  We prove that if the Hardy-Littlewood maximal operator is bounded on a
separable Banach function space $X(\mathbb{R}^n)$ and on its associate space
$X'(\mathbb{R}^n)$ and a maximally modulated Calder\'on-Zygmund singular
integral operator $T^\Phi$ is of weak type $(r,r)$ for all $r\in(1,\infty)$,
then $T^\Phi$ extends to a bounded operator on $X(\mathbb{R}^n)$. This theorem
implies the boundedness of the maximally modulated Hilbert transform on
variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R})$ under natural assumptions
on the variable exponent $p:\mathbb{R}\to(1,\infty)$. Applications of the above
result to the boundedness and compactness of pseudodifferential operators with
$L^\infty(\mathbb{R},V(\mathbb{R}))$-symbols on variable Lebesgue spaces
$L^{p(\cdot)}(\mathbb{R})$ are considered. Here the Banach algebra
$L^\infty(\mathbb{R},V(\mathbb{R}))$ consists of all bounded measurable
$V(\mathbb{R})$-valued functions on $\mathbb{R}$ where $V(\mathbb{R})$ is the
Banach algebra of all functions of bounded total variation.
