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