I strongly believe that there are many people - myself included - that will happily pay for applications to increase the functionality or entertainment value of their smart phones for no other reason than it being fair. Not "everybody", but many will.
I do agree that SOME people genuinely will pay for software even when they know how to pirate it. There are honest people who realise the need to pay the people who write the software, and to encourage the production of more software by the same people.
In fact if you'll forgive a bit of nostalgia:
Back in the glory days of shareware I used to send a fiver now and then to authors of my favourite games, even when the official free version of a game was also the full one (that system was called "donationware" I think, and it relied entirely on good will as the purchaser got absolutely nothing in return except perhaps a thank you letter).
The software authors were often touchingly honest: one shareware author in America (I think he did "Shooting Gallery" and "EGATrek"? Nils something) said he received lots of donations in foreign currency which he could not cash, but he still sent out the registered versions of the games in return, purely because he could see these people had given up some money for his sake. It didn't make business sense, but it showed what a nice guy he was.
However, this was all in the days before the internet and mass storage, before people could download a stupidly huge amount of free pirated content onto their computers unbelievably easily and at no significant cost. Good will is (I'm afraid) likely to diminish as the culture of filling your PC with torrented goodies spreads. Good apps and games are no longer seen as individual gems but just tiny fish in a huge trawler net of tens of thousands.
I know some people who regard paying for content as stupid if they can get it pirated, and don't even consider sending any authors any kind of support, no matter how small the company or how low the price. If games cost 1 euro each and had no DRM, these particular people would carry on pirating them simply because that's somehow the "cooler" thing to do. A few go beyond that and actively spread the pirated content because that's cool too.
Some people are making out piracy to be some kind of romantic stand against the system, but that's not why most people do it. Most people do it because it costs nothing.
So how does Apple sell applications worth $1 million per day from the App Store? They are DRMed but seemingly nobody cares.
...and the Wii Virtual Console, and the Xbox Live Arcade, and the PS3 Store etc. And indeed almost all console games, PC games and DVDs too. DRM is not really the issue here, people are okay with DRM if it's implemented with the average consumer's needs in mind (which it generally is on the above systems).
As Rafe says, if DRM doesn't get in the way then people won't care about it one way or the other. For example, DRM on a DVD generally doesn't get in the way because most people only play films on their television sets, and a DVD is equally suited to that task whether it has DRM or not.
If you can make commercial content much easier to obtain than pirated content, that's probably the most effective way to fight piracy.
However, piracy is becoming increasingly easy (imagine if YouTube did games and audio too), so that approach to tackling piracy is going to get more and more difficult.