The Journal of Geometric Analysis最新文献

In this paper, we study the existence of limits at infinity along almost every infinite curve for the upper and lower approximate limits of bounded variation functions on complete unbounded metric measure spaces. We prove that if the measure is doubling and supports a 1-Poincaré inequality, then for every bounded variation function f and for 1-a.e. infinite curve (gamma ), for both the upper approximate limit (f^vee ) and the lower approximate limit (f^wedge ) we have that

exist and are equal to the same finite value. We give examples showing that the conditions of the doubling property of the measure and a 1-Poincaré inequality are needed for the existence of limits. Furthermore, we establish a characterization for strictly positive 1-modulus of the family of all infinite curves in terms of bounded variation functions. These generalize results for Sobolev functions given in Koskela and Nguyen (J Funct Anal 285(11):110154, 2023).

The present paper is concerned by the study of a nonlinear elliptic equation which contains a Hardy potential, lower order term, singular term and (L^{m(cdot )} ) data. Our approach is based on approximating the initial problem with a non-singular problem that is well-posed. We then establish the necessary estimates to pass to the limit.

We are concerned with the linear problem

where (lambda ) is a positive parameter, (kappa in [0,N-2)), (N> 2) and (K:mathbb {R}^N rightarrow (0,infty )) is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue (lambda _1^{text {rad}}) and the corresponding positive eigenfunction (varphi _1) satisfies (lim nolimits _{|x|rightarrow infty }varphi _1(|x|)=frac{c}{|x|^{N-2-kappa }}) for some (c>0). As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques.

In a recent paper, Li–Ni–Zhu study the nefness and ampleness of the canonical line bundle of a compact Kähler manifold with (textrm{Ric}_kleqslant 0) and provide a direct alternate proof to a recent result of Chu–Lee–Tam. In this paper, we generalize the method of Li–Ni–Zhu to a more general setting which concerning the connection between the mixed curvature condition and the positivity of the canonical bundle. The key point is to do some a priori estimates to the solution of a Mong-Ampère type equation.

We propose a generalization of the Minkowski average of two subsets of a Riemannian manifold, in which geodesics are replaced by an arbitrary family of parametrized curves. Under certain assumptions, we characterize families of curves on a Riemannian surface for which a Brunn–Minkowski inequality holds with respect to a given volume form. In particular, we prove that under these assumptions, a family of constant-speed curves on a Riemannian surface satisfies the Brunn–Minkowski inequality with respect to the Riemannian area form if and only if the geodesic curvature of its members is determined by a function (kappa ) on the surface, and (kappa ) satisfies the inequality

where K is the Gauss curvature.

Let P be a point of a compact Riemann surface X. We study self-adjoint extensions of the Dolbeault Laplacians in hermitian line bundles L over X initially defined on sections with compact supports in (Xbackslash {P}). We define the (zeta )-regularized determinants for these operators and derive comparison formulas for them. We introduce the notion of the Robin mass of L. This quantity enters the comparison formulas for determinants and is related to the regularized (zeta (1)) for the Dolbeault Laplacian. For spinor bundles of even characteristic, we find an explicit expression for the Robin mass. In addition, we propose an explicit formula for the Robin mass in the scalar case. Using this formula, we describe the evolution of the regularized (zeta (1)) for scalar Laplacian under the Ricci flow. As a byproduct, we find an alternative proof for the Morpurgo result that the round metric minimizes the regularized (zeta (1)) for surfaces of genus zero.

This paper establishes the mapping properties of the bilinear operators on the ball Banach function spaces. The main result of this paper yields the mapping properties of the bilinear Fourier multipliers, the rough bilinear singular integrals and the bilinear Calderón–Zygmund operators on the ball Banach function spaces. As applications of the main result, we have the mapping properties of the bilinear Fourier multipliers, the rough bilinear singular integrals and the bilinear Calderón–Zygmund operators on the Morrey spaces and the Herz spaces.

In this article, we consider the Gaussian curvature problem

where (Omega ) is a bounded smooth uniformly convex domain in ({mathbb {R}}^{N}) with (Nge 2), (bin mathrm C^{infty }(Omega )) is positive in (Omega ) and may be singular or vanish on (partial Omega ), (fin C^{infty }[0, +infty )) (or (fin C^{infty }({mathbb {R}}))) is positive and increasing on ([0, +infty )) ((hbox {or } {mathbb {R}})), (gin C^{infty }[0, +infty )) is positive on ([0, +infty )). We first establish the existence and global estimates of (C^{infty })-strictly convex solutions to this problem by constructing some fine coupling (limit) structures on f and g. Our results (Theorems 2.1–2.3) clarify the influence of properties of b (on the boundary (partial Omega )) on the existence and global estimates, and reveal the relationship between the solutions of the above problem and some corresponding problems (some details see page 6–7). Then, the nonexistence of convex solutions and strictly convex solutions are also obtained (see Theorems 2.4 and C). Finally, we study the principal expansions of strictly convex solutions near (partial Omega ) by analyzing some coupling structure and using the Karamata regular and rapid variation theories.

Let (Omega subset mathbb {R}^n) be a domain and X be a ball quasi-Banach function space with some extra mild assumptions. In this article, the authors establish the extension theorem about inhomogeneous X-based Triebel–Lizorkin-type spaces (F^s_{X,q}(Omega )) for any (sin (0,1)) and (qin (0,infty )) and prove that (Omega ) is an (F^s_{X,q}(Omega ))-extension domain if and only if (Omega ) satisfies the measure density condition. The authors also establish the Sobolev embedding about (F^s_{X,q}(Omega )) with an extra mild assumption, that is, X satisfies the extra (beta )-doubling condition. These extension results when X is the Lebesgue space coincide with the known best ones of the fractional Sobolev space and the Triebel–Lizorkin space. Moreover, all these results have a wide range of applications and, particularly, even when they are applied, respectively, to weighted Lebesgue spaces, Morrey spaces, variable Lebesgue spaces, Orlicz spaces, Orlicz-slice spaces, mixed-norm Lebesgue spaces, and Lorentz spaces, the obtained results are also new. The main novelty of this article exists in that the authors use the boundedness of the Hardy–Littlewood maximal operator and the extrapolation about X to overcome those essential difficulties caused by the deficiency of the explicit expression of the norm of X.

We study the action of Bogoliubov transformations on admissible generalized one-particle density matrices arising in Hartree–Fock–Bogoliubov theory. We show that the orbits of this action are reductive homogeneous spaces, and we give several equivalences that characterize when they are embedded submanifolds of natural ambient spaces. We use Lie theoretic arguments to prove that these orbits admit an invariant symplectic form. If, in addition, the operators in the orbits have finite spectrum, then we obtain that the orbits are actually Kähler homogeneous spaces.