During East Zhou in southern China there seems to be more use of axes in warfare, and in Yunnan & Dian cultures there are quite elaborate forms with decoration and incast design.
Earlier axes from Shang seem more associated with the executioner only...for lopping off heads & sacrifice.
During Han there are effective long handled axes depicted as carried by both cavalry and infantry. This isi one weapon that does not seem common though. I havent seen many examples of 'central plains' battle axes from the Zhou & Han period beyond those in art, and a few finds that could even be mundane axes for use outside warfare (in this way simple axes of bronze and iron are common).
The weapon association for axes I have heard commented on was linked proportionately to finds and suggests groups viewed as non-Chinese favoured axes more.
I am meaning to post on CHF an example of a Yunnan style bronze battle axe I aquired recently, spring & autumn period. I have conducted some home tests and am fairly sure it is authentic.
To illustrate the basic form of these 'Yunnan' axes it will suffice either way. It is the same as examples in Y. Hongs text, even down to the art depicting mountains and clouds on the blade.
Note' some people call the dagger-axe (ge) a 'battle axe' too, but that is only an English term and not accurate.
Basically as a true ''Chinese'' weapon, at least up untill Han which is my only real area of study, the axe is not widespread..although it was used.
Edit; here is the Yunnan battle axe. I have better pictures on another PC and will elaborate on it, and issues of authenticity, on the 'archaeology' forum later. (click to see full sized image)

People with a copy of Yang Hong can compare it to the section on weapons of minority cultres. The single loop and in-cast art shows it to be the same weapon style.
The picture is not the best, and shot under artificial light. From a jpg. one commentator I sent this too expressed doubts, another thinks it looks good. My own inspection over several days, under magnification and with chemical strippers and heat tests show no sign of fakery. The only reason to be suspicious is my normal suspicious nature. If it is fake it is good, and still shows the appearnace of the axes I was meaning.
I am 70% in favour of it being a Spring & Autumn piece. I hope to get some metallurgical testing done in future. It lacks traces of wood or other such patina features as I normally select.
This is the design on it, again make comparisons to Y. Hong.
The pattern is mountain, cloud, land & sea below. I like this design very much and will recreate it on paper in a spare moment. I hope to get a tattoo done of the design to sit below my Sun Wu Kung tattoo. Mountain of fruit and flowers and all that...