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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 \(\mathbb R^2\) and \(\mathcal M\) be a submanifold in \(\Omega_L\times\Omega_R\), and assume that the varieties \(\mathcal M_x=\{y\in \Omega_R; (x,y)\in \mathcal M\}\) and \(\mathcal M_y=\{x\in \Omega_L; (x,y)\in \mathcal M\}\) are smooth immersed curves in \(\Omega_R\) and \(\Omega_L\), respectively. Let \(\chi\in C^\infty(\Omega_L\times\Omega_R)\) be compactly supported. The authors consider the generalized Radon transform \(\mathcal Rf(x)=\int_{\mathcal M_x}\chi(x,y)f(y) d\sigma(y)\), where \(d\sigma_x\) is a smooth density on \(\mathcal M_x\) depending smoothly on \(x\in \Omega_L\). They also consider the weighted generalized Radon transform \(\mathcal R_\gamma f(x)=\int_{\mathcal M_x}\chi(x,y)|J(x,y)|^\gamma f(y) d\sigma(y)\), where \(J(x,y)\) is the rotational curvature. They give two endpoint estimates for these two operators. One is: Suppose that \(\mathcal M\) satisfies a left finite type condition of degree \(n\) and a right finite type condition of degree \(m\). (i) Suppose that \((1/p,1/q)\) belongs to the closed trapezoid with corners \((0,0)\), \((1,1)\), \((\frac{m}{m+1}, \frac{m-1}{m+1})\), \((\frac{2}{n+1}, \frac{1}{n+1})\). Then \(\mathcal R\) maps \(L^p\) boundedly to \(L^q\). (ii) \(\mathcal R\) 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 \(\mathcal R_\gamma\), it is known that for \(\gamma>\frac 13\) it maps \(L^\frac 32\) boundedly to \(L^3\). They treat the case \(\gamma=\frac 13\) and \(\mathcal M\) is given by the equation \(y_2=x_2+P(x_1,y_1)\), where \(P\) is a polynomial of degree at most \(N\). In this case \(J=\frac{\partial ^2P}{\partial x_1\partial y_1}\). Their result is: For the operator \(\mathcal Af(x_1,x_2)=\int_{-\infty}^{\infty}|J|^{1/3}f(y_1,x_2+P(x_1,y_1))dy_1\), there exists a constant \(C(N)\) (independent of particular polynomial) so that for \(3/2\leq r\leq 3\) \(\|\mathcal A f\|_{L^{3,r}}\leq C(N)\|f\|_{L^{{\frac 32},r}}\). If \(\frac{\partial ^2P}{\partial x_1\partial y_1}\) does not vanish identically then \(\mathcal A\) does not map \(L^{\frac 32,r}\) to \(L^{3,s}\) for any \(s<r\). This deepens the known case where \(P\) is real analytic. Their proofs are based on multilinear arguments.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
44A12 Radon transform
35S30 Fourier integral operators applied to PDEs
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)

Citations:

Zbl 0907.35147
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References:

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