It is well-known that in Banach spaces with finite cotype, the $R$ -bounded and $\gamma $ -bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$ -boundedness implies $\gamma $ -boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$ -boundedness is stable under taking adjoints if and only if the underlying space is $K$ -convex.

We establish local higher integrability estimates for upper gradients of vector-valued parabolic quasi-minimizers in metric measure spaces, satisfying a doubling property and supporting a weak Poincaré inequality.

We show that positive$α$-stable densities are hyperbolically completely monotone if and only if$α$≤1/2. This gives a positive answer to a question raised by L. Bondesson in 1977.

This paper studies stability of the Ekman boundary layer. We utilize a new approach, developed by the authors in a precedent paper, based on Fourier transformed finite vector Radon measures, which yields exponential stability of the Ekman spiral. By this method we can also derive very explicit bounds for solutions of the linearized and the nonlinear Ekman systems. For example, we can prove these bounds to be uniform with respect to the angular velocity of rotation, which has proved to be relevant for several aspects. Another advantage of this approach is that we obtain well-posedness in classes containing nondecaying vector fields such as almost periodic functions. These outcomes give respect to the nature of boundary layer problems and cannot be obtained by approaches in standard function spaces such as Lebesgue, Bessel-potential, Hölder or Besov spaces.

In the present paper, we prove an improved Combes–Thomas estimate, viz. the Combes–Thomas estimate in trace-class norms, for magnetic Schrödinger operators under general assumptions. In particular, we allow for unbounded potentials. We also show that for any function in the Schwartz space on the reals the operator kernel decays, in trace-class norms, faster than any polynomial.