Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $L^p$ Boundedness of Hilbert Transforms Associated with Variable Plane
  Curves   Let $p\in (1,\infty)$. In this paper, for any given measurable function $u:\
\mathbb{R}\rightarrow \mathbb{R}$ and a generalized plane curve $\gamma$
satisfying some conditions, the $L^p(\mathbb{R}^2)$ boundedness of the Hilbert
transform along the variable plane curve $u(x_1)\gamma$
$$H_{u,\gamma}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-u(x_1)\gamma(t))
\,\frac{\textrm{d}t}{t}, \quad \forall\, (x_1,x_2)\in\mathbb{R}^2, $$ is
obtained. At the same time, the $L^p(\mathbb{R})$ boundedness of the
corresponding Carleson operator along the general curve $\gamma$
$$\mathcal{C}_{u,\gamma}f(x):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}e^{iu(x)\gamma
(t)}f(x-t)\,\frac{\textrm{d}t}{t}, \quad\forall\, x\in\mathbb{R}, $$ is also
obtained. Moreover, all the bounds are independent of the measurable function
$u$.

13
The categories of this paper are: quant-ph
14
The categories of this paper are: math-ph
35
The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
34
The categories of this paper are: chem-ph
29
The categories of this paper are: funct-an
6
The categories of this paper are: astro-ph
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
26
The categories of this paper are: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
30
The categories of this paper are: q-alg
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The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg