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We prove a rigidity theorem in Poisson geometry around compact Poisson submanifolds, using the Nash–Moser fast convergence method. In the case of one-point submanifolds (fixed points), this implies a stronger version of Conn’s linearization theorem [2], also proving that Conn’s theorem is a manifestation of a rigidity phenomenon; similarly, in the case of arbitrary symplectic leaves, it gives a stronger version of the local normal form theorem [7]. We can also use the rigidity theorem to compute the Poisson moduli space of the sphere in the dual of a compact semisimple Lie algebra [17].

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A sufficient condition for the Wiener regularity of a boundary point with respect to the operator (− Δ)^{μ}in$R$^{$n$},$n$≥1, is obtained, for μ∈(0,1/2$n$)/(1,1/2$n$−1). This extends some results for the polyharmonic operator obtained by Maz'ya and Maz'ya-Donchev.

It is proved in Benamara-Nikolski [1] that if the spectrum σ($T$) of a contraction$T$with finite defects (rank($I−T$^{*}$T$)=rank ($I−TT$^{*})<∞) does not coincide with $$\bar D$$ , then the contraction is similar to a normal operator if and only if $$C_1 (T) = \mathop {\sup }\limits_{\lambda \in C\backslash \sigma (T)} \parallel (T - \lambda )^{ - 1} \parallel dist(\lambda ,\sigma (T))< \infty .$$