Coorbit space theory is an abstract approach to function spaces and their atomic decompositions. The original theory developed by Feichtinger and Gröchenig in the late 1980ies heavily uses integrable representations of locally compact groups. Their theory covers, in particular, homogeneous Besov-Lizorkin-Triebel spaces, modulation spaces, Bergman spaces and the recent shearlet spaces. However, ...

Mourre’s commutator method is used together with Besov spaces, in this paper, to settle the question of uniqueness of solutions to the scattering problem for the acoustic wave propagation in perturbed stratified fluids in R with n > 2. The uniqueness of solutions is established by introducing a radiation condition in the framework of Besov spaces, and is then used to prove resolvent estimates i...

We prove that if an initial datum to the incompressible Navier-Stokes equations in any critical Besov space Ḃ −1+ 3 p p,q (R ), with 3 < p, q < ∞, gives rise to a strong solution with a singularity at a finite time T > 0, then the norm of the solution in that Besov space becomes unbounded at time T . This result, which treats all critical Besov spaces where local existence is known, generalizes...

We extend to the vector-valued situation some earlier work of Ciesielski and Roynette on the Besov regularity of the paths of the classical Brownian motion. We also consider a Brownian motion as a Besov space valued random variable. It turns out that a Brownian motion, in this interpretation, is a Gaussian random variable with some pathological properties. We prove estimates for the first momen...

In this paper we propose a function space approach to Representation Learning [3] and the analysis of the representation layers in deep learning architectures. We show how to compute a ‘weak-type’ Besov smoothness index that quantifies the geometry of the clustering in the feature space. This approach was already applied successfully to improve the performance of machine learning algorithms suc...

We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations in mixed time-space Besov spaces.

Abstract. We address the function space theory associated with the Schrödinger operatorH = −d2/dx2+V . The discussion is featured with potential V (x) = −λ(λ−1) sechx, which is called in quantum physics the Pöschl-Teller potential. Using biorthogonal dyadic system, we introduce Besov spaces and Triebel-Lizorkin spaces (including Sobolev spaces) associated with H . We then use interpolation meth...

In this note we prove that the finite time blow-up of classical solutions of the 3-D homogeneous incompressible Euler equations is controlled by the Besov space, Ḃ0 ∞,1, norm of the two components of the vorticity. For the axisymmetric flows with swirl we deduce that the blow-up of solution is controlled by the same Besov space norm of the angular component of the vorticity. For the proof of th...

We investigate the asymptotic behavior of the entropy numbers of the compact embedding B1 p1,q1(R d , α) ↪→ B2 p2,q2(R ). Here Bs p,q (R d , α) denotes a weighted Besov space, where the weight is given by wα(x) = (1 + |x |2)α/2, and B2 p2,q2 (Rd ) denotes the unweighted Besov space, respectively. We shall concentrate on the so-called limiting situation given by the following constellation of pa...

In this paper, we are concerned with the global wellposedness of 3-D inhomogeneous incompressible Navier-Stokes equations (1.2) in the critical Besov spaces with the norm of which are invariant by the scaling of the equations and under a nonlinear smallness condition on the isentropic critical Besov norm to the fluctuation of the initial density and the critical anisotropic Besov norm of the ho...