Convolution with semigroup: does this belong to the Sobolev space $W^{1,1}$?

Question

Let $X$ be a Banach space, $T(t)$ be a strongly continuous semigroup on $X$, and $f\in L^1(0,\tau;X)$. It has been implied that the integral $$v(t)=\int_0^t T(t-s)f(s)ds,\quad t\in [0,\tau]$$ is not always an element of $W^{1,1}(0,\tau;X)$. That seems odd to me. Can anyone think of an example?

Answer 1

In other words, if $A$ is the infinitesimal generator of $T$, the mild solution of the abstract inhomogeneous Cauchy problem $$\begin{cases}\dot v =A v +f\\v(0)=0\end{cases}$$ needs not to be $W^{1,1}_{loc}(\mathbb{R}_+, X)$. For instance an $f\in C^0(\mathbb{R}_+,X)$ of the form $f(t):=T(t)x$ for some $x=x(\theta)\in X$, gives $v(t)=tT(t)x=tf(t)$ which has no reason to be in $W^{1,1}_{loc}(\mathbb{R}_+, X)$.

To justify the preceding claim it is sufficient to show that $f(t)$ itself, the mild solution to the homogeneous Cauchy problem $\dot f=Af$ with $f(0)=x$, may fail to be $W^{1,1}$ (even at any point). Consider e.g.

• $X:=L^1(\mathbb{S}^1)$, $1$-periodic one variable $L^1_{loc}$ functions;

• $T :\mathbb{R}_+\times X\to X$ the left translation semigroup $T(t)x:=x(\cdot+t)$, whose infinitesimal generator is $A:=\partial_\theta$, with domain $D(A)=W^{1,1}(\mathbb{S}^1)$;

Then, for $x\in X$, $f(t):=T(t)x=x(\cdot+t)$ is, of course, the mild solution to $\dot f(t)=\partial_\theta f(t)$ with initial data $f(0)=x$, and defines a continuous path $f:\mathbb{R}\ni t \mapsto x(\cdot+t)\in X$. Saying, for some open interval $I$, that $f\in W^{1,1}(I;X)$ means there is $h\in L^1(I;X)\sim L^1(I\times \mathbb{S}^1)$ such that $f(t')-f(t)=\int_t^{t'} h(s,\cdot)ds$ in $X$, that is $x(\theta+t')-x(\theta+t)=\int_t^{t'} h(s,\theta)ds$ a.e., whence $x\in W^{1,1}(\mathbb{S}^1)$.