Publication at Faculty of Mathematics and Physics |
2023
Abstract
A mapping f : X -> Y between metric spaces is termed little Lipschitz if the function lip f : X -> [0, infinity], lip f (x) = lim inf(r -> 0) diam f(B(x, r))/r, is finite at every point. We prove that for each s > 0 the little Lipschitz mapping f satisfies the inequality H-s (f(X)) <= integral(X) (lip f)(s) dP(s) as long as {lip f = 0} is of sigma-finite measure P-s, where H-s and P-s denote the s-dimensional Hausdorff and packing measures, respectively.
We derive a dimensional in-equality for little Lipschitz mappings dim(H) f (X) <= dim(H) f <= (dim) over bar (P) X and we provide a few examples that show that these inequalities are the best possible.