Negligible Variation, Change of Variables, and a Smooth Analog of the Hobby-Rice Theorem

This dissertation concerns two topics in analysis. The rst section is an exposition
of the Henstock-Kurzweil integral leading to a necessary and su cient condition
for the change of variables formula to hold, with implications for the change
of variables formula for the Lebesgue integral. As a corollary, a necessary and suf-
cient condition for the Fundamental Theorem of Calculus to hold for the HK integral
is obtained. The second section concerns a challenge raised in a paper by O.
Lazarev and E. H. Lieb, where they proved that, given f1….,fn ∈ L1 ([0,1] ; C),
there exists a smooth function φ that takes values on the unit circle and annihilates
span {f1...., fn}. We give an alternative proof of that fact that also shows the W1,1
norm of φ can be bounded by 5πn + 1. Answering a question raised by Lazarev and
Lieb, we show that if p > 1 then there is no bound for the W1,p norm of any such
multiplier in terms of the norms of f1...., fn.