Does price cap in emergency situation (COVID-19) reduce the availability of a good? Does unregulated market increase it instead? I was discussing with one of my friend about Italy price cap on surgical mask to control speculation. He says price cap does not allow industries to invest more to increase the production that would then naturally lower the price in accordance with the "demand and offer rule". I say that, because most of the speculation happens at the retail level (pharmacies and shops) surgical masks factories are not affected by price caps and so the quantity of production is not affected.
Am I right? Can some data or real life example backup this?
Capping prices at a low level permanently would presumably lower supply in the long term, but that is not germane to this question.

I will attack a slightly narrower version of this question: does capping prices in an emergency lower investment?

I would argue that the definition of “emergency” means that it is short term, and activity is disrupted. It is extremely difficult to launch a fixed investment project during an emergency. Fixed investments take time to finish, and the emergency is likely past before completion.

What we are seeing in 2020 is existing capacity being re-purposed to new uses. E.g., distillers are switching from producing whiskey to producing hand sanitiser. I doubt a survey has been completed, but anecdotally, many manufacturers are selling at a slim or no profit. (However, they benefit from switching from non-essential to essential production, allowing them to keep workforces employed. This is a benefit, even if profit margins are lower than their usual output.) 

Most emergencies are geographically limited; e.g., hurricane. It typically makes no sense to switch production like is happening in 2020, capacity in other regions can be accessed more quickly.

The question then shifts: does a lack of controls create an incentive to have extra capacity/inventory during normal times to take advantage of an emergency?

This is perhaps plausible, but a good portion of manufacturers attempt to reach a global market. The existence or non-existence of price caps in a few countries would not radically shift their investment decisions.

Keep in mind that there is an aversion to price gouging during emergencies. Very few companies want to be publicly associated with the practice. As such, most news stories about price gouging involve largely invisible intermediaries that do no have a public “brand”. (There is an academic literature on price gouging, but I am not familiar with it. However, the reality that there is a public aversion to price gouging is non-controversial.)