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A new and fairly elementary proof is given of the result by B. Simon [S], that the potential in a Sturm-Liouville operator is determined by the asymptotics of the associated$m$-function near −∞. The proof given is based on relations between the classical transformation operators and the$m$-function.

The study of Hamiltonian graphs began with Dirac’s classic result in 1952. This was followed by that of Ore in 1960. In 1984 Fan generalized both these results with the following result: If$G$is a 2-connected graph of order$n$and max{$d$($u$),$d$($v$)}≥$n$/2 for each pair of vertices$u$and$v$with distance$d$($u$,$v$)=2, then$G$is Hamiltonian. In 1991 Faudree–Gould–Jacobson–Lesnick proved that if$G$is a 2-connected graph and |$N$($u$)∪$N$($v$)|+δ($G$)≥$n$for each pair of nonadjacent vertices$u$,$v$∈$V$($G$), then$G$is Hamiltonian. This paper generalizes the above results when$G$is 3-connected. We show that if$G$is a 3-connected graph of order$n$and max{|$N$($x$)∪$N$($y$)|+$d$($u$),|$N$($w$)∪$N$($z$)|+$d$($v$)}≥$n$for every choice of vertices$x$,$y$,$u$,$w$,$z$,$v$such that$d$($x$,$y$)=$d$($y$,$u$)=$d$($w$,$z$)=$d$($z$,$v$)=$d$($u$,$v$)=2 and where$x$,$y$and$u$are three distinct vertices and$w$,$z$and$v$are also three distinct vertices (and possibly |{$x$,$y$}∩{$w$,$z$}| is 1 or 2), then$G$is Hamiltonian.

As a tool for solving the Neumann problem for divergence-form equations, Kenig and Pipher introduced the space ${\mathcal{X}}$ of functions on the half-space, such that the non-tangential maximal function of their$L$_{2}Whitney averages belongs to$L$_{2}on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of ${\mathcal{X}}$ , and characterize the pointwise multipliers from ${\mathcal{X}}$ to$L$_{2}on the half-space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to$L$_{$p$}generalizations of the space ${\mathcal{X}}$ . Our results elaborate on the well-known duality between Carleson measures and non-tangential maximal functions.

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