Prove that for all $1<p<\infty $ there exist a constant $A_p>0$ such that for every $C^2_0(\mathbb{R}^2)$(twice continuously differentiable with compact support complex value function) such that $$||\partial{x_1}f||_{L^p}+||\partial{x_2}f||_{L^p}\leq A_p||\partial{x_1}f+i\partial{x_2}f||_{L^p}$$ Well,This problem is one of the exercise in Grafakos' GTM 249 Singular Integral Chapter(Sec 5.2 Page 354)See. It seems much easier than any other problem in this chapter,But I have no idea of it,Even after finishing the other exercise in this section,May anyone give me some hint or correct solution?Thanks in advance!!!
2026-03-25 16:44:23.1774457063
How to solve this estimate in Grafakos' book?
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Hint: $\frac{\partial f}{\partial x_j} = -R_j(R_1-i R_2)\Big(\frac{\partial f}{\partial x_1} + i \frac{\partial f}{\partial x_2}\Big)$, for $j=1,2$, where $R_j$ is the $j$-th Riesz transform.