An approach to metric space-valued Sobolev maps via weak* derivatives

We give a characterization of metric space-valued Sobolev maps in terms of weak* derivatives. More precisely, we show that Sobolev maps with values in dual-to-separable Banach spaces can be defined in terms of classical weak derivatives in a weak* sense. Since every separable metric space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X embeds isometrically into <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> </m:math> { }∞ , we conclude that Sobolev maps with values in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> X can be characterized by postcomposition with such embedding and the mentioned weak gradients. A slight variation on our definition was proposed previously by Hajłasz and Tyson. However, we show that their definition does not work in the sense that for technical reasons the arising Sobolev space is essentially empty.