ON THE COMPOSITION OF FUNCTIONS IN MULTIDIMENSIONAL BESOV SPACES

For the composition operator T-f : g -> f omicron g we find a class of functions f : R -> R for which there exists a family of positive constants c(f, t), t > 0, such that the estimate parallel to T-f (g)parallel to B-p,q((R))(s)n <= c(f, t) parallel to g parallel to B-p,q((R))(s)n holds, for all g epsilon W-infinity(1) boolean AND B-p,q(s)(Rn) satisfying parallel to del g parallel to(infinity) <= t (or g is an element of L-infinity boolean AND B-p,(s)(q)(R-n) with parallel to g parallel to(infinity) <= t and [s] = 1). We establish this assertion, for all f is an element of B-p,infinity(s1,loc) (R) with s(1) > 1+1/p, in the case 1 < p < infinity, 0 < q <= 8 and 0 < s < s(1).