  We study associate and double associate spaces of two-weighted Sobolev spaces
of the first order on real half-line and we show that unlike the notion of
duality the associativity is divided into two cases which we call "strong" and
"weak" ones with the division of the second associativity into four cases. On
the way we prove that the Sobolev space of compactly supported functions
possess weak associated reflexivity and the double weak-strong associate space
is vacuous. The case of power weights was recently characterized by reduction
to Ces\`{a}ro or Copson type spaces [18].
