I am not in mechanical engineering field. However, fluid mechanics concepts could be applied for explaining the topic. Therefore, if I am wrong, please do not laugh at me.
For an airplane to take off either from land or from HM's deck, the important thing is to generate more uplifting force than the gravitational force so the plane can still move upwards at the time it leaves the runway. Once in air, the plane can increase the speed further in the air. The Bernolli Equation can be used here to determine the velocity pressure (Hv) that is applied to the bottom surface area of the wings and plane body (A). When you ride airplanes, you could see the airplane wings are extended out to increase this area during the taking-off and landing.
The total uplifting force is the product of these two (Hv x A). Hv is determined by CxV^2/(2g) in which C is a coefficient to consider various known and unknown factors, V is the plane speed at the time it leaves the runway, and g is gravitational acceleration. C has something to do with the attack angle (or lifting angle of the HM's deck) and the wing angle control.
Therefore, in order for the airplane with a mass of m to take off (i.e., to have a positive acceleration rate in the vertical direction a), a simple form of equation becomes
a = CAV^2/(2gm) - g
Based on this equation, the variables that can play to make a taking-off successful are C, A, V and m. In increase the first part of the equation on the right, one can increase A and V, particularly V due to its being proportional to the power of 2, and decrease m.
For the transport planes or their variances such as AWCS, A is normally much larger than that of fighters and can expand easily during taking-off and landing or some other required occasions. That is why the slow impeller airplanes were used in early HM's since some of them even have double-deck wings to have a large A. To increase V, the engine should be strong or to have a long runway or both.
Normally, it is undesirable to reduce m. However, as the previous posts mentioned, the J15 is actually heavier than the AWCS, I can not see why the AWCS can not be launched from Liaoning in theory. It is true that its size is much larger than J15, it makes more difficult to control within the limited space on HM. But it is a training issue.