A Criterion for a Set and Its Image under Quasiconformal Mapping to be of α(0 < α ≦ 2)-Dimensional Measure Zero

Abstract

showed that for 0 < a S 2, any set of -? -dimensional measure zero in a Jordan

On the Distortion and Correspondence under Quasiconformal Mappings in Space

“…Take the plane passing through three points the origin, Since Φz(a) ^λa for e>1 as is stated in §2^ we get (6) Furthermore, it holds by the modulus condition that…”

“…An extremal ijf-quasiconformal mapping which induces the equality for each * in 0 < | * I < 1 in each of the above estimates has not yet been found. 6. Next, we consider a space form corresponding to the second estimate in the lemma of Schwarz.…”

“…We gave in [6] an example of the set, of positive logarithmic capacity, to which a system satisfying the condition in Theorem 8 or Corollary 9 corresponds.…”

“…we consider the distortion of mappings belonging to δ«, we shall establish the following general theorem indicating the range of such an a as S rt is empty. Form the above (5), (6) and 7, we obtain the following two relations…”

“…By using these results, the monotoneity for moduli of space rings (B. Fuglede [1]) and the extremality and estimates for moduli of the space Grδtzsch and Teichmuller rings (Gehring [2]) as our main tools, we shall establish mainly the space forms corresponding to our previous results ( [5], [6]) of plane Kquasiconformal mappings. That is to say, we are concerned with the Schwarz's lemma for space K-quasiconformal mappings, some distortions in these mappings (under a certain normalization at the origin) and a criterion for two sets corresponding to each other by such mappings to be of the same dimensional Hausdorff measure zero, where it is remarked that even if one of such mappings transforms a plane domain into another plane domain, its correspondence between these plane domains does not always induce a plane Tf-quasiconformal mapping.…”