An \(L^p-L^q\) estimate for Radon transforms associated to polynomials
type: Item
instance of: scholarly article
title: An \(L^p-L^q\) estimate for Radon transforms associated to polynomials
zbMATH Open document ID: 0980.42008
DOI: 10.1215/S0012-7094-00-10125-1
author: Jong-Guk Bak
published in: Duke Mathematical Journal
publication date: 2002-02-19T00:00:00Z
review text: Let \(S(x,y)\) be a real polynomial of degree \(n\geq 2\) defined by \[ S(x,y)=\sum_{d=0}^{n}\sum_{j+k=d} a_{jk}x^jy^k. \] The Radon transforms \(R\), and \(T\) of \(f\) are defined by \[ Rf(t,x)=\int_{-\infty}^{\infty}f(t+S(x,y),y)\psi(t,x,y) dy, \] and \[ Tf(t,x)=\int_{-\infty}^{\infty}f(t+S(x,y),y) dy, \] respectively, where \(\psi\in C_c^\infty(\mathbb R^3)\) is a cutoff function. Let \(\Delta\) 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 \(a_{1, n-1}\neq 0\) and \(a_{n-1, 1}\neq 0\), Phong and Stein proved that \(R\) is bounded from \(L^p(\mathbb R^2)\) to \(L^q(\mathbb R^2)\), if \((1/p, 1/q)\) is in the set \(\Delta\) minus the half-open segments \((O, A]\) and \([A', O')\). They also proved that for \(R\) to be bounded from \(L^p(\mathbb R^2)\) to \(L^q(\mathbb R^2)\), it is necessary that \((1/p, 1/q)\) is in \(\Delta\), and in the case \(n=2, 3\) it is known that \(R\) is bounded precisely for \((1/p, 1/q)\in\Delta\). The author gives a positive result for endpoint estimates in the case \(n\geq 4\), and more. His main result is (for not necessarily homogeneous polynomials): if \(a_{1, n-1}\neq 0\), then there is \(C_n>0\) such that \[ \|Tf\|_{L^{n+1}}\leq C_n|a_{1,n-1}|^{-1/(n+1)}\|f\|_{L^{(n+1)/2}}, \] 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.
reviewed by: Kôzô Yabuta
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B15
Mathematics Subject Classification ID: 42B30
Mathematics Subject Classification ID: 44A12
zbMATH DE Number: 1442142
zbMATH Keywords: Radon transform
zbMATH Keywords: \(L^p-L^q\) estimate
MaRDI profile type: Publication
cites work: On the Restriction of The Fourier Transform to Curves: Endpoint Results and The Degenerate Case
cites work: Q4235427
cites work: Q4728804
cites work: Multilinear proofs for two theorems on circular averages
cites work: Q4847895
cites work: Q3984203
cites work: Models of degenerate Fourier integral operators and Radon transforms
cites work: The Newton polyhedron and oscillatory integral operators
cites work: Degenerate Fourier integral operators in the plane
cites work: Radon transforms and finite type conditions
cites work: Oscillatory integrals related to Radon-like transforms
full work available at URL: s0012-7094-00-10125-1
OpenAlex ID: W2017306751
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description: scientific article; zbMATH DE number 1442142