# Bruhat-Schwartz functions and derivatives in p-adic numbers

First of all, I am not an expert in neither classical, nor $p$-adic functional analysis, but anyway, I stumbled over the following lately:

Let $\varphi:\mathbb{Q}_p\rightarrow\mathbb{C}$.

Canonically, I would define a derivative of such a function in the following way: $\varphi'(x):=\lim\limits_{h\rightarrow 0} \frac{\varphi(x+h)-\varphi(x)}{|h|_p},$

provided of course that the limit exists. Is this the usual definition of the derivative?

In real functional analysis, there is the notion of Schwartz functions, which are rapidly decreasing functions supported on $\mathbb{R}^n$, i.e. functions, where for multiindices $\alpha,\beta$

$\sup_{x\in\mathbb{R}^n}|x^\alpha D^\beta (f)(x)|<\infty.$

To my knowledge, the term Bruhat-Schwartz function generalizes this to the case where the domain is a locally compact abelian group, e.g. the field of $p$-adic numbers. Thus there should be an analogue notion where one considers $p$-adic derivatives.

Now I found in some sources, that a Bruhat-Schwartz function on $\mathbb{Q}_p^n$ is a locally constant function with compact support. Unfortunately, I don't see at all how you get from the one definition to the other.

• Contrary to the situation over the reals, the delta function (up to multiplicative constant) is the only distribution supported at the origin. There is no notion of derivatives in the sense of distributions of \delta and therefore there is no good notion of local derivative. One can define fractional derivatives via Fourier but these are nonlocal. This issue is also related to the classification of homogeneous distributions and the appearance of poles with respect to the homogeneity parameter. – Abdelmalek Abdesselam May 27 '13 at 18:39

There is no good (nontrivial) notion for a derivative on functions on a totally disconnected group $G$ with values in $\mathbb{R}$ or $\mathbb{C}$. Assume (wlog) that $G$ is compact, then there exists a countable, ordered set of open, normal subgroups $(N_i)$ with the projective limit of $G/N_i$ being canonically isomorphic to $G$. So $G$ is a projective limit of finite groups. More generally this works for locally compact almost connected groups and Lie groups instead of finite groups. This was the motivation for Bruhat to introduce this notion for p-adic groups.

Another natural reason for why we say this is the analogue of a Schwartz space is that the Fourier trandform on $\mathbb{Q}_p$ transforms compactly supported and being locally constant into each other.

I suggest to have a look at Paul Sally survey on complex functions on $p$-adic groups. http://link.springer.com/article/10.1023%2FA%3A1007583108067#page-1

• It’s important to know that that article of Paul’s is rife with typos (introduced in the editing process, not by Paul). I think most of them are pretty obvious, though. – LSpice Jan 1 '19 at 20:53

Your formula for the derivative isn't even correct in the most classical setting : why do you divide by the norm of the increment instead of the increment itself?

Schwartz-Bruhat functions for totally disconnected spaces are generally defined to be the locally constant ones ; at least that's what one takes when considering distributions in a $p$-adic setting.

• Without the norm bars, the numerator is a real or complex number, the denominator is $p$-adic. – Lubin May 27 '13 at 15:40
• Ah, indeed ; still my remark stands : there is no absolute value in the usual case, and there is one here, so it can't be correct. – Julien Puydt May 27 '13 at 16:33
• I see, p-adic analysis is not as straight forward as originallyI thought. Thanks a lot for your help! – MHMertens May 28 '13 at 15:54