Many tools in analysis and combinatorics, such as integration by parts, summation by parts, the fundamental theorem of calculus, or telescoping series, can create undesirable "boundary" terms when the integral or sum is cut off. When faced with this problem, one can sometimes solve it by averaging over all choices of the parameter that is generating this boundary, or (equivalently) by choosing the parameter randomly and using the probabilistic method.
Problem (Non-endpoint Sobolev inequality) Let , and let be such that . Establish the Sobolev inequality
for all smooth compactly supported functions , where denotes a constant that depends on .
Solution The strategy here is to obtain some pointwise bound on in terms of an integral involving other values of and , so that one can appeal to known inequalities, and specifically Young's inequality. Indeed, if we can get a pointwise bound of the form
for two kernels which are bounded in , where , then we will be done by Young's inequality and the triangle inequality.
From the fundamental theorem of calculus, we can write
for any , , and , so by the triangle inequality
This does indeed express in terms of values of and at other places, but it is not of the form (1); the measures that one is convolving by here are far too singular. But we can do better by averaging the parameters. If we first average over all directions in the unit sphere, we obtain
which upon converting from polar coordinates back to Cartesian, becomes
The second term is in the desired form (1), but the first term is still problematic (one is convolving here with a singular measure supported on a sphere of radius , rather than an function to which Young's inequality can be applied). But we can average this problem away, by integrating over all choices of from to (say). To simplify the average we will replace the constraint by the slightly looser constraint , and end up with
or, upon a further conversion from polar to Cartesian coordinates,
Thus, one can bound pointwise by the convolution of with , plus the convolution of with . One can check that one now has the desired representation (1).
This technique is closely related to that of using smoothing sums. Indeed, one can view a smoothed sum as an average of an unsmoothed sum.
Another related tactic is to use the pigeonhole principle to select a good choice of boundary term.