Young equations with singularities

Abstract

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 $(S(t){)}_{t\ge 0}$ on a Banach space $X$, $x$ is a real-valued $\eta$-Hölder 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 $D(A)$.