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We show how the Fourier transform for distributional sections of vector bundles over symmetric spaces of non-compact type $��G�/�K�$ can be used for questions of solvability of systems of invariant differential equations in analogy to Hörmander's proof of the Ehrenpreis–Malgrange theorem. We get complete solvability for the hyperbolic plane $�H^2$ and partial results for products $��H^2�\times �\cdots �\times �H^2�$ and the hyperbolic 3-space $�H^3$.

This paper studies a diffusive competition model with degeneracy and Holling-II functional response in spatially heterogeneous environment. First, we discuss the structures and stability of steady-state bifurcation solutions. Then, the existence, nonexistence, and multiplicity of steady-state solutions are established. We conclude that there exist two critical values induced by the spatial degeneracy and the functional response, respectively, such that when the growth rate of one of the competition species is between these two critical values, the model behaves drastically and some qualitative changes occur, which is in sharp contrast to the well-studied classical models. In addition, it is found that the boundary condition also has important effects on the critical value. These show that not only degeneracy but also the combination of functional response and boundary condition have important influences on the model, especially on the structures of bifurcations and the existence of steady-state solutions. Finally, the asymptotic behavior and global attractor of positive solutions for the parabolic system are investigated, which enrich the study of dynamical behavior for the model.

The authors introduce generalized Campanato space with regularized condition over non-homogeneous space, and study its basic properties including the John–Nirenberg inequality and equivalent characterizations. As applications, the boundedness of fractional type Marcinkiewicz integral and its commutator on generalized Morrey space over non-homogeneous space is obtained.

A new family of Volterra-type operators $��V_{�\alpha �,�\beta �}^{\phi }��(�·�)��$ based on bona fide fractional calculus is introduced in [12] by constructing analytic paraproducts acting on $��H�\left(�D�\right)�$ and their boundedness between Hardy spaces is characterized for certain parameter ranges there. This paper is a natural companion to [12] in the sense that it characterizes those $�\phi$’s such that $�V_{�\alpha �,�\beta �}^{\phi }$ is bounded from weighted Bergman spaces $��L_a^p��\left(�d�A_{\gamma }�\right)��$ to Hardy spaces $�H^q$ for the range

In this paper, we study the Dirichlet series that enumerates proper equivalence classes of full-rank sublattices of a given quadratic lattice in a hyperbolic plane—that is, a nondegenerate isotropic quadratic space of dimension 2. We derive explicit formulas for the associated zeta functions and obtain a combinatorial way to compute them. Their analytic properties lead to the intriguing consequence that a large proportion of proper classes are one-lattice classes.

We study an indefinite weighted eigenvalue problem for an operator of mixed-type (that includes both the classical $�p$-Laplacian and the fractional $�p$-Laplacian) in a bounded open subset $��\Omega �\subset �R^N���(�N�\ge �2�)��$ with Lipschitz boundary $��\partial �\Omega �$, which is given by