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While there are many open problems in Real Analysis like Khabibullin's conjecture or Lehmer's conjecture, those are big enough to take an expert's life for several years, let alone some graduate student that may haven't had any research done yet.

Are there any problems in Real Analysis which has not been solved yet or otherwise have still some rooms to generalize, on the other hand easy enough to consider it as a project for a research-degree student to finish it in a few years?

Suggestions on this would be very much appreciated!

PS - Migrated from math.SE

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    $\begingroup$ Picking a research problem from the list of answers you're going to get seems kind of risky. How many other graduate students are going to look at this list and pick the same problem you did? $\endgroup$
    – bof
    Commented Jun 14, 2015 at 11:39
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    $\begingroup$ Yes. Many people get PhDs in analysis, so looking at their theses will give you examples. $\endgroup$
    – Kimball
    Commented Jun 14, 2015 at 12:45
  • $\begingroup$ @bof In addition, most people won't want to give away their best ideas for problems to grad students who aren't theirs. $\endgroup$
    – Kimball
    Commented Jun 14, 2015 at 12:54

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Make Weyl-van der Corput estimates explicit:

Let $f:[a,b]\rightarrow\mathbb{R}$ be a somewhat smooth function, and assume that you have some bounds on certain derivatives of $f$. Then give an upper bound for $\sum_{n=N}^{2N} e^{2\pi i f(n)}$.

There are deep qualitative bounds, however, not much work has been done on the constants involved in these bounds. While it is trivial to get some constants, getting good constants can become quite demanding.

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