THE RELATION BETWEEN SSC(F, K) AND Q(M) CONDITIONS

Abstract

Let ???? − be some family of curves in ℂ. A nonnegative Borel function ???? is called
the admissibility metric of the family ???? if
∫ ????(????)|???????? |
???? ≥ 1,
for every ???? ∈ ???? . The modulus of the family ???? is defined by the formula
???? (????) = 1
2????∬|????(????)| 2 ???????? ∧ ????????
ℂ
where inf is taken over the class of all possible metrics ???? . For a pair of continums ???? 1 , ???? 2 ⊂ ???? , the symbol ???? (???? 1 , ???? 2 , ???? ) denotes the modulus of the family of all curves
in ???? connecting ???? 1 and ???? 2 . The direction of the square ???? will be called an angle ???? , 0 ≤ ???? ≤ ????
2 such that when turning to an angle ???? , the square ???? is transformed into
a square whose sides are parallel to the coordinate axes.