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We investigate here the nonlinear elliptic equations − Δ u = | x | α e u and − Δ u = | x | α | u | p − 1 u with α > − 2 , p > 1 and N ⩾ 2 . In particular, we prove some Liouville type theorems for weak solutions with finite Morse index in the low dimensional Euclidean spaces or half spaces.

International audience; We prove bounded stability for strongly coupled critical elliptic systems in the inhomogeneous context of a compact Riemannian manifold when the potential of the operator is less, in the sense of bilinear forms, than the geometric threshold potential of the conformal Laplacian.

14 pages; Journal of Functional Analysis; International audience; We consider operators $H_\mu$ of convolution with measures $\mu$ on locally compact groups. We characterize the spectrum of $H_\mu$ by constructing auxiliary operators whose kernel contain the pure point and singular subspaces of $H_\mu$, respectively. The proofs rely on commutator methods.

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let $\M$ be a von Neumann algebra equipped with a normal faithful semifinite trace $\t$, and let $E$ be an r.i. space on $(0, \8)$. Let $E(\M)$ be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from $E(\M)$ into a Hilbert space $\mathcal H$ corresponds a positive norm one functional $f\in E_{(2)}(\M)^*$ such that $$\forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*),$$ where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for $E(\M)$ when $E$ is either 2-concave or 2-convex and $q$-concave for some $q, Comment: 14 pages. To appear in J. Funct. Anal

We extend the Trotter–Kato–Chernoff theory of strong approximation of C 0 semigroups on Banach spaces to operator-norm approximation of analytic semigroups with error estimate. As application we obtain a criterion for the operator-norm convergence of the Trotter product formula on Banach spaces with error estimate n −1 log n , provided one of the generators has a bounded H ∞ functional calculus. For both results, we present versions for approximation in operator-ideal-norms such as the trace norm or the Hilbert–Schmidt norm. Finally, we give some remarks on the operator-norm convergence of the Trotter product formula for semigroups acting on a scale of L p -spaces.