I have a conjecture that's motivated by the following example.
Let be a positive, function that is supported in the interval , normalized so that , and satisfies .
(When I write " " I always mean the Lebegsue integral over the entire real line.)
The functions defined by
are sported in the intervals and their integrals satisfy
Finally consider the function defined by
Since the summands are supported on pairwise disjoint intervals, the integral satisfies
which converges. By construction for all , so that .
Thus is an example of a non-negative, integrable function (i.e., an -function) which does not vanish at infinity. Its derivative, however, is not so well behaved: by the chain rule and substitution,
and the pairwise disjointness of the supports implies that
Conjecture. If both and are integrable on the real line, then . (Let's assume that is at least .)
I've stated this for functions, but I'd be equally happy to see a proof for the class. There one might be able to apply Cauchy-Schwartz and integration by parts in some slick way. Any ideas?