  Using truncated variation techniques we obtain an improved version of the
Loeve-Young inequality for the Riemann-Stieltjes integrals driven by rough
paths. This allowed us to strenghten some result on the existence of solutions
of integral equations driven by moderately irregular signals. We introduce also
a new Banach space, containing as a proper subspace the paths with finite
$p$-variation, and develop, in a systematic way, several parallel results for
the paths from this space, obtained so far for the paths with finite
$p$-variation. We start the paper with a general theorem on the existence of
the Riemann-Stieltjes integral.
