اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدA sharp estimate for the Hilbert transform along finite order lacunary sets of directions
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Let $D$ be a nonnegative integer and ${mathbf{Theta}}subset S^1$ be a lacunary set of directions of order $D$. We show that the $L^p$ norms, $1<p<infty$, of the maximal directional Hilbert transform in the plane $$ H_{{mathbf{Theta}}} f(x):= sup_{vin {mathbf{Theta}}} Big|mathrm{p.v.}int_{mathbb R }f(x+tv)frac{mathrm{d} t}{t}Big|, qquad x in {mathbb R}^2, $$ are comparable to $(log#{mathbf{Theta}})^frac{1}{2}$. For vector fields $mathsf{v}_D$ with range in a lacunary set of of order $D$ and generated using suitable combinations of truncations of Lipschitz functions, we prove that the truncated Hilbert transform along the vector field $mathsf{v}_D$, $$ H_{mathsf{v}_D,1} f(x):= mathrm{p.v.} int_{ |t| leq 1 } f(x+tmathsf{v}_D(x)) ,frac{mathrm{d} t}{t}, $$ is $L^p$-bounded for all $1<p<infty$. These results extend previous bounds of the first author with Demeter, and of Guo and Thiele.
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