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Convolution with measures on flat curves in low dimensions. (English) Zbl 1203.42024

Let \(\gamma\) be a curve in \(\mathbb R^d\) given by \[ \gamma (t)=(t, \frac {t^2}{2}, \dots, \frac{t^{d-1}}{(d-1)!}, \phi (t)), \] where \(\phi \in C^d(a,b)\), where \(\phi^{(j)}(t)>0\) for \(t \in (a,b)\) and \(j=0, 1, 2, \dots, d\), and where \(\phi^{(d)}\) is nondecreasing. Such curves are termed simple. In the paper under review the author proves \(L^p \to L^q\) convolution estimates for the affine arclength measure \(\lambda\) on \(\gamma\), given by \( d\lambda = \phi^{(d)}(t)^{2/(d^2+d)}dt\), when \(d=2, 3, 4\). For \(d=2, 3\), he also establishes certain related Lorentz space estimates.

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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[1] Bennett, J.; Seeger, A., The Fourier extension operator on large spheres and related oscillatory integrals, Proc. Lond. Math. Soc., 98, 45-82 (2009) · Zbl 1159.42005
[2] Bak, J.-G.; Oberlin, D.; Seeger, A., Two endpoint bounds for Radon transforms in the plane, Rev. Mat. Iberoamericana, 18, 231-247 (2002) · Zbl 1015.42007
[3] Bak, J.-G.; Oberlin, D.; Seeger, A., Restriction of Fourier transforms to curves II: some classes with vanishing curvature, J. Aust. Math. Soc., 85, 1-28 (2008) · Zbl 1154.42004
[4] Choi, Y., Convolution operators with the affine arclength measure on plane curves, J. Korean Math. Soc., 36, 193-207 (1999) · Zbl 0927.42009
[5] Choi, Y., The \(L^p - L^q\) mapping properties of convolution operators with the affine arclength measure on space curves, J. Aust. Math. Soc., 75, 247-261 (2003) · Zbl 1037.42012
[6] Christ, M., Convolution, curvature and combinatorics. A case study, Int. Math. Res. Not. IMRN, 19, 1033-1048 (1998) · Zbl 0927.42008
[7] M. Christ, Quasi-extremals for a Radon-like transform, preprint; M. Christ, Quasi-extremals for a Radon-like transform, preprint
[8] Dendrinos, S.; Laghi, N.; Wright, J., Universal \(L^p\) improving for averages along polynomial curves in low dimensions, J. Funct. Anal., 257, 1355-1378 (2009) · Zbl 1177.42009
[9] Drury, S. W., Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc., 108, 89-96 (1990) · Zbl 0708.42011
[10] Drury, S. W.; Marshall, B., Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc., 97, 111-125 (1985) · Zbl 0567.42009
[11] Drury, S. W.; Marshall, B., Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc., 101, 541-553 (1987) · Zbl 0645.42015
[12] Oberlin, D. M., Convolution estimates for some measures on curves, Proc. Amer. Math. Soc., 99, 56-60 (1987) · Zbl 0613.43002
[13] Oberlin, D. M., Convolution with affine arclength measures in the plane, Proc. Amer. Math. Soc., 127, 3591-3592 (1999) · Zbl 0972.42008
[14] Oberlin, D. M., Some convolution inequalities and their applications, Trans. Amer. Math. Soc., 354, 2541-2556 (2002) · Zbl 0996.42005
[15] Stovall, Betsy, Endpoint bounds for a generalized Radon transform, J. Lond. Math. Soc., 1-18 (2009) · Zbl 1174.42010
[16] Betsy Stovall, Endpoint \(L^p \to L^q\) bounds for integration along polynomial curves, preprint; Betsy Stovall, Endpoint \(L^p \to L^q\) bounds for integration along polynomial curves, preprint · Zbl 1209.44003
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