Don't do anything if it's going to be a display piece!
Only repair it if it's going to be used.
If it's old enough the body of the axe will be wrought iron and of very low carbon content indeed! So no HAZ issues. Wrought is a bit trickier to weld using modern methods but can be done and done well. You may want to experiment welding on a piece of scrap wrought iron first.
If it's more modern it may all be a carbon steel; preheat, weld, postheat---keep the blade edge cool with wet rags and don't get excessive with the temps and you will have no problems with it.
So you gonna use it or just look at it?
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Thomas
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