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Estimates for Hilbert transforms along variable general curves. (English) Zbl 1462.44004

Summary: We consider the problem of proving \(L^2(\mathbb{R}^2)\) boundedness and single annulus \(L^p(\mathbb{R}^2)\) estimate for the Hilbert transform along variable general curve \((t, u_1(x_1) t + u_2(x_1) \gamma(t))\) \[ H_{u_1, u_2, \gamma} f(x_1, x_2) := \text{p. v.} \int\limits_{-\infty}^\infty f(x_1 - t, x_2 - u_1(x_1) t - u_2(x_1) \gamma(t)) \frac{\text{d}t}{t}, \quad \forall (x_1, x_2) \in \mathbb{R}^2, \] where \(p \in (1, \infty), \gamma\) is a general curve on \(\mathbb{R}, u_1 : \mathbb{R} \to \mathbb{R}\) and \(u_2 : \mathbb{R} \to \mathbb{R}\) are measurable functions. Moreover, all the bounds are independent of the measurable functions \(u_1\) and \(u_2\). For any given \(p \in (1, \infty)\), we also obtain the \(L^p (\mathbb{R})\) boundedness of the corresponding Carleson operator \[ \mathcal{C}_{N_1, N_2, \gamma} f(x) := \sup_{N_1, N_2 \in \mathbb{R}} \left|\text{p. v.} \int\limits_{-\infty}^\infty e^{iN_1 t + iN_2 \gamma (t)} f (x - t) \frac{\text{d}t}{t}\right|, \quad \forall x \in \mathbb{R}. \]

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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