  On a real analytic manifold M, we construct the linear subanalytic
Grothendieck topology Msal together with the natural morphism of sites $\rho$
from Msa to Msal, where Msa is the usual subanalytic site. Our first result is
that the derived direct image functor by $\rho$ admits a right adjoint,
allowing us to associate functorially a sheaf (in the derived sense) on Msa to
a presheaf on Msa satisfying suitable properties, this sheaf having the same
sections that the presheaf on any open set with Lipschitz boundary. We apply
this construction to various presheaves on real manifolds, such as the
presheaves of functions with temperate growth of a given order at the boundary
or with Gevrey growth at the boundary. On a complex manifold endowed with the
subanalytic topology, the Dolbeault complexes associated with these new sheaves
allow us to obtain various sheaves of holomorphic functions with growth. As an
application, we can endow functorially regular holonomic D-modules with a
filtration, in the derived sense.
