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An L p -L q estimate for Radon transforms associated to polynomials. (English) Zbl 0980.42008

Let S(x,y) be a real polynomial of degree n2 defined by

S(x,y)= d=0 n j+k=d a jk x j y k ·

The Radon transforms R, and T of f are defined by

Rf(t,x)= - f(t+S(x,y),y)ψ(t,x,y)dy,

and

Tf(t,x)= - f(t+S(x,y),y)dy,

respectively, where ψC c ( 3 ) is a cutoff function. Let Δ be the closed convex hull of the points O=(0,0), A=(2/(n+1), 1/(n+1)), A ' =(n/(n+1), (n-1)/(n+1)), and O ' =(1,1). When S(x,y) is a homogeneous polynomial, and a 1,n-1 0 and a n-1,1 0, Phong and Stein proved that R is bounded from L p ( 2 ) to L q ( 2 ), if (1/p,1/q) is in the set Δ minus the half-open segments (O,A] and [A ' ,O ' ). They also proved that for R to be bounded from L p ( 2 ) to L q ( 2 ), it is necessary that (1/p,1/q) is in Δ, and in the case n=2,3 it is known that R is bounded precisely for (1/p,1/q)Δ. The author gives a positive result for endpoint estimates in the case n4, and more. His main result is (for not necessarily homogeneous polynomials): if a 1,n-1 0, then there is C n >0 such that

Tf L n+1 C n |a 1,n-1 | -1/(n+1) f L (n+1)/2 ,

where C n is independent of the coefficients a jk . Using this, he gets endpoint estimates, not treated by Phong and Stein, in the homogeneous polynomial case.

MSC:
42B20Singular and oscillatory integrals, several variables
42B15Multipliers, several variables
42B30H p -spaces (Fourier analysis)
44A12Radon transform