  In this paper we prove existence and uniqueness of a mild solution to the
Young equation $dy(t)=Ay(t)dt+\sigma(y(t))dx(t)$, $t\in[0,T]$, $y(0)=\psi$.
Here, $A$ is an unbounded operator which generates a semigroup of bounded
linear operators $(S(t))_{t\geq 0}$ on a Banach space $X$, $x$ is a real-valued
$\eta$-H\"older continuous. Our aim is to reduce, in comparison to [4] and [1]
(see also [2,5]) in the bibliography, the regularity requirement on the initial
datum $\psi$ eventually dropping it.
  The main tool is the definition of a sewing map for a new class of increments
which allows the construction of a Young convolution integral in a general
interval $[a,b]\subset \mathbb R$ when the $X_\alpha$-norm of the function
under the integral sign blows up approaching $a$ and $X_{\alpha}$ is an
intermediate space between $X$ and $D(A)$.
