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

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
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### References:

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