I am looking for a differential analogue to the Lebesgue spaces $\mathcal{L}^p$ in the following sense:
The integral behaves badly for functions with too much mass i.e. such whose tails decay too slow and/or whose singularities diverge too slowly. This leads $\int f d \mu = \infty$ and it makes it impossible to compare two functions (by their integral). However, if we make some modifications $f$ to $f_p$ for varying $p$ s.t. at least for some $p$ we have $\int f_p d \mu < \infty$ and keep track of the modification, then we can work properly with it. This is what the $p$-norm captures. The modification in question is of course raising the absolute value to the power $p$.
Functions in $\mathcal{L}^p$ i.e. with finite $p$-norm should then be seen as "modification $p$ makes the integral of $f$ finite". In the case of measure spaces were there are no sets with arbitrary small, positive measure or in the case of spaces with finite measure we have the embedding theorems. In those cases, this modifications become linearly ordered, making it more reminiscent of a valuation. In general one at least has the interpolation embedding: $\mathcal{L}^p \cap \mathcal{L}^q \subseteq \mathcal{L}^r$ for any $r \in [p, q]$.
I would like to consider a similar modification scheme for differentiability. Thats is, I am looking for a gradual way to smooth out functions, of which I am able to keep track. My first approach was to consider convolution with functions of certain regularity and then keep track of the regularity of that function. But that seems to be too strong, since convoluting any function with a $C^k$ function yields something in $C^k$. However, I would like to have something like "up to regularity $\alpha$ the convolution is still not differentiable, but convoluting with anything above regularity $\alpha$ makes the convolution differentiable". But I think that convolution with special functions yields the right notion.
I am sure something like this has already been explored by someone at some point? Does it sound familiar to you?
I would very much appreciate any pointers/references.