In the present study we consider the solution of the Dirichlet problem
in conical domain. For general elliptic problems in non Hilbertian
Sobolev spaces built on $L^{p},1<p<\infty$ the theory of sums of
operators developed by Dore-Venni $\left[8\right]$ provides an optimal
result. Holder spaces, as opposed to LP spaces, are not UMD. Using
the results of Da Prato-Grisvard $\left[6\right]$ and Labbas $\left[14\right]$
we cope with the singular behaviour of the solution in the framework
of H$\ddot{\textrm{o}}$lder and little H$\ddot{\textrm{o}}$lder
spaces.
