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Following the recent work [13] fulfilled in the discrete case, we pro- videin this paper new intertwining relations for semigroups of one-dimensionaldiffusions. Various applications of these results are investigated, among themthe famous variational formula of the spectral gap derived by Chen and Wang[15] together with a new criterion ensuring that the logarithmic Sobolevinequality holds. We complete this work by revisiting some classical examples,for which new estimates on the optimal constants are derived.

Intertwining relations for one-dimensional diffusions and application to functional inequalities