What is the *standard duality argument? What is the standard duality argument? I saw this foor exemplo in the following statement. The case $p < 2$ follows from the standard duality argument. To prove

Theorem: [Calderón Zigmund] If $u$ is a solution of
  \begin{equation}
\Delta u = f \quad \mbox{in} \quad B_2
\end{equation}
  then
  \begin{equation}
\int_{B_2} | D^2u|^p \le \Bigl(\int_{B_2} |f|^p + \int_{B_2} |u|^p \Bigr) \quad \mbox{for any} 1<p<  + \infty.
\end{equation}

 A: Let $A_p = -\Delta$, $A_p : W^{1,p} \to (W^{1,p'})'$ for $1 < p < \infty$, where $p'$ is the adjoint exponent with $1 = 1/p + 1/p'$.
First, you prove that $A_p$ is invertible for $p \in [2,\infty)$. The duality argument simply means that the adjoint of $A_p$ is $A_{p'}$. Then, the (bounded) invertibility for $A_{p'}$ follows from the (bounded) invertibility of $A_p$.
A: Alberto Calderón and Antoni Zygmund have nothing to do with this theorem that cannot be true without specifying some boundary condition at $\partial B_2$. The missing boundary condition is of course a terrible blunder making it easy, for example, to establish that the subspace of all harmonic functions in $W^{2,p}(B_2)$ is finite-dimensional, which is obviously absurd. There are only two options to correct the blunder and relate the estimate to the Calderón–Zygmund theorem: replace the ball $B_2$ by the whole space $\mathbb{R}^n$, or otherwise replace $B_2$ in the left-hand side by a smaller ball, say $B_1\subset B_2\,$, inserting in either case a constant factor $C_p\,$ into the right-hand side of the esimate. Without such factor, depending on $p$, the estimate cannot be made valid for any $p\in (1,\infty)$, since due to 
$C_p=1$  solutions would necessarily stay bounded while $p\to 1\,$ or $\,p\to \infty\,$, which is definitely wrong.
