Endpoint \(L^p \rightarrow L^q\) bounds for integration along certain polynomial curves. (English) Zbl 1209.44003
Let \(P:\mathbb{R}\to \mathbb{R}^d\) be a polynomial and let \(T_P\) be the operator defined by
\[ T_P f(x)=\int_I f(x-P(s))d\sigma_P(s),\tag{1} \]
where \(I\) is an interval and \(d\sigma_P(s)\) represents affine arclength measure along \(P\),
\[ d\sigma_P(s)=|\mathrm{det}(P'(s),P''(s),\ldots,P^{(d)}(s))|^{2/{d(d+1)}}ds. \]
The goal of the article is to establish \(L^p\to L^q\) bounds for \(T_P\) in the full conjectured range of exponents in dimension \(d\geq 4\) (together with a slight improvement in Lorentz spaces).
The main result of the paper is the following.
Theorem 1. Let \(d\geq 4\), let \(P:\mathbb{R}\to \mathbb{R}^d\) be a polynomial of degree \(N\), and let \(T_P\) be the operator defined by (1). Let \(p_d=(d+1)/2\) and \(q_d=d(d+1)/2 (d-1)\). Then \(T_P\) maps \(L^p\to L^q\) if \((p,q)=(p_d,q_d)\) or \( (q'_d,p'_d)\), with bounds depending only on \(d,N\). Moreover, \(T_P\) maps the Lorentz space \(L^{p_d,u}(\mathbb{R}^d)\) boundedly into \(L^{q_d,v}(\mathbb{R}^d)\) and \(L^{q'_d,v'}(\mathbb{R}^d)\) into \(L^{p'_d,u'}(\mathbb{R}^d)\) whenever \(u<q_d\), \(v>p_d\), and \(u<v\).
This result has already been established in dimension \(2\) by D. M. Oberlin [Math. Scand. 90, No. 1, 126–138 (2002; Zbl 1034.42012)] and in dimension \(3\) by S. Dendrinos, N. Laghi and J. Wright [J. Funct. Anal. 257, No. 5, 1355–1378 (2009; Zbl 1177.42009)]. In the case when \(I\) has infinite length and \(d\sigma_P\not\equiv 0\), Theorem 1 is sharp up to the Lorentz space endpoints. A proof of this in the case \(P(t)=(t,t^2,\dots, t^d)\) is given by B. Stovall [J. Lond. Math. Soc., II. Ser. 80, No. 2, 357–374 (2009; Zbl 1174.42010)].
\[ T_P f(x)=\int_I f(x-P(s))d\sigma_P(s),\tag{1} \]
where \(I\) is an interval and \(d\sigma_P(s)\) represents affine arclength measure along \(P\),
\[ d\sigma_P(s)=|\mathrm{det}(P'(s),P''(s),\ldots,P^{(d)}(s))|^{2/{d(d+1)}}ds. \]
The goal of the article is to establish \(L^p\to L^q\) bounds for \(T_P\) in the full conjectured range of exponents in dimension \(d\geq 4\) (together with a slight improvement in Lorentz spaces).
The main result of the paper is the following.
Theorem 1. Let \(d\geq 4\), let \(P:\mathbb{R}\to \mathbb{R}^d\) be a polynomial of degree \(N\), and let \(T_P\) be the operator defined by (1). Let \(p_d=(d+1)/2\) and \(q_d=d(d+1)/2 (d-1)\). Then \(T_P\) maps \(L^p\to L^q\) if \((p,q)=(p_d,q_d)\) or \( (q'_d,p'_d)\), with bounds depending only on \(d,N\). Moreover, \(T_P\) maps the Lorentz space \(L^{p_d,u}(\mathbb{R}^d)\) boundedly into \(L^{q_d,v}(\mathbb{R}^d)\) and \(L^{q'_d,v'}(\mathbb{R}^d)\) into \(L^{p'_d,u'}(\mathbb{R}^d)\) whenever \(u<q_d\), \(v>p_d\), and \(u<v\).
This result has already been established in dimension \(2\) by D. M. Oberlin [Math. Scand. 90, No. 1, 126–138 (2002; Zbl 1034.42012)] and in dimension \(3\) by S. Dendrinos, N. Laghi and J. Wright [J. Funct. Anal. 257, No. 5, 1355–1378 (2009; Zbl 1177.42009)]. In the case when \(I\) has infinite length and \(d\sigma_P\not\equiv 0\), Theorem 1 is sharp up to the Lorentz space endpoints. A proof of this in the case \(P(t)=(t,t^2,\dots, t^d)\) is given by B. Stovall [J. Lond. Math. Soc., II. Ser. 80, No. 2, 357–374 (2009; Zbl 1174.42010)].
Reviewer: Vitaly Vladimirovich Volchkov (Donetsk)
MSC:
| 44A12 | Radon transform |
| 42A85 | Convolution, factorization for one variable harmonic analysis |
| 44A35 | Convolution as an integral transform |
Keywords:
generalized Radon transform; affine arclength; polynomial curves; convolution; Lorentz spacesReferences:
| [1] | Bak, J.-G.; Oberlin, D. M.; Seeger, A., Restriction of Fourier transforms to curves. II. Some classes with vanishing torsion, J. Aust. Math. Soc., 85, 1, 1-28 (2008) · Zbl 1154.42004 |
| [2] | Bak, J.-G.; Oberlin, D. M.; Seeger, A., Restriction of Fourier transforms to curves and related oscillatory integrals, Amer. J. Math., 131, 2, 277-311 (2009) · Zbl 1166.42006 |
| [3] | Choi, Y., Convolution operators with the affine arclength measure on plane curves, J. Korean Math. Soc., 36, 1, 193-207 (1999) · Zbl 0927.42009 |
| [4] | 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, 2, 247-261 (2003) · Zbl 1037.42012 |
| [5] | Christ, M., On the restriction of the Fourier transform to curves: endpoint results and the degenerate case, Trans. Amer. Math. Soc., 287, 1, 223-238 (1985) · Zbl 0563.42010 |
| [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] | S. Dendrinos, M. Folch-Gabayet, J. Wright, An affine invariant inequality for rational functions and applications in harmonic analysis, preprint.; S. Dendrinos, M. Folch-Gabayet, J. Wright, An affine invariant inequality for rational functions and applications in harmonic analysis, preprint. · Zbl 1202.42020 |
| [9] | Dendrinos, S.; Laghi, N.; Wright, J., Universal \(L^p\) improving for averages along polynomial curves in low dimensions, J. Funct. Anal., 257, 5, 1355-1378 (2009) · Zbl 1177.42009 |
| [10] | S. Dendrinos, J. Wright, Fourier restriction to polynomial curves I: a geometric inequality, preprint.; S. Dendrinos, J. Wright, Fourier restriction to polynomial curves I: a geometric inequality, preprint. · Zbl 1203.42015 |
| [11] | Drury, S. W., Degenerate curves and harmonic analysis, Math. Proc. Cambridge Philos. Soc., 108, 1, 89-96 (1990) · Zbl 0708.42011 |
| [12] | Drury, S. W.; Marshall, B. P., Fourier restriction theorems for curves with affine and Euclidean arclengths, Math. Proc. Cambridge Philos. Soc., 97, 1, 111-125 (1985) · Zbl 0567.42009 |
| [13] | Drury, S. W.; Marshall, B. P., Fourier restriction theorems for degenerate curves, Math. Proc. Cambridge Philos. Soc., 101, 3, 541-553 (1987) · Zbl 0645.42015 |
| [14] | Gressman, P. T., Convolution and fractional integration with measures on homogeneous curves in \(R^n\), Math. Res. Lett., 11, 5-6, 869-881 (2004) · Zbl 1129.42345 |
| [15] | Gressman, P. T., \(L^p\)-improving properties of averages on polynomial curves and related integral estimates, Math. Res. Lett., 16, 6, 971-989 (2009) · Zbl 1204.42022 |
| [16] | Guggenheimer, H. W., Differential Geometry (1963), McGraw-Hill Book Co., Inc.: McGraw-Hill Book Co., Inc. New York, San Francisco, Toronto, London · Zbl 0116.13402 |
| [17] | Laghi, N., A note on restricted X-ray transforms, Math. Proc. Cambridge Philos. Soc., 146, 3, 719-729 (2009) · Zbl 1180.44002 |
| [18] | Littman, W., \(L^p - L^q\)-estimates for singular integral operators arising from hyperbolic equations. Partial differential equations, (Proc. Sympos. Pure Math., vol. XXIII, Univ. California. Proc. Sympos. Pure Math., vol. XXIII, Univ. California, Berkeley, CA, 1971 (1973), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI), 479-481 · Zbl 0263.44006 |
| [19] | Oberlin, D. M., Convolution with measures on flat curves in low dimensions, preprint · Zbl 1203.42024 |
| [20] | Oberlin, D. M., Convolution estimates for some measures on curves, Proc. Amer. Math. Soc., 99, 1, 56-60 (1987) · Zbl 0613.43002 |
| [21] | Oberlin, D. M., A convolution estimate for a measure on a curve in \(R^4\), Proc. Amer. Math. Soc., 125, 5, 1355-1361 (1997) · Zbl 0972.42006 |
| [22] | Oberlin, D. M., A convolution estimate for a measure on a curve in \(R^4\). II, Proc. Amer. Math. Soc., 127, 1, 217-221 (1999) · Zbl 0972.42007 |
| [23] | Oberlin, D. M., Convolution with measures on polynomial curves, Math. Scand., 90, 1, 126-138 (2002) · Zbl 1034.42012 |
| [24] | Pan, Y., \(L^p\)-improving properties for some measures supported on curves, Math. Scand., 78, 1, 121-132 (1996) · Zbl 0871.42011 |
| [25] | Shafarevich, I. R., Basic Algebraic Geometry (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0362.14001 |
| [26] | Sjölin, P., Fourier multipliers and estimates of the Fourier transform of measures carried by smooth curves in \(R^2\), Studia Math., 51, 169-182 (1974) · Zbl 0291.42003 |
| [27] | Stovall, B., Endpoint bounds for a generalized Radon transform, J. Lond. Math. Soc. (2), 80, 2, 357-374 (2009) · Zbl 1174.42010 |
| [28] | Tao, T.; Wright, J., \(L^p\) improving bounds for averages along curves, J. Amer. Math. Soc., 16, 3, 605-638 (2003) · Zbl 1080.42007 |
| [29] | Weyl, H., The Classical Groups (1946), Princeton Univ. Press: Princeton Univ. Press Princeton NJ · JFM 65.0058.02 |
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