Geodesic
Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves
Canadian mathematical bulletin, Tome 43 (2000) no. 1, pp. 17-20

Voir la notice de l'article provenant de la source Cambridge University Press

Suppose that $\gamma \in {{C}^{2}}\left( \left[ 0,\infty\right]) \right)$ is a real-valued function such that $\gamma \left( 0 \right)\,=\,{\gamma }'\left( 0 \right)\,=\,0$ , and ${\gamma }''\left( t \right)\,\approx \,{{t}^{m-2}}$ , for some integer $m\,\ge \,2$ . Let $\Gamma \left( t \right)\,=\,\left( t,\,\gamma \left( t \right) \right),\,t\,>\,0$ , be a curve in the plane, and let $d\text{ }\!\!\lambda\!\!\text{ }\,\text{=}\,dt$ be a measure on this curve. For a function $f$ on ${{\mathbf{R}}^{2}}$ , let $$Tf\left( x \right)\,=\,\left( \text{ }\lambda \text{ }*f \right)\left( x \right)=\int_{0}^{\infty }{f\left( x-\Gamma \left( t \right) \right)dt,\,\,x\in {{\mathbf{R}}^{2}}}.$$ An elementary proof is given for the optimal ${{L}^{p}}-{{L}^{q}}$ mapping properties of $T$ .
DOI : 10.4153/CMB-2000-002-2
Mots-clés : 42A85, 42B15 
Bak, Jong-Guk. Multilinear Proofs for Convolution Estimates for Degenerate Plane Curves. Canadian mathematical bulletin, Tome 43 (2000) no. 1, pp. 17-20. doi: 10.4153/CMB-2000-002-2
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[B] [B] Bak, J.-G., Sharp convolution estimates for measures on flat surfaces. J.Math. Anal. Appl. 193 (1995), 756–771. Google Scholar

[BMO] [BMO] Bak, J.-G., McMichael, D. and Oberlin, D., Convolution estimates for some measures on flat curves. J. Funct. Anal. 101 (1991), 81–96. Google Scholar

[C1] [C1] Christ, M., On the restriction of the Fourier transform to curves: endpoint results and the degenerate case. Trans. Amer.Math. Soc. 287 (1985), 223–238. Google Scholar

[C2] [C2] Christ, M., Endpoint bounds for singular fractional integral operators. Preprint. Google Scholar

[D1] [D1] Drury, S., A survey of k-plane transform estimates. Commutative harmonic analysis (Canton, NY, 1987), 43–55; Contemp.Math. 91, Amer. Math. Soc. Providence, Rhode Island, 1989. Google Scholar

[D2] [D2] Drury, S., Degenerate curves and harmonic analysis. Math. Proc. Cambridge Philos. Soc. 108 (1990), 89–96. Google Scholar

[O1] [O1] Oberlin, D., Convolution estimates for some measures on curves. Proc.Amer.Math. Soc. 99 (1987), 56–60. Google Scholar

[O2] [O2] Oberlin, D., Multilinear proofs for two theorems on circular averages. Colloq.Math. 63 (1992), 187–190. Google Scholar

[O3] [O3] Oberlin, D., A convolution estimate for a measure on a curve in R4, II. Proc. Amer. Math. Soc. (to appear). Google Scholar

[RS] [RS] Ricci, F. and Stein, E. M., Harmonic analysis on nilpotent groups and singular integrals III: fractional integration along manifolds. J. Funct. Anal. 86 (1989), 360–389. Google Scholar

[Se] [Se] Secco, S., Fractional integration along homogeneous curves in R3. Preprint. Google Scholar

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