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The Riesz $R_i$ transform on $\mathbb{R}^n$ is defined by $$ R_if(x)= \int_{\mathbb{R}^n} \frac{t_i-x_i}{\vert x-t \vert^{n+1}}f(t) dt$$ for a Schwartz function $f$ on $\mathbb{R}^n$. Can you please tell me, how can one show that they admit a bounded extension on $L^2(\mathbb{R}^n)$?

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Using the Fourier transform and the Plancherel identity is the quickest way for this particular question, but more generally I would recommend Stein's book "Singular integrals and differentiability properties of functions" for the relevant theory for singular integrals such as the Riesz transforms. – Terry Tao Feb 8 '13 at 3:28
Thank you very much! – nicolas Feb 8 '13 at 14:30
@nicolas You could write an answer. – Davide Giraudo Feb 15 '13 at 15:55

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