This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on [0, L] 2 , and a force g ∈ D(A α−12 ), we show there is a fixed strip about the real time axis on which a uniform bound A α u < mανκ α 0 holds for each α ∈ N. Here ν is viscosity, κ 0 = 2π L , and mα is explicitly given in terms of g and α. We show that if any element in A is in D(A α ), then all of A is in D(A α ), and likewise with D(A α ) replaced by C ∞ (Ω). We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class.

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Inner functions are an important and popular object of study in the field of complex function theory. We look at meromorphic inner functions with a given spectrum and provide sufficient conditions for them to have uniformly bounded derivative on the real line. This question was first studied by Louis de Branges in 1968 and was later revived by Anton Baranov in 2011.

Abstract. An asymptotically orthonormal sequence is a sequence which is "nearly" orthonormal in the sense that it satis es the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized reproducing kernels for model spaces and de Branges-Rovnyak spaces.

Meromorphic Inner Functions (MIFs) on the upper half plane play an important role in applications to spectral problems for differential operators. In this paper, we survey some recent results concerning function theoretic properties of MIFs and show their connections with spectral problems for the Schrödinger operator.