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Let Ω be an open subset of$R$^{$d$},$d$≥2, and let$x$∈Ω. A$Jensen measure$for$x$on Ω is a Borel probability measure μ, supported on a compact subset of Ω, such that ∫$u$$d$μ≤$u$($x$) for every superharmonic function$u$on Ω. Denote by$J$_{$x$}(Ω) the family of Jensen measures for$x$on Ω. We present two characterizations of ext($J$_{$x$}(Ω)), the set of extreme elements of$J$_{$x$}(Ω). The first is in terms of finely harmonic measures, and the second as limits of harmonic measures on decreasing sequences of domains.

We prove the following extension of the Wiener–Wintner theorem and the Carleson theorem on pointwise convergence of Fourier series: For all measure-preserving flows ($X$,μ,$T$_{$t$}) and$f$∈$L$^{$p$}($X$,μ), there is a set$X$_{$f$}⊂$X$of probability one, so that for all$x$∈$X$_{$f$}, $$\lim_{s\downarrow0}\int_{s<|t|<1/s}e^{i\theta t} f(\textup{T}_tx)\,\frac{dt}t\quad\text{exists for all}\ \theta.$$ The proof is by way of establishing an appropriate oscillation inequality which is itself an extension of Carleson’s theorem.

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We define a partition of Z into intervals {$I$_{j}} and prove the Littlewood-Paley inequality ‖$f$‖_{$p$}≦$C$_{p}‖$Sf$‖_{$p$}, 2≦$p$<∞. Here$f$is a function on [o, 2π) and $$Sf = (\sum |\Delta _j |^2 )^{1/2} , \hat \Delta j = \hat f\chi _{Ij} $$ . This is a new example of a partition having the Littlewood-Paley property since the {$I$_{j}} are not of the type obtained by iterating lacunary partitions finitely many times.