One-sided jet limits from uniform derivative bounds

Statement

Let $f\in C^\infty(U\times(0,T))$ , $T<\infty$ . Suppose every mixed derivative is uniformly bounded on $K\times(T-\delta_K,T)$ for each compact $K\subset U$ , with some $\delta_K>0$ . Then each $\partial_x^\alpha\partial_t^j f$ has a locally uniform limit as $t\uparrow T$ . These limits form a compatible smooth endpoint jet: if $F_j(x)=\lim_{t\uparrow T}\partial_t^jf(x,t)$ , then

F_j\in C^\infty(U),\qquad \partial_x^\alpha F_j=\lim_{t\uparrow T}\partial_x^\alpha\partial_t^jf.

Time Cauchy estimate and compatibility

A bound for $\partial_t(\partial_x^\alpha\partial_t^jf)$ gives a uniform Lipschitz estimate in time, hence the endpoint limit. On small rectangular boxes inside $U$ , the fundamental theorem of calculus along spatial coordinate segments identifies the spatial derivatives of each limit. Passing to the endpoint in the time identity gives

\partial_x^\alpha\partial_t^jf(x,t) =\partial_x^\alpha F_j(x)-\int_t^T\partial_x^\alpha\partial_t^{j+1}f(x,s)\,ds.

This records compatibility of consecutive normal derivatives. A bound for $f$ alone is insufficient, as a bounded oscillatory function near $T$ need not have a limit.