space F;,,(8) with q = 00; however, the arguments of the paper shall be independent of these identifications. Secondly, we study the boundedness of the pointwise multiplication operator (1, g) + fg in our spaces. We shall prove that if (1.1) co > l / p , , 1/p2 > a/n and lip = l/pl + l/p2 -a/n or if l/pl = l/p2 = l/p = u/n 2 1 then, for certain k, there exists a constant c independent of B such that the inequality Ilfg; c;k(Q)ll 5 c Ilf; c; : (Q)I l Ilg; qWII holds for all B and for all f~ C::(B) and g E C::(B), where 11. ; C;k(B)I) denotes the quasinorm of the space C>'(Q). In particular, C;'(B) with l/p = a/n 1 1 is an algebra with respect to the pointwise multiplication. We shall also consider the multiplication of functions in C;'(B) for l/p < a/n by putting a restriction on 8. Thirdly, we consider the factorization problem, i.e., the problem that whether a given f~ C;k(B) can be written as f = gh (the pointwise product) with g E C; : (B) and h E C;:(B), where p, pl, p2, and a are fixed and satisfy (1.1) and k is also fixed and satisfies a certain condition. To this problem we give a weak affirmative answer and a negative answer. The afirmative one states that if p S 1 then every f~ C",.(B) can be written as f = gjhj (an infinite sum) with gj E C; : ( 8) and hi E C ;: ( 8 ) and with an appropriate estimate of the quasinorms. This result shows that our result on the multiplication of functions in C:k(B) for lip > a/n is, in a sense, j 14 Math. Nachr.. Bd. 176