Related papers: On Qian's problem for $\mathcal{L}_{\infty}$-space…
The discrete functional $L_p$ Minkowski problem is posed and solved. As a consequence, the general affine P\'{o}lya-Szeg\"{o} principle and the general affine Sobolev inequalities are established.
By using Liu's $q$-partial differential equations theory, we prove that if an analytic function in several variables satisfies a system of $q$-partial differential equations, if and only if it can be expanded in terms of homogeneous…
In this paper, we study the Cauchy problem of the 3-dimensional (3D) generalized incompressible Navier-Stokes equations (gNS) in Triebel-Lizorkin space $\dot{F}^{-\alpha,r}_{q_\alpha}(\mathbb{R}^3)$ with…
We consider quasi-polynomial spaces of differential forms defined as weighted (with a positive weight) spaces of differential forms with polynomial coefficients. We show that the unisolvent set of functionals for such spaces on a simplex in…
In this paper we present explicit formulas for the *-product on quantum spaces which are of particular importance in physics, i.e., the q-deformed Minkowski space and the q-deformed Euclidean space in 3 and 4 dimensions, respectively. Our…
A variant of Li-Tam theory, which associates to each end of a complete Riemannian manifold a positive solution of a given Schr\"odinger equation on the manifold, is developed. It is demonstrated that such positive solutions must be of…
We consider a manifestly Lorentz invariant form $\mathbb L$ of the biquaternion algebra and its generalization to the case of curved manifold. The conditions of $\mathbb L$-differentiability of $\mathbb L$-functions are formulated and…
A complex space $X$ is in class ${\mathcal Q}_G$ if it is a semistable quotient of the complement to an analytic subset of a Stein manifold by a holomorphic action of a reductive complex Lie group $G$. It is shown that every pseudoconvex…
For $0 < p \leq 1 < q < \infty$ and $\gamma > 0$, we introduce the Calder\'on-Hardy spaces $\mathcal{H}^{p}_{q, \gamma}(\mathbb{H}^{n})$ on the Heisenberg group $\mathbb{H}^{n}$, and show for every $f \in H^{p}(\mathbb{H}^{n})$ that the…
We consider the Cauchy problem for the Gross-Pitaevskii infinite linear hierarchy of equations on $\mathbb{R}^n.$ By introducing a (F)-norm in certain Sobolev type spaces of sequences of marginal density matrices, we establish local…
For prime powers q, let s(q) denote the probability that a randomly-chosen principally-polarized abelian surface over the finite field F_q is not simple. We show that there are positive constants B and C such that for all q, B (log…
The Cauchy problem for the Chern-Simons-Higgs system in the (2+1)-dimensional Minkowski space in temporal gauge is globally well-posed in energy space improving a result of Huh. The proof uses the bilinear space-time estimates in…
We consider the initial value Cauchy problem for a class of evolution equations whose Hamiltonian is the Weyl quantization of a homogeneous quadratic form with non-negative definite real part. The solution semigroup is shown to be strongly…
A sharp version of the Balian-Low theorem is proven for the generators of finitely generated shift-invariant spaces. If generators $\{f_k\}_{k=1}^K \subset L^2(\mathbb{R}^d)$ are translated along a lattice to form a frame or Riesz basis for…
The parabolic integro-differential Cauchy problem with spatially dependent coefficients is considered in generalized Bessel potential spaces where smoothness is defined by L\'evy measures with O-regularly varying profile. The coefficients…
We prove that the Cauchy problem for the 2D quintic defocusing biharmonic Schr\"odinger equation is globally well-posed in the Sobolev spaces $H^s(\mathbb{R}^2)$ for $\frac{8}{7}<s<2$. Our main ingredient to establish the result is the…
Complementing and extending the Inventiones work of Benson, Grodal, Henke [Group cohomology and control of p-fusion, Invent. Math. 197 (2014), 491--507] we give criteria for a space to have cohomology (strongly) F-isomorphic in the sense of…
In this paper, we prove that the cubic nonlinear Schr\"odinger equation with the fractional Laplacian on the unit disk is globally well-posed for certain radial initial data below the energy space. The result is proved by extending the…
Using a global version of the equivariant Chern character, we describe the complexified twisted equivariant K-theory of a space with a compact Lie group action in terms of fixed-point data. We apply this to the case of a compact Lie group…
In this paper we classify the solutions to the geometric Neumann problem for the Liouville equation in the upper half-plane or an upper half-disk, with the energy condition given by finite area. As a result, we classify the conformal…