The "fractional Leibniz rule" asserts that $$\Vert D^s(fg)\Vert_{L^r}\lesssim\Vert D^sf\Vert_{L^{p_1}}\Vert g\Vert_{L^{q_1}}+\Vert f\Vert_{L^{p_2}}\Vert D^sg\Vert_{L^{q_2}}$$ where $$\frac{1}{p_i}+\frac{1}{q_i}=r,\quad r\in(1,\infty),\quad p_i,q_i\in (1,\infty]$$ Does there exists a variant of such estimates in which the $L^p$ norms are replaced by the more general Lorentz (quasi)norms?
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1$\begingroup$ The proof that I know basically uses Hölder's inequality, the square function estimate, and the (vector) maximal function estimate. I know Hölder's inequality extends to Lorentz spaces. If you can extend the other estimates to Lorentz spaces, then it seems likely. It looks like these questions have been studied, so you can probably find the answer if you search around in some references. $\endgroup$ – JCM Apr 14 '18 at 16:05