Simultaneous approximation by smooth functions in Sobolev and Lebesgue spaces

We show how to approximate functions defined on smooth bounded domains by smooth functions in such a way that the approximations are bounded and converge in both Lebesgue spaces and L2-based Sobolev spaces simultaneously. We prove an abstract result referred to fractional power spaces of positive, self-adjoint, compact-inverse operators on Hilbert spaces, and then obtain our main result by identifying explicitly these fractional power spaces for the Dirichlet Laplacian and Dirichlet Stokes operators.

This is joint work with Charles Fefferman (Princeton) and Karol Hajduk (Warwick).