The Bandwidth theorem of B\"ottcher, Schacht and Taraz gives a condition on
the minimum degree of an n-vertex graph G that ensures G contains every
r-chromatic graph H on n vertices of bounded degree and of bandwidth
o(n), thereby proving a conjecture of Bollob\'as and Koml\'os. In this paper
we prove a version of the Bandwidth theorem for locally dense graphs. Indeed,
we prove that every locally dense n-vertex graph G with δ(G)>(1/2+o(1))n contains as a subgraph any given (spanning) H with bounded
maximum degree and sublinear bandwidth.Comment: 35 pages. Author accepted version, to appear in Forum of Mathematics,
Sigm