# Convolution with an analytic semigroup

Let $$e^{At}$$ denote an analytic semigroup on Hilbert space $$X$$ generate by $$A:D(A)\to X$$. Also, let $$f\in L^1(0,\tau;X)$$. I want to show that the convolution $$g(t)=\int_0^t e^{A(t-s)}f(s)ds$$ belongs to $$W^{1,1}(0,\tau;X)\cap L^1(0,\tau;D(A))$$. If I set $$f(s)=e^{As}x$$ where $$x\in X$$ then $$g(t)=te^{At}x$$. Because of analyticity of $$e^{At}$$ we have $$\sup_{t>0}\|tAe^{At}\|_{L(X)}<\infty$$ thus $$g(t)\in L^{\infty}(0,\tau;D(A))$$ and $$\dot{g}(t)=e^{At}x+tAe^{At}x\in L^{\infty}(0,\tau;X)$$. Does this prove it?

• Did you try with $A$ compact and normal – reuns Dec 4 at 0:11