MILD WELL-POSEDNESS OF SECOND ORDER DIFFERENTIAL EQUATION ON THE REAL LINE

Abstract

We study the $(W^{2,p},W^{1,p})$-mild well-posedness of the second order differential equation $(P_2): u'' = Au+f$ on the real line $\mathbb{R}$, where $A$ is a densely defined closed operator on a Banach space $X$. We completely characterize the $(W^{2,p}, W^{1,p})$-mild well-posedness of $(P_2)$ by $L^p$-Fourier multiplier defined by the resolvent of $A$.