The Journal of Geometric Analysis最新文献

It is demonstrated that a form of Rubio de Francia’s hitherto unresolved Littlewood-Paley Type Conjecture from the year 1985 is valid for the weighted-(L^{2}left( {mathbb {R}}right) ) space corresponding to any even (A_{1}left( {mathbb {R}}right) ) weight. Otherwise expressed, we show that if (omega ) is any even (A_{1}left( {mathbb {R}}right) ) weight, C is an (A_{1}left( {mathbb {R}}right) ) weight constant for (omega ), ( fin ) (L^{2}left( {mathbb {R}},omega left( tright) dtright) ), and (left{ J_{k}right} _{kge 1}) is any finite or infinite sequence of disjoint intervals of ({mathbb {R}}), then the following estimate holds for the corresponding Littlewood-Paley Type square function defined by (left{ S_{J_{k}}left( fright) right} _{kge 1})(where the symbol (S_{_{J_{k}} }) denotes the indicated partial sum projection for the context of ({mathbb {R}})):

where (omega ^*) is the decreasing rearrangement of (omega ). A corollary of this even (A_{1}left( {mathbb {R}}right) )-weighted theorem is obtained which provides a related variant thereof in the setting of any (not necessarily even) (A_{1}left( {mathbb {R}}right) ) weight.

We are interested in the existence of normalized solutions to the problem

in the so-called at least mass critical regime. We utilize recently introduced variational techniques involving the minimization on the (L^2)-ball. Moreover, we find also a solution to the related curl–curl problem

which arises from the system of Maxwell equations and is of great importance in nonlinear optics.

We prove that a maximally modulated singular oscillatory integral operator along a hypersurface defined by ((y,Q(y))subseteq mathbb {R}^{n+1}), for an arbitrary non-degenerate quadratic form Q, admits an a priori bound on (L^p) for all (1<p<infty ), for each (n ge 2). This operator takes the form of a polynomial Carleson operator of Radon-type, in which the maximally modulated phases lie in the real span of ({p_2,ldots ,p_d}) for any set of fixed real-valued polynomials (p_j) such that (p_j) is homogeneous of degree j, and (p_2) is not a multiple of Q(y). The general method developed in this work applies to quadratic forms of arbitrary signature, while previous work considered only the special positive definite case (Q(y)=|y|^2).

Let X be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan–Li. Given a semipositive form (theta ) with positive volume, we define the Monge–Ampère operator for unbounded (theta )-psh functions and prove that it is continuous with respect to convergence in capacity. We then develop pluripotential tools to study degenerate complex Monge–Ampère equations in this context, extending recent results of Tosatti–Weinkove, Kolodziej–Nguyen, Guedj–Lu and many others who treat bounded solutions.

We study the existence of minimal networks in the unit sphere ({textbf{S}}^d) and the unit ball ({textbf{B}}^d) of ({textbf{R}}^d) endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of (theta )-networks in ({textbf{S}}^d) and triods in ({textbf{B}}^d), jointly with the Lusternik–Schnirelmann category.

We study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincaré inequality. We show that the limits of these nonlocal functionals are comparable to the total variation (Vert DfVert (Omega )) or the Sobolev semi-norm (int _Omega g_f^p, dmu ), which extends Euclidean results to metric measure spaces. In contrast to the classical setting, we also give an example to show that the limits are not always equal to the corresponding total variation even for Lipschitz functions.

We prove a generalization of a well-known theorem of Rado for continuous CR functions on a class of bihololomorphically invariant hypersurfaces that are considerably larger than convex ones of finite type and strictly pseudoconvex hypersurfaces.

In this paper, we are concerned with normalized solutions ((u,lambda )in W^{1,N}(mathbb {R}^N)times mathbb {R}^+) to the following N-Laplacian problem

satisfying the normalization constraint (int _{mathbb {R}^N}|u|^Ntextrm{d}x=c^N). The nonlinearity f(s) is an exponential critical growth function, i.e., behaves like (exp (alpha |s|^{N /(N-1)})) for some (alpha >0) as (|s| rightarrow infty ). Under some mild conditions, we show the existence of normalized mountain pass type solution via the variational method. We also emphasize the normalized ground state solution has a mountain pass characterization under some further assumption. Our existence results in present paper also solve a Soave’s type open problem (J Funct Anal 279(6):108610, 2020) on the nonlinearities having an exponential critical growth.

In this note we establish an expression for the Steklov spectrum of warped products in terms of auxiliary Steklov problems for drift Laplacians with weight induced by the warping factor. As an application, we show that a compact manifold with connected boundary diffeomorphic to a product admits a family of Riemannian metrics which coincide on the boundary, have fixed volume and arbitrarily large first non-zero Steklov eigenvalue. These are the first examples of Riemannian metrics with these properties on three-dimensional manifolds.

In this paper, we introduce an anisotropic geometric quantity (mathbb {W}_{p,q;k} ) which involves the weighted integral of k-th elementary symmetric function. We first show the monotonicity of ({mathbb {W}}_{p,1;k}) and ({mathbb {W}}_{0,q;k}) along a class of inverse anisotropic curvature flows, and then prove the generalization of anisotropic Alexandrov–Fenchel type inequalities. On the other hand, an extension of anisotropic Hsiung–Minkowski formula is derived. Therefore, we at last obtain an extension of the Alexandrov–Fenchel type inequality, which involve the general (mathbb {W}_{p,q;k}). In terms of the above inequalities, we have also demonstrated some other meaningful conclusions on convex body geometry, such as generalized (L^p)-Minkowski inequality and estimates of anisotropic p-affine surface area.