In this paper, we prove the equidistribution of the Hecke eigenvalues of Maass forms over an arbitrary number field at a fixed prime ideal, with respect to the Sato–Tate measure. As an application, we obtain that the proportion of Maass forms that do not satisfy the Ramanujan–Petersson conjecture at a fixed prime ideal is 0.
In this paper, we study the following fourth-order equation of the Kirchhoff type
In this paper, we prove the functional Minkowski inequality for -torsional rigidity under suitable regularity assumptions, where -torsional rigidity can be formulated as the weak solution of an elliptic boundary value problem of the -Laplacian. We also establish the Orlicz Brunn–Minkowski and Orlicz Minkowski inequalities for -torsional rigidity, which are extensions of the Brunn–Minkowski and Minkowski inequalities for torsional rigidity. Finally, we describe the equivalence between the Orlicz Brunn–Minkowski inequality and the Orlicz Minkowski inequality.
We study resonances for Jacobi operators on the half lattice with matrix-valued coefficients and finitely supported perturbations. We describe a forbidden domain, the geometry of resonances and their asymptotics when the main coefficient of the perturbation (determining its length of the support) goes to zero. Moreover, we show that
In this paper, we investigate the asymptotic behavior as infinity of a class of degenerate Monge–Ampère equations in half spaces. Meanwhile, we establish the Schauder estimates on a class of degenerate linear elliptic equations.
We study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbation. We introduce and study the class of relatively operator Lipschitz functions. An essential role is played by double operator integrals. We also consider the class of resolvent Lipschitz functions. Then we obtain a trace formula in the case of relatively trace class perturbations and show that the maximal class of function for which the trace formula holds in the case of relatively trace class perturbations coincides with the class of relatively operator Lipschitz functions. Our methods also give us a new approach to the inequality for the spectral shift function in the case of relatively trace class perturbations.
The following problems related to almost complex structures on a twistor space are discussed: (1) when two almost complex structures defined by a morphism of the twistor space and two conformal metrics on the base manifold coincide; (2) when the Atiyah–Hitchin–Singer or Eells–Salamon almost complex structures on the twistor space defined by two metric connections on the base Riemannian manifold coincide.
This paper is concerned with a class of bounded pesudoconvex Hartogs domains, which is defined by the inequality
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