Endpoint regularity of Radon transforms associated with real analytic phases in the plane

The full range of Sobolev endpoint regularity of Radon transform is given. We prove the failure of endpoint regularity for real analytic phases. The critical endpoint regularity $L^p(\bR^2)\rightarrow L^p_s(\bR^2)$ is true if only and only if $p=2$. This generalizes a result of M. Christ. In particular, it implies that the Radon transform is not bounded from $L^p$ into $L^p_{1/n}$, where phase functions are $S(x,y)=a_1x^{n-1}y+a_2x^{n-2}y^2+\cdots+a_{n-1}xy^{n-1}$ with $n\geq 3$, the coefficients are real numbers satisfying $a_1a_{n-1}\neq0$, and $p$ is equal to $\frac{n}{n-1}$ or $n$. This answers a problem of Phong and Stein in \cite{PS1994}.

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