ABSTRACT This paper completes the classification of seven‐dimensional nilpotent Lie groups endowed with a left‐invariant purely coclosed ‐structure, initiated in Bazzoni et al. [ Mathematische Nachrichten 296 no. 6 (2023): 2236–2257], the authors provided the classification of decomposable seven‐dimensional nilpotent Lie groups and of the indecomposable ones up to step 4 of nilpotency. Here, we address the case of indecomposable 5‐ and 6‐step nilpotent Lie groups
ABSTRACT Let be equipped with the left Haar measure . We study maximal averages associated with three basic motions on : horizontal translations, vertical dilations, and fixed hyperbolic geodesics in the upper half‐space model. The translation maximal operator is the Euclidean Hardy–Littlewood maximal operator on each horizontal slice. The Haar‐compatible dilation maximal operator is of weak type and bounded on for , but it is not strongly bounded on . By contrast, the unweighted Lebesgue dilation average is unbounded on every finite and is not of weak type . For fixed hyperbolic geodesic averages, the large‐time part is strongly bounded on because of modular exponential decay. The small‐time part is a finite‐type parabolic maximal problem. Using the corresponding local finite‐type endpoint estimate for the geodesic slice, we prove in weak Orlicz form, together with the strong bounds for . We also show that the strong endpoint fails. Finally, we record a discrete random‐walk maximal inequality whose sufficient condition is expressed through the modular drift , where is the probability measure defining the right random walk.
ABSTRACT We consider a Dirac system on with , , and study the asymptotic behavior of its fundamental solutions as the spectral parameter tends to infinity in the half‐plane , where . We obtain detailed asymptotic formulas and, as an application, derive new half‐plane asymptotics for fundamental systems of solutions to Sturm–Liouville equations with singular potentials.
ABSTRACT In this paper, we study properties of the infinitesimal generators of ‐semigroups of composition operators on Hardy spaces of Dirichlet series. We also characterize the ‐semigroups of weighted composition operators on by using abelian intertwiners of multiplication operators. Moreover, we establish a necessary and sufficient condition for embedding a weighted composition operator into a ‐semigroup on .
ABSTRACT The goal of this work is to generalize the intersection product formulas for the Fulton–Johnson, Schwartz–MacPherson, and Milnor classes, originally obtained in the work of Callejas‐Bedregal, Morgado, and Seade, to a broader framework of characteristic classes. More precisely, we extend these formulas to the classes introduced by Schürmann, which are characteristic classes relative to constructible functions.
ABSTRACT In this paper, we study a class of elliptic problems in the presence of a nonlocal term and a parameter. A careful analysis of the influence of the referred nonlocal term and the parameter on the existence and nonexistence of solution was carried out, where we considered different scenarios. Among our main contributions are: a identity for a class of elliptic integro‐differential problems (which produces some results of nonexistence of solutions) and an extensive study of the critical nonlocal problem, which includes the study of the double critical case. Some of the main tools used in the research were global and local minimization, mountain pass theorem, concentration compactness principle, sub‐supersolution method, and a Carl–Heikkilä fixed point theorem.
ABSTRACT In this paper, the finite spectrum problem of Dirac operators with an eigenparameter contained in the boundary conditions is studied. First, the existence of finite spectrum under certain conditions is proved, that is, for each nonnegative integer , we construct a class of regular Dirac operators, each of which has at most eigenvalues. Second, we identify a class of Dirac equations such that every Dirac operator formed by such an equation together with eigenparameter‐dependent boundary conditions can be reformulated as an equivalent finite‐dimensional matrix eigenvalue problem. Third, for any matrix eigenvalue problem of a prescribed type paired with eigenparameter‐dependent boundary conditions, we construct a family of Dirac operators equipped with the specified boundary conditions, each of which is equivalent to the given matrix eigenvalue problem. Here, equivalence signifies that the two problems share identical sets of eigenvalues.
ABSTRACT In this paper, we establish lower bounds for the Cheeger isoperimetric constant and the first Dirichlet eigenvalue of bounded domains in a complete minimal hypersurface immersed in the unit sphere. These bounds are expressed in terms of the squared norm of the second fundamental form of .
ABSTRACT This paper presents a comprehensive study of hyperbolic Ricci solitons on sequential doubly warped product manifolds (SDWPM). We establish a complete geometric framework for SDWPM, deriving explicit formulas for the Levi‐Civita connection, Riemann curvature tensor, Ricci tensor, and scalar curvature. Our main results include necessary and sufficient conditions for the existence of both gradient and non‐gradient hyperbolic Ricci solitons on these manifolds, complete classification theorems, and rigidity results for compact cases. We prove several novel existence theorems and provide explicit constructions of hyperbolic Ricci solitons on various SDWPM configurations. The relationship between hyperbolic Ricci solitons and Einstein metrics on SDWPM is thoroughly investigated, revealing deep connections between soliton structures and the underlying Riemannian geometry. Applications to cosmological models and geometric evolution equations demonstrate the physical relevance of our theoretical framework. The work significantly extends previous results on warped product manifolds and provides powerful new tools for studying geometric flows in complex geometric settings.
ABSTRACT In this paper, we first characterize the polar decomposition of unbounded weighted composition operator pairs in an ‐space. Based on this characterization, we introduce the ‐spherical mean transform for . We then investigate the dense definiteness of . As an application, we provide an example of a ‐hyponormal operator whose Aluthge transform is densely defined, while its ‐mean transform has a trivial domain. Furthermore, we establish the relationship between the dense definiteness of and , based on the notion of powers for operator pairs in the sense of Müller and Soltysiak. We also give a characterization of spherically quasinormal weighted composition operator pairs via the ‐spherical mean transform, revealing some properties that differ from the single operator case. Finally, we characterize a class of spherically ‐hyponormal weighted composition operators on discrete measure spaces. As a corollary, we present corresponding results on the spherical ‐hyponormality of unbounded 2‐variable weighted shifts and theirs ‐spherical mean transforms.
ABSTRACT We consider a two by two system of inequalities which include fractional powers of the Laplace operator, coupled through a semilinear term of derivative type. We prove the nonexistence of global‐in‐time weak solutions for powers below the critical curve and possibly on the critical curve.
ABSTRACT This paper investigates a class of ‐Laplacian‐viscoelastic wave equations with logarithmic source terms, defined on a bounded domain . First, we prove the local existence of solutions by using the Faedo–Galerkin method. Next, by combining analytical techniques, including the potential well method and energy estimates, we establish the global existence of solutions. Furthermore, employing the Komornik inequality, we prove the asymptotic stability of these solutions. Finally, through an integration of the potential well framework with concavity arguments, we demonstrate the occurrence of blow‐up phenomena under conditions of positive initial energy and derive an upper bound for the blow‐up time.
ABSTRACT In this paper, we establish interpolation theorems for generalized Morrey‐type spaces for a broad class of nonlinear operators, namely, Urysohn‐type operators. Using these results, we prove Marcinkiewicz‐ and Stein–Weiss‐type interpolation theorems for such operators.
ABSTRACT In this paper, we introduce the concept of nonuniform ‐exponential dichotomies to formulate an asymptotic behavior weaker than the known nonuniform ‐exponential dichotomy. We discuss the roughness of such dichotomies on the whole line and the half line separately without global invertibility of the evolution family. Our results extend the known results to nonuniform ‐exponential dichotomies in the case that or .
We consider the radial Schr & ouml;dinger operator on the half-line with angular momentum . Under some conditions on the potentials, we show that the eigenvalues and resonances uniquely determine the radial Schr & ouml;dinger operator. The methods of proof are based on reducing the inverse resonance problems to the inverse spectral or inverse scattering problems.
The purpose of this paper is to establish convergence results of the approximate solutions for control systems of generalized multiobjective games and optimal control problems driven by generalized multiobjective games. First, we revisit the control systems of generalized multiobjective games. Then, using a sequence of -converging mappings, we establish the upper convergence, lower convergence and convergence of the solution sets for such problems. Second, we investigate sufficient conditions for the convergence of approximate solutions to optimal control problems. Finally, as a real-world application, we consider the special case of control systems of economic equilibrium models. The results presented in the paper are new and improve some of the main results given in the literature.
ABSTRACT A class of vector bundles of rank on an ‐dimensional complex projective variety , spanned by global sections, with positive top Chern class and ample determinant is investigated. The standard relation between Chern and Segre polynomials allows us to decide about the bigness of a bundle closely related to . The size , an integer measuring the deviation of from bigness, is introduced and studied. The structure of , in connection with the behavior of the size, yields a useful tool to produce several examples. Among them the pair emerges, of which two characterizations are provided in terms of and in terms of the degree of curves representing with respect to the determinant.
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.
This paper concerns the two-dimensional nonhomogeneous micropolar system with density-dependent viscosities and vacuum. Based on the energy method and the structural characteristics of the model, we establish the global well-posdeness and exponential decay of strong solutions provided that with .
In this paper, we prove uniqueness results for weak solutions to a class of Neumann problems, whose prototype is where is a bounded open subset of with Lipschitz boundary, is a real number , the coefficients and belong to suitable Lebesgue spaces and is an element of the dual space of the Sobolev space having a suitable summability. Finally, and are positive constants which belong to suitable intervals specified in Theorems 2.3, 2.6, and 2.8.Uniqueness results for weak solutions are proved under smallness assumptions on the coefficients or .
