It is well known that the Steklov average of a function $f:(0, T)\to\mathbb{R}$ is defined by $$ f_h(t)=\frac{1}{h}\int_{t}^{t+h}f(s)\,ds\text{ if }0<t<T-h $$ and $f_h=0$ if $t\geq T-h$. Such a type of average is very important to justify the time derivative in parabolic PDEs. I want to know if a similar average exists for both the time and space variable which can be used to justify both the time and space derivative in a PDE, for example, to choose test functions depending on the solution of the PDE itself? If so, what is the exact definition?
Any help would be very much appreciated. Thanks.