I am currently studying parameter dependent symbols, $s(t,x,\xi)$, where $t\in [0,1],x\in \Omega, \xi \in \mathbb{R^n}$. I wanted to know how the low regularity (for example, $s$ is just continuous w.r.t. $t$) of symbol w.r.t. parameter affects further study of symbols and the corresponding operators.
To be specific, I need this to study hyperbolic operators which are low regular in time. All the books which I have referred till now assume smoothness w.r.t. parameter.
Thanking you in advance.
There exists a vast literature on this topic. The general philosophy is that one can compensate for less regular coefficients in time by a greater regularity of the initial data. For analytic data, typically you only need $L^1_{loc}$ in time coefficients; Hölder coefficients are suited for problem posed in Gevrey classes; and so on. One of the pioneering papers in this line of research is due to De Giorgi, Colombini and Spagnolo (link). The Italian and Japanese schools were rather active on this topic throughout the 80s and 90s. Look for papers by Spagnolo, Colombini, Kajitani and their students, using "weakly hyperbolic" as keyword (indeed, with very smooth data strict hyperbolicity can be relaxed). More recently, M.Ruzhansky and his students are pushing forward research on this topic.
