M. Dauge proved in [1] the regularity property $\Delta u \in (W^1_p)^*$ $\Rightarrow$ $u \in W^1_p$ for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p>3$. (see Corollary 3.10). Then the author stated: “By a duality argument it is easy to deduce from previous statement that the Laplace operator… is an isomorphism …. when $3/2-\epsilon <p<3+\epsilon$ (see Rematk 3.11). What is “a duality argument” and which theorem should be used in this case?

[1] Dauge, M. Neumann and mixed problems on curvilinear polyhedra. Integral Equations Operator Theory 15 (1992), no. 2, 227–261.

M. Dauge proved in [1] the regularity property "$\Delta u \in (W^1_{p'})^*$ $\Rightarrow$ $u \in W^1_p$" for Dirichlet and Neumann problem in domains with piecewise smooth boundaries, for $p>3$. (See Corollary 3.10.) Then the author states (Remark 3.11):

By a duality argument it is easy to deduce from previous statement that the Laplace operator [between $W^1_p$ and $(W^1_{p'})^*$] is an isomorphism [...] when $3/2-\epsilon <p<3+\epsilon$ for a $\epsilon > 0$.

What is “a duality argument” and which theorem should be used in this case?

[1] Dauge, M. Neumann and mixed problems on curvilinear polyhedra. Integral Equations Operator Theory 15 (1992), no. 2, 227–261.