ABSTRACT In this paper, we consider a Schrödinger operator with a non‐negative potential satisfying a reverse Hölder inequality and its associated critical radius function . Given a weight in a Muckenhoupt class related to , we characterize the space with Carleson measures associated to .

Abstract We prove a joint value equidistribution statement for Hecke–Maaß cusp forms on the hyperbolic three‐space . This supports the conjectural statistical independence of orthogonal cusp forms.

Abstract We consider the quasi‐geostrophic equation with its principal part for in with . We show that for every initial data with and , there exists a unique solution in the class of the maximal Lorentz–Besov regularity theorem such that for with and , where . If is sufficiently small, then we may take . Notice that both classes of initial data and solutions are scaling invariant.

Abstract Let be the unit cube in , be Banach spaces and , the associate spaces of and . In this paper, we will show that the existence in , the space of all ‐dimensional fields in , of solutions of the equation for is closely related to the problem of obtaining a Sobolev‐type embedding , where is the Sobolev space of functions vanishing on the boundary whose gradient belongs to .

Abstract 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.

Abstract 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.

Abstract We study continuity of the multiplier operator acting on Gelfand–Shilov spaces, where is a polynomial on of degree at least two with real coefficients. In the parameter quadrant for the spaces, we identify a wedge that depends on the polynomial degree for which the operator is continuous. We also show that in a large part of the complement region the operator is not continuous in dimension one. The results give information on well‐posedness for linear evolution equations that generalize the Schrödinger equation for the free particle.

Abstract We consider an even‐order ordinary differential operator with a spectral parameter and integral conditions. An a priori estimate of the solutions to the problem is obtained for sufficiently large values of the spectral parameter. The discreteness and the sectoral structure of the spectrum of the corresponding operators are proved.

Abstract Motivated by the ‐operational equation for Rogers–Szegö polynomials [Sci. China Math. 66 (2023), no. 6, 1199–1216], it is natural to ask whether some general ‐polynomials exist, which are solutions of certain ‐operational equations, ‐difference equations, and ‐partial differential equations. In this paper, based on the importance of little ‐Jacobi polynomials, we define two homogeneous little ‐Jacobi polynomials and search their corresponding ‐operational equations, ‐difference equations, and ‐partial differential equations by the technique of noncommutative ‐binomial theorem and recurrence relations. In addition, we deduce some generating functions for homogeneous little ‐Jacobi polynomials by methods of ‐operational equation, ‐difference equation, and ‐partial differential equation. Moreover, we consider recurrence relations for homogeneous little ‐Jacobi polynomials.

Abstract We study lattice summing operators between Banach spaces focusing on two classes, ‐summing and strongly ‐summing operators, which are generated by Orlicz sequence lattices . For the class of strongly ‐summing operators, we prove the domination theorem, which complements Pietsch's fundamental domination theorem for ‐summing operators. Based on this result, we show that strongly ‐summing operators are Dunford–Pettis. As a consequence, we show that these classes are, in general, distinct. We also demonstrate that the class of strongly ‐summing operators between Hilbert spaces coincides with the Hilbert–Schmidt class when is a separable Orlicz space. Finally, we consider generalized nuclear operators, and using a factorization description, we prove that ‐nuclear operators are ‐summing when is separable.