\(L^ p\) bounds for Hilbert transforms along convex curves. (English) Zbl 0641.42009

Cordoba, A.; Nagel, A.; Vance, J.; Wainger, S.; Weinberg, D.

Invent. Math. 83, 59-71 (1986).

Let \(\Gamma: {\mathbb{R}}\to {\mathbb{R}}^ n\) be a curve in \({\mathbb{R}}^ n\) with \(\Gamma (0)=0\), \(n\geq 2\). For an appropriate function f, associate to \(\Gamma\) the Hilbert transform operator \({\mathcal H}\) defined by the principal-value integral \[ {\mathcal H}f(x)=p.v.\int^{\infty}_{- \infty}f(x-\Gamma (t))\frac{dt}{t}(x\in {\mathbb{R}}^ n). \] The authors consider the problem of determining curves \(\Gamma\), and indices p for which one has the \(L^ p\) bound (*) \(\| {\mathcal H}f\|_ p\leq A_ p\| f\|_ p,\) where \(A_ p\) is a constant depending only on \(\Gamma\) and p, and not on f. They prove: Theorem. Suppose \(\Gamma (t)=(t,\gamma (t)),\) \(t\in {\mathbb{R}}\), is a continuous curve with \(\gamma: {\mathbb{R}}\to {\mathbb{R}}\) convex for \(t\geq 0\), \(\gamma (0)=0\), \(\gamma'(0)^+=0\), and \(\gamma\) either even or odd. Suppose also that \(\gamma'\) has bounded doubling time, i.e., there exists a constant \(C>1\) with \(\gamma'(CT)^+\geq 2\gamma'(t)^-\) for \(t\geq 0\). Then \({\mathcal H}\) is bounded on \(L^ p({\mathbb{R}}^ 2)\) for \(4/3<p<4\).

Reviewer: B.P.Duggal

MSC:

42A38	Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42A50	Conjugate functions, conjugate series, singular integrals

Keywords:

Fourier transform; distribution; convex curve; L p-bound; Hilbert transform operator

References:

[1]	[BP] Benedek, A., Panzone, R.: The spacesL p, with mixed norm. Duke Math. J.28, 301-324 (1961) · Zbl 0107.08902
[2]	[Ch] Christ, M.: Hilbert transforms along curves, II. A flat case (Preprint)
[3]	[CF] Cordoba, A., Fefferman, R.: On the equivalence between the boundedness of certain classes of maximal and multiplier operators in Fourier analysis. Proc. Natl. Acad. Sci. USA74, 423-425 (1977) · Zbl 0342.42003
[4]	[NSW1] Nagel, A., Stein, E.M., Wainger, S.: Differentiation in lacunary directions. Proc. Natl. Acad. Sci. USA75, 1060-1062 (1978) · Zbl 0391.42015
[5]	[NSW2] Nagel, A., Stein, E.M., Wainger, S.: Hilbert transforms and maximal functions related to variable curves, I. Proc. Symp. Pure Math., vol. 35, Am. Math. Soc. Providence, R.I. 1979 · Zbl 0463.42008
[6]	[NVWW1] Nagel, A., Vance, J., Wainger, S., Weinberg, D.: Hilbert transforms for convex curves. Duke Math. J.50, 735-744 (1983) · Zbl 0524.44001
[7]	[NVWW2] Nagel, A., Vance, J., Wainger, S., Weinberg, D.: The Hilbert transform for convex curves inR n. Am. J. Math. (To appear) · Zbl 0589.42014
[8]	[NW] Nagel, A., Wainger, S.: Hilbert transforms associated with plane curves. Trans. Am. Math. Soc.223, 235-252 (1976) · Zbl 0341.44005
[9]	[Ne] Nestlerode, W.C.: Singular integrals and maximal functions associated with highly monotone curves. Trans. Am. Math. Soc.267, 435-444 (1981) · Zbl 0488.42016
[10]	[S] Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton, N.J.: Princeton Univ. Press 1970 · Zbl 0207.13501
[11]	[SW] Stein, E.M., Wainger, S.: Problems in harmonic analysis related to curvature. Bull. Am. Math. Soc.84, 1239-1295 (1978) · Zbl 0393.42010
[12]	[SWe] Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton, N.J.: Princeton Univ. Press 1971 · Zbl 0232.42007
[13]	[Wn] Weinberg, D.: The Hilbert transform and maximal function for approximately homogeneous curves. Trans. Am. Math. Soc.267, 295-306 (1981) · Zbl 0484.42005
[14]	[Z] Zygmund, A.: Trigonometric Series, vol. 1, 2nd ed. London: Cambridge Univ. Press 1959 · Zbl 0085.05601