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Duality dispute for elliptic regularity Answer

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Duality dispute for elliptic regularity

M. Dauge proved in [1] the regularity property “$Delta u in (W^1_{p’})^*$ $Rightarrow$ $u in W^1_p$” for Dirichlet and Neumann downside in domains with piecewise flush boundaries, for $p>3$. (See Corollary 3.10.) Then the writer states (Remark 3.11):

By a duality dispute it’s simple to infer from earlier assertion 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 ought to breathe used on this illustration?

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

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