Young equations with singularities

In this paper we prove existence and uniqueness of a mild solution to the Young equation dy(t)=Ay(t)dt+σ(y(t))dx(t), t∈[0,T], y(0)=ψ. Here, A is an unbounded operator which generates a semigroup of bounded linear operators (S(t))t≥0 on a Banach space X, x is a real-valued η-Hölder continuous. Our aim is to reduce, in comparison to Gubinelli et al. (2006) and Addona et al. (2022) (see also Deya et al. (2012) and Gubinelli and Tindel, (2010)), the regularity requirement on the initial datum ψ 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]⊂R when the Xα-norm of the function under the integral sign blows up approaching a and Xα is an intermediate space between X and D(A).