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We consider smooth area-preserving flows (also known as locally Hamiltonianflows) on surfaces of genus $g\geq 1$ and study ergodic integrals of smoothobservables along the flow trajectories. We show that these integrals display a\emph{power deviation spectrum} and describe the cocycles that lead the purepower behaviour, giving a new proof of results by Forni (Annals 2002) andBufetov (Annals 2014) and generalizing them to observables which are non-zeroat fixed points. This in particular completes the proof of the originalformulation of the Kontsevitch-Zorich conjecture. Our proof is based onbuilding suitable \emph{correction operators} for cocycles with logarithmicsingularities over a full measure set of interval exchange transformations(IETs), in the spirit of Marmi-Moussa-Yoccoz work on piecewise smooth cocyclesover IETs. In the case of symmetric singularities, exploiting former work ofthe second author (Annals 2011), we prove a tightness result for a finitecodimension class of observables. We then apply the latter result to prove theexistence of ergodic infinite extensions for a full measure set of locallyHamiltonian flows with non-degenerate saddles in any genus $g\geq 2$.

On the asymptotic growth of Birkhoff integrals for locally Hamiltonian flows and ergodicity of their extensions