This is an addition to the above material. It is of course impossible to give an all-embracing answer to your question—there are so many identities. But the general answer is a resounding YES. The proofs are usually very simple, using a density argument—distributions can be approximated by smooth functions in their natural topologies so one can take limits of the classical versions for smooth functions. By the way this means that one can avoid some of the complications involved in using minimal smoothness assumptions.

There are, of course, some caveats (already hinted at above).

1. If products are involved there can be difficulties. However, all is not lost—avoid the common fallacy that the fact that one cannot always multiply distributions means that one can NEVER multiply them. Usually one can! There are many situations where one can a rigorous and elementary definition—for starters the product of a smooth scalar function and distributional vector field (or vice versa) is well-defined and the usual rules (product formula, etc.) hold.

2. The question of integration (e.g., in the Stokes theorem). Distributions, say on the line, always have primitives but not necessarily definite integrals. Again this doesn’t mean that they NEVER have—again they “usually do in practice” and have the sort of properties you would expect from the classical case. These concepts were developed decades ago using elementary methods (no functional analysis or duality theory for locally convex spaces. Not that I have anything against the latter—they just don’t usually occur in the toolbox of many mathematicians who want to use distributions).

3. I could go on and on about this but I will give two further remarks: Evaluation at points: it is not true that every distribution has a value at every point but again this does NOT mean that one can NEVER evaluate a distribution at a point. Again this concept was investigated in detail (in the 50’s and 60’s of the last century) using elementary methods and most distributions of practical relevance can be evaluated at most points. Composition with functions. Again symbols like $T\circ f$, i.e., the substitution of a function in a distribution, are not always well-defined. However, there are general situations where it can be rigorously defined and then it has the familiar properties.

The theory of distributions was developed by Sobolev and Schwartz, the latter using the language and techniques of duality for locally convex spaces. A number of mathematicians were then motivated to develop the theory from a more direct and elementary standpoint (i.e., at the level of an advanced calculus, or in germanophone countries “Analysis”, course). However, most texts use the Schwartz approach and the latter seems to have slipped into oblivion.

All these approaches are on record and easily available online (some names: Mikusinski, Heinz König, Sebastião e Silva, Sikorski). I would be happy to supply more details.

This is an addition to Dirk's answer. It is of course impossible to give an all-embracing answer to your question—there are so many identities. But the general answer is a resounding YES. The proofs are usually very simple, using a density argument—distributions can be approximated by smooth functions in their natural topologies so one can take limits of the classical versions for smooth functions. By the way this means that one can avoid some of the complications involved in using minimal smoothness assumptions.

There are, of course, some caveats (already hinted at above).

1. If products are involved there can be difficulties. However, all is not lost—avoid the common fallacy that the fact that one cannot always multiply distributions means that one can NEVER multiply them. Usually one can! There are many situations where one can a rigorous and elementary definition—for starters the product of a smooth scalar function and distributional vector field (or vice versa) is well-defined and the usual rules (product formula, etc.) hold.

2. The question of integration (e.g., in the Stokes theorem). Distributions, say on the line, always have primitives but not necessarily definite integrals. Again this doesn’t mean that they NEVER have—again they “usually do in practice” and have the sort of properties you would expect from the classical case. These concepts were developed decades ago using elementary methods (no functional analysis or duality theory for locally convex spaces. Not that I have anything against the latter—they just don’t usually occur in the toolbox of many mathematicians who want to use distributions).

3. I could go on and on about this but I will give two further remarks: Evaluation at points: it is not true that every distribution has a value at every point but again this does NOT mean that one can NEVER evaluate a distribution at a point. Again this concept was investigated in detail (in the 50’s and 60’s of the last century) using elementary methods and most distributions of practical relevance can be evaluated at most points. Composition with functions. Again symbols like $T\circ f$, i.e., the substitution of a function in a distribution, are not always well-defined. However, there are general situations where it can be rigorously defined and then it has the familiar properties.

The theory of distributions was developed by Sobolev and Schwartz, the latter using the language and techniques of duality for locally convex spaces. A number of mathematicians were then motivated to develop the theory from a more direct and elementary standpoint (i.e., at the level of an advanced calculus, or in germanophone countries “Analysis”, course). However, most texts use the Schwartz approach and the latter seems to have slipped into oblivion.

All these approaches are on record and easily available online (some names: Mikusinski, Heinz König, Sebastião e Silva, Sikorski). I would be happy to supply more details.