Seminar on theory of function spaces

Vector-valued Sobolev spaces based on Banach function spaces

It is known that for Banach valued functions there are several approaches to de ne a Sobolev class. We compare the usual de nition via weak derivatives with the Reshetnyak-Sobolev space and with the Newtonian space; in particular, we provide sucient conditions when all three agree. As well we revise the di erence quotient criterion and the property of Lipschitz mapping to preserve Sobolev space when it acting as a superposition operator.