Axler–Zheng Type Theorem on a Class of Domains in {\mathbb{C}^n}

We prove a version of Axler–Zheng’s Theorem on smooth bounded pseudoconvex domains in ${\mathbb{C}^n}$ on which the ${\overline{\partial}}$ -Neumann operator is compact.     

Bilinear Pseudo-differential Operators with Symbols in $$BS^{m}_{1,1}$$ on Triebel–Lizorkin Spaces

In this paper, bilinear pseudo-differential operators with symbols in the bilinear Hörmander symbol class \(BS^{m}_{1,1}\) on Triebel–Lizorkin spaces are discussed. As a result, we can obtain the Kato–Ponce inequality in local Hardy spaces.     

Poletsky–Stessin Hardy Spaces on Complex Ellipsoids in $$\mathbb {C}^{n}$$

We study Poletsky–Stessin Hardy spaces on complex ellipsoids in \(\mathbb {C}^{n}\). Different from one variable case, classical Hardy spaces are strictly contained in Poletsky–Stessin Hardy spaces on complex ellipsoids so boundary values are not automatically obtained in this case. We have showed that functions belonging to Poletsky–Stessin Hardy spaces have boundary values and they can be approached through admissible approach regions in the complex ellipsoid case. Moreover, we have obtained that polynomials are dense in these spaces. We also considered the composition operators acting on Poletsky–Stessin Hardy spaces on complex ellipsoids and gave conditions for their boundedness and compactness.     

Hardy spaces associated to the Schrödinger operator on strongly Lipschitz domains of {\mathbb{R}^d}

Let L = ?Δ + V be a Schrödinger operator and Ω be a strongly Lipschitz domain of ${\mathbb R^{d}}Let L = −Δ + V be a Schr?dinger operator and Ω be a strongly Lipschitz domain of \mathbb R^d{\mathbb R^{d}} , where Δ is the Laplacian on \mathbb R^d{\mathbb R^{d}} and the potential V is a nonnegative polynomial on \mathbb R^d{\mathbb R^{d}} . In this paper, we investigate the Hardy spaces on Ω associated to the Schr?dinger operator L.     

A Payne�CRayner Type Inequality for the Robin Problem on Arbitrary Minimal Surfaces in {\mathbb{R}^N}

We prove a Payne?CRayner type inequality for the first eigenfunction of the Laplacian with Robin boundary condition on any compact minimal surface with boundary in ${\mathbb{R}^N}$ . We emphasize that no topological condition is necessary on the boundary.     

On the prescribing �� 2 curvature equation on {\mathbb S^4}

Prescribing ?? _k curvature equations are fully nonlinear generalizations of the prescribing Gaussian or scalar curvature equations. For a given a positive function K to be prescribed on the 4-dimensional round sphere, we obtain asymptotic profile analysis for potentially blowing up solutions to the ?? ₂ curvature equation with the given K; and rule out the possibility of blowing up solutions when K satisfies a non-degeneracy condition. Under the same non-degeneracy condition on K, we also prove uniform a priori estimates for solutions to a family of ?? ₂ curvature equations deforming K to a positive constant; and under an additional, natural degree condition on a finite dimensional map associated with K, we prove the existence of a solution to the ?? ₂ curvature equation with the given K using a degree argument involving fully nonlinear elliptic operators to the above deformation.     

A Tauberian theorem for ℓ-adic sheaves on \mathbb{A} 1

Let K ∈ L ¹(?) and let f ∈ L ^∞(?) be two functions on ?. The convolution $$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$ can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if $$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$ for some constant A, then $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$ We prove the following ?-adic analogue of this theorem: Suppose K, F, G are perverse ?-adic sheaves on the affine line $ \mathbb{A} $ over an algebraically closed field of characteristic p (p ≠ ?). Under suitable conditions, if $ \left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty } $ , then $ F|_{\eta _\infty } \cong G|_{\eta _\infty } $ , where η _∞ is the spectrum of the local field of $ \mathbb{A} $ at ∞.     

One-radius results for supermedian functions on {\mathbb R^d} , d ≤ 2

A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c ₀, where c ₀ = 1, if d = 2, whereas 2.50 < c ₀ < 2.51, if d = 1.     

Zeros of H ∞-Functions on Hyperplanes in \mathbb{B} n

Let ⁿ be the unit ball in ℂⁿ, n ≥ 2. Let T_α = {z ∈ ⁿ : (z, a) = |a|²} for a ∈ ⁿ and denote for a discrete set A in ⁿ. We find a sharp necessary condition for a set A to be a part of the zero-set for a function in H^∞( ⁿ). Bibliography 4 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 303, 2003, pp. 272–278.     

Schatten class Hankel and $$\overline{\partial }$$ ∂ ¯ -Neumann operators on pseudoconvex domains in $$\mathbb {C}^{n}$$ C n

Monatshefte für Mathematik - Let $$\Omega $$ be a $$C^2$$ -smooth bounded pseudoconvex domain in $$\mathbb {C}^n$$ for $$n\ge 2$$ and let $$\varphi $$ be a holomorphic function on $$\Omega $$...     

Vojta’s conjecture on blowups of $${\mathbb{P}^n}$$, greatest common divisors,and the abc conjecture

We will prove some cases of Vojta’s conjecture on blowups of \({\mathbb{P}^n}\), using Schmidt’s subspace theorem. The results can be stated as inequalities of greatest common divisors. Moreover, from Vojta’s conjecture on one further blowup at an infinitely near point, we derive a still-open special case of the abc-conjecture.     

Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere :

Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets

This paper investigates the Lipschitz equivalence of generalized {1,3,5}-{1,4,5} self-similar sets D=(r_1D)∪(r_2D (1 r_1-r_2-r_3)/2)∪(r_3D 1 r_3) and E=(r_1E)∪(r_2E 1-r_2- r_3)∪(r_3E 1-r_3),and proves that D and E are Lipschitz equivalent if and only if there are positive integers m and n such that r_1~m=r_3~n.     

${R}_{0}$代数中的滤子格

Abstract In the present paper, some basic properties of MP filters of Ro algebra M are investigated. It is proved that（FMP（M）,包含,′∧^-∨^-,{1},M）is a bounded distributive lattice by introducing the negation operator ′, the meet operator ∧^-, the join operator ∨^- and the implicati on operator → on the set FMP（M） of all MP filters of M. Moreover, some conditions under which （FMP（M）,包含,′∨^-,→{1},M）is an Ro algebra are given. And the relationship between prime elements of FMP （M） and prime filters of M is studied. Finally, some equivalent characterizations of prime elements of .FMP （M） are obtained.     

Pareto–Fenchel {\epsilon} -subdifferential sum rule and {\epsilon} -efficiency

The paper is centered around a sum rule for the efficient (Pareto) ${\epsilon}$ -subdifferential of two convex vector mappings, having the property to be exact under a qualification condition. Such a formula has not been explored previously. Our formula which holds under the Attouch?CBrézis as well as Moreau?CRockafellar conditions, reveals strangely a primordial presence of the convex (Fenchel) ${\epsilon}$ -subdifferential. This appearance turns out to be rather favorable. This effectively permits to derive approximate efficiency conditions in terms of Pareto subgradient and vectorial normal cone, which completely characterizes an ${\epsilon}$ -efficient solution in constrained convex vector optimization in (partially) ordered spaces. Our sum rule also allows a fundamental deduction of relation between Pareto and Fenchel ${\epsilon}$ -subdifferentials, which, in reality, brings out a certain gap linking ${\epsilon}$ -efficiency with ${\epsilon}$ -optimality. Scalarization approaches in connection with ${\epsilon}$ -subdifferentials are first established by simple proofs. This principle has contributed for a large part, not only for discovering the sum formula, but also for establishing some punctual necessary and/or sufficient conditions for Pareto ${\epsilon}$ -subdifferentiability.     

广义{1,3,5}-{1,4,5}自相似集的Lipschitz 等价

讨论广义{1,3,5}-{1,4,5}自相似集D=(r₁D)∪(r₂D+ (1+r₁-r₂-r₃)/2)∪(r₃ D+1-r₃)以及E=(r₁E)∪(r₂E+1-r₂-r₃)∪(r₃E+1-r₃)的Lipschitz等价性, 并证明了D与E为Lipschitz等价当且仅当存在正整数m和n, 使得 r₁^m=r₃ⁿ.     

r模同态的{1}-逆与{2}-逆

本文探讨模同态广义逆在环模理论中的应用．利用模同态的{1}-逆与{2}-逆，分别给出了一类环及一类重要模的特征刻画．     

Метод многопараметрической интерполяции и теоремы вложения пространств БесоваB_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)

In this paper we determine the method of multi-parameter interpolation and the scales of Lebesgue spaces $B_{\vec p} \left[ {0,2\pi } \right)$ and Besov spaces $B_{\vec p}^{\vec \alpha } \left[ {0,2\pi } \right)$ , which are generalizations of the Lorentz spacesL _pq [0, 2π) and Besov spacesB _pq ^α [0, 2π). We also prove imbedding theorems.     

Teichmüller curves in genus three and just likely intersections in $\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}$

We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).These constraints are akin to those that appear in Zilber and Pink’s conjectures on unlikely intersections in diophantine geometry. However, in our case one is lead naturally to the intersection of a surface with a family of codimension two algebraic subgroups of \(\mathbf{G}_{m}^{n}\times\mathbf{G}_{a}^{n}\) (rather than the more standard \(\mathbf{G}_{m}^{n}\)). The ambient algebraic group lies outside the scope of Zilber’s Conjecture but we are nonetheless able to prove a sufficiently strong height bound.For the generic stratum \(\Omega\mathcal{M}_{3}(1,1,1,1)\), we obtain global torsion order bounds through a computer search for subtori of a codimension-two subvariety of \(\mathbf{G}_{m}^{9}\). These torsion bounds together with new bounds for the moduli of horizontal cylinders in terms of torsion orders yields finiteness in this stratum. The intermediate strata are handled with a mix of these techniques.