zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 $n\ge 2$ 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 $\psi\in C_c^\infty(\Bbb R^3)$ 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 $a_{1, n-1}\ne 0$ and $a_{n-1, 1}\ne 0$, Phong and Stein proved that $R$ is bounded from $L^p(\Bbb R^2)$ to $L^q(\Bbb R^2)$, 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 $L^p(\Bbb R^2)$ to $L^q(\Bbb R^2)$, 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 $n\ge 4$, and more. His main result is (for not necessarily homogeneous polynomials): if $a_{1, n-1}\ne 0$, then there is $C_n>0$ such that $$\|Tf\|_{L^{n+1}}\le C_n|a_{1,n-1}|^{-1/(n+1)}\|f\|_{L^{(n+1)/2}},$$ 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.

42B20Singular and oscillatory integrals, several variables
42B15Multipliers, several variables
42B30$H^p$-spaces (Fourier analysis)
44A12Radon transform
Full Text: DOI
[1] J.-G. Bak, Multilinear proofs for convolution estimates for degenerate plane curves , to appear in Canad. Math. Bull. (1999).
[2] M. Christ, On the restriction of the Fourier transform to curves: Endpoint results and the degenerate case , Trans. Amer. Math. Soc. 287 (1985), 223--238. · Zbl 0563.42010 · doi:10.2307/2000407
[3] --. --. --. --., Convolution, curvature, and combinatorics: A case study , Internat. Math. Res. Notices 1998 , 1033--1048. · Zbl 0927.42008 · doi:10.1155/S1073792898000610
[4] S. Drury, “A survey of $k$-plane transform estimates” in Commutative Harmonic Analysis (Canton, N.Y., 1987) , Contemp. Math. 91 , Amer. Math. Soc., Providence, 1989, 43--55. · Zbl 0679.44004
[5] D. Oberlin, Multilinear proofs for two theorems on circular averages , Colloq. Math. 63 (1992), 187--190. · Zbl 0765.42008 · eudml:210144
[6] D. H. Phong, “Singular integrals and Fourier integral operators” in Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, N.J., 1991) , Princeton Math. Ser. 42 , Princeton Univ. Press, Princeton, 1995, 286--320. · Zbl 0836.42009
[7] D. H. Phong and E. M. Stein, Radon transforms and torsion , Internat. Math. Res. Notices 1991 , 49--60. · Zbl 0761.46033 · doi:10.1155/S1073792891000077
[8] --. --. --. --., Models of degenerate Fourier integral operators and Radon transforms , Ann. of Math. (2) 140 (1994), 703--722. · Zbl 0833.43004 · doi:10.2307/2118622
[9] --. --. --. --., The Newton polyhedron and oscillatory integral operators , Acta Math. 179 (1997), 105--152. · Zbl 0896.35147 · doi:10.1007/BF02392721
[10] A. Seeger, Degenerate Fourier integral operators in the plane , Duke Math. J. 71 (1993), 685--745. · Zbl 0806.35191 · doi:10.1215/S0012-7094-93-07127-X
[11] --. --. --. --., Radon transforms and finite type conditions , J. Amer. Math. Soc. 11 (1998), 869--897. JSTOR: · Zbl 0907.35147 · doi:10.1090/S0894-0347-98-00280-X · http://links.jstor.org/sici?sici=0894-0347%28199810%2911%3A4%3C869%3ARTAFTC%3E2.0.CO%3B2-U&origin=euclid
[12] E. M. Stein, “Oscillatory integrals related to Radon-like transforms” in Proceedings of the Conference in Honor of Jean-Pierre Kahane (Orsay, 1993) , J. Fourier Anal. Appl. 1995 , special issue, CRC, Boca Raton, Fla., 535--551. · Zbl 0971.42009