Young equations with singularities.

In this paper we prove existence and uniqueness of a mild solution to the Young equation d y (t) = A y (t) d t + σ (y (t)) d x (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 X α -norm of the function under the integral sign blows up approaching a and X α is an intermediate space between X and D (A). [ABSTRACT FROM AUTHOR]