Bornitude et continuité de la transformation de Lévy en analyse

our previous papers (Adv. in Math. 138 (1) (1998) 182; Potential Anal. 12 (2000) 419), we have obtained a decomposition of | f |, where f is a function defined on Rn, that is analogous to the one proved by H. Tanaka for martingales (the so-called “Tanaka formula”). More precisely, the decomposition has the form | f |=f̃+D∗0( f ), where D∗0( f ) is (a variant of ) the density of the area integral associated with f. This functional (introduced by R.F. Gundy in his 1983 paper (The density of area integral, Conference on Harmonic Analysis in Honor of Antoni Zygmund. Wadsworth, Belmont, CA, 1983, pp. 138–149.)) can be viewed as the counterpart of the local time in Euclidean harmonic analysis. In this paper, we are interested in boundedness and continuity properties of the mapping f↦f̃ (which we call the Lévy transform in analysis) on some classical function or distribution spaces. As was shown in [4,5], the above (non-linear) decomposition is bounded in Lp for every p∈[1,+∞[, i.e. one has || f̃ ||p⩽Cp|| f||p, where Cp is a constant depending only on p. Nevertheless our methods (roughly speaking, the Calderón–Zygmund theory in [4], stochastic calculus and martingale inequalities in [5]) both gave constants Cp whose order of magnitude near 1 is O(1/(p−1)). The aim of this paper is two-fold: first, we improve the preceding result and we answer a natural question, by proving that the best constants Cp are bounded near 1. Second, we prove that the Lévy transform f↦f̃ is continuous on the Hardy spaces Hp with p>n/(n+1).