Why studying, I repeatedly see people use the following result. That is there exists $C > 0$ such that $$\|\nabla \Delta^{-1}\nabla \times u\|_p \le C \|u\|_p$$ for every $u \in C^1_c(\mathbb{R},\mathbb{R}^3)$. Here $p > 1$.
The reference I got so far for the above estimate is the book "Singular Integrals and Differentiability Properties of Functions" of Stein, E.M., especially chapter II. However, I cannot find exactly which results there to apply.
Would anyone give me some hints? Thank you very much in advanced!
In terms of Stein: Chapter III is all about Riesz transforms. Note that the formal Fourier transform of the operator $ \nabla \triangle^{-1} \nabla\times$ is a multiplier of the form $x_i x_j / |x|^2$ and is a homogeneous function of degree 0.
In terms of general theory, the $L^p$ boundedness of Riesz transforms is a standard/motivating application of Calderon-Zygmund Theory and can be viewed as a result of the Mikhlin-Hormander Multiplier Theorem. If you like to see a different approach in getting this result, you can look at Terry Tao's old Math 254A notes from the Winter 2001 term. Weak 3 covers Littlewood-Paley theory and then derives some portion of Calderon-Zygmund theory including the multiplier theorem.