$\Delta u = f, f \in L^q \Rightarrow u \in W^{2,q}$ References I'm looking for references for the following theorem. I will very grateful

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: You could try Gilbard-Trudinger chapter 9, in particular section 9.6. 
A: The original method of obtaining the $L^p$-type regularity estimate should be the paper On the existence of certain singular integrals by Calderón and Zygmund in 1952.
Gilbard and Trudinger established the result of Theorem 9.9(first edition) by a slightly different approach than Calderón and Zygmund's singular kernel, which is kernel with non-integrable singularities. C-Z's paper considered:
$$
D^2 u = \int_{B_2} K(x,y)f(y)dy,
$$
where $K(x,y)$ is that singular kernel. Please see the Notes section of chapter 9 in Gilbard and Trudinger.
A: Despite being concise, formulation of the "cited" theorem does contain two  mistakes. One mistake is just grave: a missing constant factor, depending on $p$, in the right-hand side of the estimate forcing solutions to stay bounded while $p\to 1$ or $p\to \infty$ which is well known to be wrong. Another mistake is not just very grave, it is fantastically grave. All integrals in the estimate are taken over one and the same ball $B_2$ which makes the estimate impossible without imposing some boundary condition at $\partial B_2$. Among other terrible things, this impossible estimate implies, for example,  that the subspace of all harmonic functions in $W^{2,p}(B_2)$ is finite-dimensional, which is obviously absurd. Needless to say, Alberto Calderón  and Antoni Zygmund could not be the authors of this "theorem" already previously "cited" in another question What is the *standard duality argument? 
