Let D be a quaternion algebra over a totally real number field F that splits exactly at one infinite place of F. The associated Shimura curve Sh is canonically defined over F. Let 𝔭_0 | p be a prime of F such that D_𝔭_0 is a division algebra. Let K⊂ (D ⊗𝔸_f)^× be an open compcact subgroup such that K_𝔭_0 is maximal. Sh_K has a unique stable model over O_F_𝔭_0 . The p-adic unifomization describes completion of this model by quotients of the p-adic upper halfplane Ω̂_F_𝔭_0^2 by actions of discrete subgroups. It was discovered by Cherednik. Our result generalizes a more precise theorem of Drinfeld in the case F = ℚ .
We investigate the Monge–Ampère equation on a compact Hermitian manifold. We begin by refining the challenging zeroth order estimate previously established for the equation when the right-hand side is a smooth, positive function. Subsequently, we tackle the more complex case where the right-hand side depends explicitly on the solution and its gradient, under various hypotheses. A primary contribution of this work is a comprehensive exposition of all necessary estimates, with particular emphasis on the crucial gradient and Hessian estimates. In our specific framework, the gradient estimate is a prerequisite for achieving the necessary Laplacian estimation. Furthermore, the Hessian estimate is essential either for the application of the Evans–Krylov theory to obtain Hölder estimates for the second order derivatives or to generalize Calabi estimates for third-order derivatives.
We study nonlinear Choquard equations with general radially non-increasing convolution kernels 𝒥(|x|) that interpolate between Riesz potentials I_α and I_β of order 0<α≤β
We study positive supersolutions of quasilinear elliptic problems driven by the (p, q)-Laplace operator which arise in models with nonstandard and unbalanced growth. Our approach is based on an explicit construction of radial solutions and torsion functions that serve as comparison functions. This gives a fundamental pointwise estimate for positive supersolutions, reducing the analysis to an integral condition involving a weight and a nonlinearity evaluated along a suitable radial infimum profile. As an application, we derive explicit upper bounds for the extremal parameter associated with nonlinear Dirichlet problems in bounded domains. In exterior domains, we establish optimal Liouville-type nonexistence results for (p, q)-problems. The method extends naturally to multi-phase operators, providing a unified framework for pointwise estimates and Liouville theorems in problems with nonstandard growth.
In this paper, we establish interior BMO regularity for generalized De Giorgi classes via a perturbation technique. As an application, we obtain interior BMO regularity for linear elliptic equations in divergence form under nearly optimal assumptions on the coefficients.
