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Two endpoint bounds for generalized Radon transforms in the plane. (English) Zbl 1015.42007
Let ${{\Omega }}_{L}$ and ${{\Omega }}_{R}$ be open sets in ${ℝ}^{2}$ and $ℳ$ be a submanifold in ${{\Omega }}_{L}×{{\Omega }}_{R}$, and assume that the varieties ${ℳ}_{x}=\left\{y\in {{\Omega }}_{R};\left(x,y\right)\in ℳ\right\}$ and ${ℳ}_{y}=\left\{x\in {{\Omega }}_{L};\left(x,y\right)\in ℳ\right\}$ are smooth immersed curves in ${{\Omega }}_{R}$ and ${{\Omega }}_{L}$, respectively. Let $\chi \in {C}^{\infty }\left({{\Omega }}_{L}×{{\Omega }}_{R}\right)$ be compactly supported. The authors consider the generalized Radon transform $ℛf\left(x\right)={\int }_{{ℳ}_{x}}\chi \left(x,y\right)f\left(y\right)d\sigma \left(y\right)$, where $d{\sigma }_{x}$ is a smooth density on ${ℳ}_{x}$ depending smoothly on $x\in {{\Omega }}_{L}$. They also consider the weighted generalized Radon transform ${ℛ}_{\gamma }f\left(x\right)={\int }_{{ℳ}_{x}}\chi \left(x,y\right){|J\left(x,y\right)|}^{\gamma }f\left(y\right)d\sigma \left(y\right)$, where $J\left(x,y\right)$ is the rotational curvature. They give two endpoint estimates for these two operators. One is: Suppose that $ℳ$ satisfies a left finite type condition of degree $n$ and a right finite type condition of degree $m$. (i) Suppose that $\left(1/p,1/q\right)$ belongs to the closed trapezoid with corners $\left(0,0\right)$, $\left(1,1\right)$, $\left(\frac{m}{m+1},\frac{m-1}{m+1}\right)$, $\left(\frac{2}{n+1},\frac{1}{n+1}\right)$. Then $ℛ$ maps ${L}^{p}$ boundedly to ${L}^{q}$. (ii) $ℛ$ maps the Lorentz space ${L}^{\frac{n+1}{2},n+1}$ to ${L}^{n+1}$ and ${L}^{\frac{m+1}{m}}$ to ${L}^{\frac{m+1}{m-1},\frac{m+1}{m}}$ (formulations of left and right finite type conditions are given in [A. Seeger, “Radon transforms and finite type conditions”, J. Am. Math. Soc. 11, No. 4, 869-897 (1998; Zbl 0907.35147)]. As for ${ℛ}_{\gamma }$, it is known that for $\gamma >\frac{1}{3}$ it maps ${L}^{\frac{3}{2}}$ boundedly to ${L}^{3}$. They treat the case $\gamma =\frac{1}{3}$ and $ℳ$ is given by the equation ${y}_{2}={x}_{2}+P\left({x}_{1},{y}_{1}\right)$, where $P$ is a polynomial of degree at most $N$. In this case $J=\frac{{\partial }^{2}P}{\partial {x}_{1}\partial {y}_{1}}$. Their result is: For the operator $𝒜f\left({x}_{1},{x}_{2}\right)={\int }_{-\infty }^{\infty }{|J|}^{1/3}f\left({y}_{1},{x}_{2}+P\left({x}_{1},{y}_{1}\right)\right)d{y}_{1}$, there exists a constant $C\left(N\right)$ (independent of particular polynomial) so that for $3/2\le r\le 3$ ${\parallel 𝒜f\parallel }_{{L}^{3,r}}\le C\left(N\right){\parallel f\parallel }_{{L}^{\frac{3}{2},r}}$. If $\frac{{\partial }^{2}P}{\partial {x}_{1}\partial {y}_{1}}$ does not vanish identically then $𝒜$ does not map ${L}^{\frac{3}{2},r}$ to ${L}^{3,s}$ for any $s. This deepens the known case where $P$ is real analytic. Their proofs are based on multilinear arguments.
##### MSC:
 42B20 Singular and oscillatory integrals, several variables 44A12 Radon transform 35S30 Fourier integral operators 47A30 Operator norms and inequalities 46E30 Spaces of measurable functions