\(L^ p\) bounds for Hilbert transforms along convex curves
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 \(4/3<p<4\).
scientific article; zbMATH DE number 4044424
scientific article