Elliptic operators and Muckenhoupt weights

Under different conditions on a domain $\Omega$, we prove some weighted a priori estimates for solutions to the Dirichlet problem when the weight is in the Muckenhoupt class $A_p(R^n)$. In particular, we are interested in the case of weights which are powers of the distance to the boundary of $\Omega$ and we study for which powers these weights belongs to $A_p(R^n)$.
These results can be extended to a metric measure space $(X;d;\mu)$
satisfying the so called Ahlfors condition, which is a particular case of space of homogeneous type. We also obtain a new family of weights in the class $A_p(X;d;\mu)$ for a general metric measure space.



This seminar is organized within the PRIN 2012 Research project «Equazioni alle derivate
parziali di tipo ellittico e parabolico: aspetti geometrici, disuguaglianze collegate, e applicazioni
- Partial Differential Equations and Related Analytic-Geometric Inequalities» Grant Registration
number 2012TC7588_003, funded by MIUR – Project coordinator Prof. Filippo Gazzola