Bergman spaces with exponential type weights

For $1\le p<\infty $ 1 ≤ p < ∞ , let $A^{p}_{\omega }$ A ω p be the weighted Bergman space associated with an exponential type weight ω satisfying $$ \int _{{\mathbb{D}}} \bigl\vert K_{z}(\xi ) \bigr\vert \omega (\xi )^{1/2} \,dA(\xi ) \le C \omega (z)^{-1/2}, \quad z\in {\mathbb{D}}, $$ ∫ D | K z ( ξ ) | ω ( ξ ) 1 / 2 d A ( ξ ) ≤ C ω ( z ) − 1 / 2 , z ∈ D , where $K_{z}$ K z is the reproducing kernel of $A^{2}_{\omega }$ A ω 2 . This condition allows us to obtain some interesting reproducing kernel estimates and more estimates on the solutions of the ∂̅-equation (Theorem 2.5) for more general weight $\omega _{*}$ ω ∗ . As an application, we prove the boundedness of the Bergman projection on $L^{p}_{\omega }$ L ω p , identify the dual space of $A^{p}_{\omega }$ A ω p , and establish an atomic decomposition for it. Further, we give necessary and sufficient conditions for the boundedness and compactness of some operators acting from $A^{p}_{\omega }$ A ω p into $A^{q}_{\omega }$ A ω q , $1\le p,q<\infty $ 1 ≤ p , q < ∞ , such as Toeplitz and (big) Hankel operators.

On Riemann problem in weighted Smirnov classes with general weight

Weighted Smirnov classes in bounded and unbounded domains are defined in this work. Nonhomogeneous Riemann problems with a measurable coefficient whose argument is a piecewise continuous function are considered in these classes. A Muckenhoupt type condition is imposed on the weight function and the orthogonality condition is found for the solvability of nonhomogeneous problem in weighted Smirnov classes, and the formula for the index of the problem is derived. Some special cases with power type weight function are also considered,and conditions on degeneration order are found.

Polynomials and Reciprocals of Eisenstein series

Hardy and Ramanujan introduced the Circle Method to study the Fourier expansion of certain meromorphic modular forms on the upper complex half-plane. These led to asymptotic results for the partition numbers and proven and unproven formulas for the coefficients of the reciprocals of Eisenstein series [Formula: see text], especially of weight 4. Berndt et al. finally proved them all. Recently, Bringmann and Kane generalized Petersson’s approach via Poincaré series, to handle the general case. We introduce a third approach. We attach recursively defined polynomials to reciprocals of Eisenstein series. This provides easy access to the signs of the Fourier coefficients of reciprocals of Eisenstein series, sheds some light on reciprocals of [Formula: see text] of general weight, and provides some upper and lower bounds for their growth.

Nonexistence for hyperbolic problems on Riemannian manifolds1

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

A NEW STUDY ON ASYMPTOTIC OPTIMALITY OF LEAST SQUARES MODEL AVERAGING

In this article, we present a comprehensive study of asymptotic optimality of least squares model averaging methods. The concept of asymptotic optimality is that in a large-sample sense, the method results in the model averaging estimator with the smallest possible prediction loss among all such estimators. In the literature, asymptotic optimality is usually proved under specific weights restriction or using hardly interpretable assumptions. This article provides a new approach to proving asymptotic optimality, in which a general weight set is adopted, and some easily interpretable assumptions are imposed. In particular, we do not impose any assumptions on the maximum selection risk and allow a larger number of regressors than that of existing studies.

The Application of Teachings by the International Tribunal for the Law of the Sea

Scholars have examined the role of ‘teachings’ (or ‘literature’, ‘doctrine’ or ‘scholarship’) in various international courts and tribunals, but never the International Tribunal for the Law of the Sea (ITLOS). This article analyses the general weight ITLOS judges assign to teachings, how the judges distinguish between more and less significant teachings, and how and why different judges use teachings differently. ITLOS judges generally seem to assign teachings low weight, albeit with some exceptions. Some teachings are seen as more important, on the basis of their quality and on the fact that multiple writers agree. Judges treat teachings somewhat differently, with Judge Laing being a significant outlier, responsible for roughly half of all citations.