Let $X=\left\{f\in L^1(\mathbb R^3): \int_{\mathbb R^3} f\,dx=0, \langle x\rangle f\in L^1\right\}$, where $\langle x\rangle :=(1+|x|^2)^\frac12$ is the Japanese bracket. Consider the operator $Tf=\nabla_x(-\Delta_x)^{-1} P_{\leq 1} f$, where $P$ is the Littlewood-Paley projection. For all $p\in (1,\infty]$, show that $$\|Tf\|_{L^p}\lesssim_p\|\langle x\rangle f\|_{L^1},\qquad \forall f\in X.$$
This question comes from page 23 of this paper. I can only prove the $p\in [2,\infty]$ case.
By definition $\widehat{Tf}(\xi)=\frac{i\xi}{|\xi|^2}\hat f(\xi)\chi(\xi)$, where $\chi$ corresponds to the Littlewood-Paley projection and we can assume without loss of generality that $\text{supp }\chi\subset B(0,1)$. Since $f$ is mean-zero, we have $\hat f(0)=0$ and thus by the mean-value theorem $$|\widehat{Tf}(\xi)|\leq \frac1{|\xi|}|\chi(\xi)||\hat f(\xi)|\leq |\chi(\xi)|(\sup|\nabla\hat f|)\lesssim |\chi(\xi)|\|\langle x\rangle f\|_{L^1}.$$
Now, for $p=\infty$ we have $$\|Tf\|_{L^\infty}\lesssim \|\widehat{Tf}\|_{L^1}\lesssim\|\langle x\rangle f\|_{L^1}\|\chi\|_{L^1}\lesssim \|\langle x\rangle f\|_{L^1}.$$ For $p=2$ we have $$\|Tf\|_{L^2}\lesssim \|\widehat{Tf}\|_{L^2}\lesssim \|\langle x\rangle f\|_{L^1}\|\chi\|_{L^2}\lesssim \|\langle x\rangle f\|_{L^1}.$$ By the interpolation, we get the desired estimates for $p\in[2,\infty]$.
But I don't know how to handle the case where $1<p<2$, which is the exact part to be applied in the paper. Any help would be appreciated!