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I have read that these phenomena do not have an interaction.
However, I would think if CFCs cause the ozone layer to thin, and more UV radiation enters the Earth system, the Greenhouse Effect would cause global warming would increase.
So:
Will the infrared energy emitted by the Earth increase with a higher input of ultraviolet radiation?
As shown in this Chemistry.SE answer, ozone breaks upon incidence of various energies of UV radiation to form oxygen, which itself is somewhat effective at blocking UV radiation at other energies. THe key is that the incident UV radiation transfers its energy to the reaction products in order to break the bonds holding ozone (or oxygen) together.
As opposed to protecting us from UV, the only way for ozone to protect us from global warming is for it to reflect the incident UV radiation back into space. That is not what it does. UV that enters the Earth's atmosphere and interacts with the ozone stays in the Earth's atmosphere.
Therefore, your question premise is flawed when you say "if ozone layers gets smaller, more UV radiation enters on Earth." That is not true, because UV radiation that that enters the Earth will be absorbed by something, whether it is ozone or the rest of the atmosphere or the ground.
Furthermore, 'global warming' is primarily concerned with radiation heading outwards in the infrared band. IR radiation emitted from the ground is blocked by CO$_2$ in the atmosphere and prevented from escaping into space. So the effect of ozone on UV radiation will have little to no effect on 'global warming'.
Ozone absorbs nearly all incoming solar radiation at wavelengths less than .25 microns (and in so doing, protects us from skin cancer), but also absorbs in the infrared / terrestrial wavelengths. And this absorption band does not intersect the nearby absorption band for H2O (9 vs 10 to 20 microns for water vapor) or the primary wavelengths where CO2 absorbs. See http://www.astronomy.ohio-state.edu/~pogge/Ast161/Unit5/atmos.html for a discussion of the Greenhouse Effect and the atmospheric window.
This means that a depletion of the ozone would allow the earth to radiate to space around the 9 micron wavelength where the presence of water vapor and CO2 makes no difference. The present "hole" in the ozone over the Antarctic should be keeping the atmosphere cooler (especially the stratosphere) than otherwise by allowing radiation around 9 microns to escape to space.
A thinning of the ozone in the upper atmosphere will allow more UV energy to reach the surface, but will also allow more IR to escape back to space. Clear as mud, right?
