Let $\Omega $ a bounded open set of $\mathbb R^n$ and let $\{ {e_n}\} _{n = 1}^\infty $ an orthonormal basis of ${L^2}(\Omega )$, let $\{ {\lambda _n}\} _{n = 1}^\infty $ the eigenvalues of the Laplacian Dirichlet operator.
It is well know that if $u \in H_0^1(\Omega )$ then $u = \sum\limits_{n \ge 1} {{\lambda _n}(} {e_n},u){e_n}$ where $(\cdot,\cdot)$ is the inner product in ${L^2}(\Omega )$, and for $f \in {H^{ - 1}}(\Omega )$ we have $f = \sum\limits_{n \ge 1} {{1 \over {{\lambda _n}}}(} {e_n},f){e_n}$.
Now I want to estimate the duality bracket $ \langle f,u \rangle $ by using spectral decomposition. We have:$$ \langle f,u \rangle = \sum\limits_{n \ge 1} ( {e_n},f)({e_n},u)$$ How can I prove the continuity of $f$ on $H_0^1(\Omega )$? My second question: Let $${\partial _x}{(I - \partial _x^2)^{ - 1}}{\partial _x}:{L^2}(0,1) \to {L^2}(0,1)$$ How can I use spectral decomposition to prove the continuity of this operator ? Thank you.
For the first one, we have \begin{align} |\langle f,u \rangle| &= \Big|\sum\limits_{n \ge 1} \frac1{\lambda_n^{\frac12}}( {e_n},f)\lambda_n^{\frac12}({e_n},u)\Big|\\ &\le \Big(\sum\limits_{n \ge 1} \frac1{\lambda_n}( {e_n},f)^2\Big)^{\frac12} \Big(\sum\limits_{n \ge 1} \lambda_n({e_n},u)^2\Big)^{\frac12} =\|f\|_{H^{-1}}\|u\|_{H^1_0} \end{align} For the second one, we note we have $e_n=\sqrt{2}\sin(n\pi\cdot)$ under the zero Dirichlet condition.