Linear equation with data in non standard spaces

Abstract

Given a finite family of Banach function spaces $V_{\alpha }$ over a bounded set $\Omega$, $V={\prod }_{\alpha }{V}_{\alpha }$, and let $T$ an element of the dual of the Sobolev space $W^{2}V$. We discuss the existence, uniqueness and regularity of the solution of the linear equation $Lu=T$ under the Dirichlet or Neumann condition on the boundary of $\Omega$.

Our results extend recent works on very weak solution with data in weighted distance space or Lorentz space.