We introduce a full scale of Lorentz-BMO spaces BMO_{$L$}^{$p,q$}on the bidisk, and show that these spaces do not coincide for different values of$p$and$q$. Our main tool is a detailed analysis of Carleson's construction in [C].
The spectral order on$R$^{$n$}induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree$n$. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^{∞}of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$ -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.
Kwokwai ChanThe Chinese University of Hong KongNaichung Conan LeungThe Chinese University of Hong KongQin LiSouthern University of Science and Technology
We prove that if a complete, properly embedded, finite-topology minimal surface in S2×R contains a line, then its ends are asymptotic to helicoids, and that if the surface is an annulus, it must be a helicoid.