AbstractLet {\boldsymbol{L}} be a second order uniformly elliptic operator, and
consider the equation {\boldsymbol{L}u=f} under the boundary
condition {u=0}. We assume data f in generical
subspaces of continuous functions {D_{\overline{\omega}}} characterized by a
given modulus of continuity{\overline{\omega}(r)}, and show that the
second order derivatives of the solution u belong to
functional spaces {D_{\widehat{\omega}}}, characterized by a modulus of
continuity{\widehat{\omega}(r)} expressed in terms of {\overline{\omega}(r)}.
Results are optimal. In some cases, as for Hölder spaces,
{D_{\widehat{\omega}}=D_{\overline{\omega}}}. In this case we say that full regularity
occurs. In particular, full regularity occurs for the new class of
functional spaces {C^{0,\lambda}_{\alpha}(\overline{\Omega})} which includes, as a
particular case, the classical Hölder spaces
{C^{0,\lambda}(\overline{\Omega})=C^{0,\lambda}_{0}(\overline{\Omega})}.
Few words, concerning the possibility of generalizations and
applications to non-linear problems, are expended at the end of the
introduction and also in the last section.
Asymptotic estimates in Weighted Hölder spaces for a class of elliptic scale-covariant second order operators
