Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

Let Ω be a strongly Lipschitz domain of R n. Consider an elliptic second order divergence operator L (including a boundary condition on ∂Ω) and define a Hardy space by imposing the non-tangential maximal function of the extension of a function f via the Poisson semigroup for L to be in L 1. Under suitable assumptions on L, we identify this maximal Hardy space with atomic Hardy spaces, namely with H 1 (R n) if Ω = R n, H 1 r(Ω) under the Dirichlet boundary condition, and H1 z (Ω) under the Neumann boundary condition. In particular, we obtain a new proof of the atomic decomposition for H1 z (Ω). A version for local Hardy spaces is also given. We also present an overview of the theory of Hardy spaces and BMO spaces on Lipschitz domains with proofs.

The classical criterion for a circle diffeomorphism to be topologically conjugate to an irrational rigid rotation was given by A. Denjoy [1]. In [5] one of us gave a new criterion. There is an example satisfying Denjoy’s bounded variation condition rather than [5]’s Zygmund condition and vice versa. This paper will give the third criterion which is implied by either of the above criteria. 1.

Abstract. The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane–Emden type, including the following two model problems: −∆pu = u q + µ, Fk[−u] = u q + µ, u ≥ 0, on R n, or on a bounded domain Ω ⊂ R n. Here ∆p is the p-Laplacian defined by ∆pu = div (∇u|∇u | p−2), and Fk[u] is the k-Hessian defined as the sum of k × k principal minors of the Hessian matrix D 2 u (k = 1, 2,...,n); µ is a nonnegative measurable function (or measure) on Ω. The solvability of these classes of equations in the renormalized (entropy) or viscosity sense has been an open problem even for good data µ ∈ Ls (Ω), s> 1. Such results are deduced from our existence criteria with the sharp exponents s = n(q−p+1) pq for the first equation, and s = n(q−k) 2kq for the second one. Furthermore, a complete characterization of removable singularities is given. Our methods are based on systematic use of Wolff’s potentials, dyadic models, and nonlinear trace inequalities. We make use of recent advances in potential theory and PDE due to Kilpeläinen and Mal´y, Trudinger and Wang, and Labutin. This enables us to treat singular solutions, nonlocal operators, and distributed singularities, and develop the theory simultaneously for quasilinear equations and equations of Monge-Ampère type. 1.

Abstract. The problem of the boundedness of the Riesz potential Iα, 0 < α < n in local Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted Lp-spaces on the cone of non-negative nonincreasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. 1.

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Abstract: We consider the non-linear VPFP system with a coulombian repulsive interaction potential and a generic confining potential in space dimension d ≥ 3. Using spectral and kinetic methods we prove the existence and uniqueness of a mild solution with bounds uniform in time in weighted spaces, and for small total charge we find an explicit exponential rate of convergence toward the equilibrium in terms of the Witten Laplacian associated to the linear equation. Résumé: On considère le système de Vlasov-Poisson-Fokker-Planck avec un potentiel Coulombien répulsif et un potentiel confinant générique en dimension d ≥ 3. Avec des méthodes spectrales et cinétiques on prouve l’existence et l’unicité d’une solution douce dans des espaces à poids, bornée uniformément en temps, et pour petite charge totale on trouve un taux de retour exponentiel explicite vers l’équilibre en fonction du Laplacien de Witten associé à l’équation linéaire.

Abstract. Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H p-boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.

On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on R n and related estimates