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?

Jerry