I need to prove the continuity of the generalized differential operator $ D^{(\alpha)}:W^{m+|\alpha|}(\Omega)\rightarrow W^m(\Omega).$
I suppose this should be done by the use of the continuity of the inner product, Sobolev spaces being Hilbert spaces, or proving the differential operator is bounded. The generalized differential operator is defined through an inner product: $$ < D^{(\alpha)}f, \phi>=(-1)^{|\alpha|}<f, D^{\alpha}\phi > , \,\forall \phi \in \mathcal{D}(\Omega).$$ Hier $\mathcal{D}$ means the set of $C^\infty $ functions with compact support, $D^{(\alpha)} \in L^2(\Omega), |\alpha| $ multiindices $ \alpha, $ and $W^{m}(\Omega)=\{f\in L^2(\Omega): D^{(\alpha)}f \in L^2(\Omega)$ exists $\forall |\alpha|\leqslant m \}.$
Thanks for any suggestion or proposal.
Let $\alpha,\beta$ be to multiindices. Then $$ \|D^{\alpha+\beta}u\|_{L^p} = \|D^\alpha D^\beta u\|_{L^p} \le \|D^\beta u\|_{W^{|\alpha|,p}}\le \|u\|_{W^{|\alpha|+|\beta|,p}}. $$