Well the short answer is YES. The lens will perform better on a larger sensor than on a small one. And there are scientific reasons for this. In fact the very same lens – if one could place it – on a cell phone sensor would perform worse than it does on a micro-four-thirds camera.. which would be worse than crop sensor and worse again than full frame.
In simple terms it is one of the disadvantages of smaller sensors, although not the only one. On a Nikon camera as an example, a 24MP DX sensor is more demanding of the lens than is a 24MP FX sensor. Essentially the sensor is attempting to grab that same amount of detail out of a smaller area of the glass and any defects or lens degradation will appear larger and appear more obvious in the smaller sensor. To get the same results on a smaller sensor you simply need higher quality lens glass or the image will not be as good.
Now some will initially say that a lens sharpness will soften as you get towards the, therefore by the same lens on a crop sensor body the softest areas of the lens are avoided and the image is better.
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To begin with the sharpness overall on a full frame sensor will be at a 50% higher level – this includes the centre portion as well as the edges and corners. Any lens and sensor combination will resolve to a specific number of lines per millimetre on the focal plane of the sensor. This can be clearly seen on DXOMark as one changes the camera body. An example of this would be the Nikon 200mm f/2 lens which resolves to 33 megapixels on the D810. Yet the same lens resolves to 15 megapixels on the D500. Only 15 megapixels? Yes! Take a look at the Nikon 200-400mm f/4 which resolves to megapixels on the D810 and 9 on the D500. Same lens and different sensor. Okay so the D810 is a higher resolution camera – lets try the D750. The first lens, the 200mm prime lens on the D750 resolves to 24 megapixels and the second the 200-400mm zoom resolves to 19 megapixels. Put those same lenses on the D7100 and you are at 17 megapixels and 10 megapixels. It does make a difference.
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Now the soft edges. First it is important to note that not all lenses are softer in the corners. Often wide angle lenses are – but even then some of the top lenses do an amazing job in this area. Once you get into longer lenses generally 85mm or 100mm and longer you can expect excellent sharpness pretty much right across the whole surface – at least 90% of it. Second the point of interest is generally in the centre and the edges are of little consequence if a tiny bit softer. The added sharpness of full frame will more than offset this.
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Full frame cameras are generally able to capture more light, often a full stop and some a stop and a half more. This means for the same ISO and shutter speed you will be stopped down a full stop which will make the image sharper in most cases and increase depth of field, at least until you get the impact of diffraction at the very tiny aperture sizes.
Not saying don’t use a full frame lens on a crop sensor camera. I do it all the time, sharing lenses with the D500 and full frame cameras. But just saying in actual use it will not be as sharp. Will it matter? Generally not, unless you are a critical pixel peeper. Just use it and enjoy.
Note: when one takes those same statements to very small sensor – a very small sensor as in a cell phone camera – those same areas of problems show up even more dramatically.