Sagher, Yoram

Using C. Fefferman's embedding of a charge space in a measure space allows us to apply standard interpolation theorems to the establishment of norm inequalities for Besicovitch almost periodic functions. This yields a significant improvement to the results of A. Avantaggiati, G. Bruno and R. Iannacci.

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.

Assume that {φn} is an orthonormal uniformly bounded (ONB) sequence of complex-valued functions de ned on a measure space (Ω,Σ,µ), and f ∈ L1(Ω,Σ,µ). Let
be the Fourier coefficients of f with respect to {φn} .
R.E.A.C. Paley proved a theorem connecting the Lp-norm of f with a related norm of the sequence {cn}. Hardy and Littlewood subsequently proved that Paley’s result is best possible within its context. Their results were generalized by Dikarev, Macaev, Askey, Wainger, Sagher, and later by Tikhonov, Li yand, Booton and others.The present work continues the generalization of these results.