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Two endpoint bounds for generalized Radon transforms in the plane. (English) Zbl 1015.42007
Let Ω L and Ω R be open sets in 2 and be a submanifold in Ω L ×Ω R , and assume that the varieties x ={yΩ R ;(x,y)} and y ={xΩ L ;(x,y)} are smooth immersed curves in Ω R and Ω L , respectively. Let χC (Ω L ×Ω R ) be compactly supported. The authors consider the generalized Radon transform f(x)= x χ(x,y)f(y)dσ(y), where dσ x is a smooth density on x depending smoothly on xΩ L . They also consider the weighted generalized Radon transform γ f(x)= x χ(x,y)|J(x,y)| γ f(y)dσ(y), where J(x,y) is the rotational curvature. They give two endpoint estimates for these two operators. One is: Suppose that 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), (m m+1,m-1 m+1), (2 n+1,1 n+1). Then maps L p boundedly to L q . (ii) maps the Lorentz space L n+1 2,n+1 to L n+1 and L m+1 m to L m+1 m-1,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 γ , it is known that for γ>1 3 it maps L 3 2 boundedly to L 3 . They treat the case γ=1 3 and 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= 2 P x 1 y 1 . Their result is: For the operator 𝒜f(x 1 ,x 2 )= - |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 3/2r3 𝒜f L 3,r C(N)f L 3 2,r . If 2 P x 1 y 1 does not vanish identically then 𝒜 does not map L 3 2,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:
42B20Singular and oscillatory integrals, several variables
44A12Radon transform
35S30Fourier integral operators
47A30Operator norms and inequalities
46E30Spaces of measurable functions