Speed limit operators for oscillating speed functions (Q1804698)

Let \(\mathcal F\) denote the class of step functions on the real line \(R\), so that, in particular, elements of \(\mathcal F\) are expressible as linear sums of functions of the form \(f_0\), where \(f_0(t)=a\sum^n_{j=1}\psi_{(a,b)}(t)\), \(\psi_{(a,b)}(t)=1\) if \(a<t<b\), (\(=0\) otherwise). Speed limit operators \(T_t\), \(t>0\), are defined in terms of a speed function \(\phi\) in the basic form \[ T_t(f)(x)=f(u(x,t)) {\phi(u(x,t))\over\phi(x)};\;{\phi(u(x,t))\over\phi(x)}=\Biggl({\partial\over\partial x}\Biggr) u(x,t), \] so that \(\{T_t\mid t>0\}\) is a semigroup of operators. The results of this paper involve the construction of speed limit operators, starting with the definitions of \(T_t(f)\) for functions \(f\) in \(\mathcal F\), so that the operators constitute a bounded semigroup of operators in \(L^p(R^+)\) for \(1\leq p<\infty\), the operators are integral preserving so that \(\int T_t(f)d\mu=\int f d\mu\), where \(\mu\) denotes the Lebesgue measure, and the operators are order preserving (positive) and positively homogeneous.