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My question is rather simple. Given $f$ a $L^2(\mathbb{R}^2)$ function with zero mean but supported in the whole plane, is there a bound of the form $$ \|\nabla(\Delta)^{-1} f\|_{L^2(\mathbb{R}^2)}\leq C\|f\|_{L^2(\mathbb{R}^2)}?. $$ Any comment or reference is very welcome!

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may be using Fourier Transform... –  Pocho la pantera Sep 3 '13 at 21:16
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Consider $f_r(x)=f(rx)$ then $\| f_r\|_{L^2}=r^{-n/2}\| f\|_{L^2}$ and $\| \nabla (\Delta^{-1})f_r\|_{L^2}=r^{-n/2-1}\| \nabla(\Delta^{-1})f\|_{L^2}$. –  Jose27 Sep 3 '13 at 21:23
    
But how can you bound the singularity of the multiplier at zero? Using the zero mean you have $$ \frac{i \xi}{|\xi|^2}\left(\hat{f}(\xi)-\hat{f}(0)\right)= \int_0^1\nabla_\xi\hat{f}(\lambda\xi)d\lambda $$ –  guacho Sep 3 '13 at 21:24
    
Is $f$ compactly supported? –  Shuhao Cao Sep 3 '13 at 21:31
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@Rafa: $\Delta^{-1} f=u$ is the solution of $\Delta u=f$ so that $u_r=r^{-2}u(rx)=\Delta^{-1}f_r$. Take the gradient and change variables. –  Jose27 Sep 3 '13 at 21:39

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