On Supnorm Estimates for $\bar{\partial}$ on Infinite Type Convex Domains in ℂ2

In this paper, we study the \(\bar{\partial}\) equation on some convex domains of infinite type in ?². In detail, we prove that supnorm estimates hold for infinite exponential type domains, provided the exponent is less than 1.     

Limits on an extension of Carleson’s\bar \partial - Theorem

We study a theorem essentially due to Carleson about solving the $\bar \partial - equation$ on the unit disk. We show that this theorem generalizes to bordered Riemann surfaces with finitely generated fundamental groups. However, our main result is that the constant appearing in the generalized theorem cannot be taken to be independent of the bordered Riemann surface in question. We exhibit a sequence of (topologically equivalent) Riemann surfaces on which the constant tends to ∞. Since Carleson’s $\bar \partial - theorem$ depends on the notion of a Carleson measure, we also discuss Carleson measures at some length in order to define them appropriately on arbitrary Riemann surfaces.     

An Extension of Mercer’s Theorem via Pontryagin Spaces

We extend Mercer’s theorem to a composition of the form RS, in which R and S are integral operators acting on a space L ²(X) generated by a locally finite measure space (X, ν). The operator R is compact and positive while S is continuous and having spectral decomposition based on well distributed eigenvalues. The proof is based on a Pontryagin space structure for L ²(X) constructed via the operators R and S themselves.     

$$\partial \overline{\partial }$$-Complex symplectic and Calabi–Yau manifolds: Albanese map,deformations and period maps

Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms.     

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 $$\Omega $$...     

图$(\bigcup\limits_{i{\in}A}P_i){\bigcup}(\bigcup\limits_{j{\in}B}U_j)$伴随唯一的充要条件

设$h(G; x) =h(G)$和$[G]_h$分别表示图$G$的伴随多项式和伴随等价类. 文中给出了$[G]_h$的一个新应用. 利用$[G]_h$, 给出了图$H{\;}(H \cong G)$伴随唯一的充要条件, 其中$H=(\bigcup_{i{\in}A}P_i){\bigcup}(\bigcup_{j{\in}B}U_j)$, $A \subseteq A^{'}=\{1,2,3,5\} \bigcup \{2n|n \in N, n \geq 3\}$, $B \subseteq B^{'}     

Schrödinger Operators with A∞$A_{\infty }$ Potentials

We study the heat kernel p(x,y,t) associated to the real Schrödinger operator H=?Δ + V on \(L^{2}(\mathbb {R}^{n})\), n≥1. Our main result is a pointwise upper bound on p when the potential \(V \in A_{\infty }\). In the case that \(V\in RH_{\infty }\), we also prove a lower bound. Additionally, we compute p explicitly when V is a quadratic polynomial.     

Notes on the Kazhdan–Lusztig Theorem on Equivalence of the Drinfeld Category and the Category of $\boldsymbol{U}_{\!\boldsymbol q}{\boldsymbol {\mathfrak g}}$-Modules

We discuss the proof of Kazhdan and Lusztig of the equivalence of the Drinfeld category \({\mathcal D}({\mathfrak g},\hbar)\) of \({\mathfrak g}\)-modules and the category of finite dimensional \(U_q{\mathfrak g}\)-modules, \(q=e^{\pi i\hbar}\), for \(\hbar\in{\mathbb C}\setminus{\mathbb Q}^*\). Aiming at operator algebraists the result is formulated as the existence for each \(\hbar\in i{\mathbb R}\) of a normalized unitary 2-cochain \({\mathcal F}\) on the dual \(\hat G\) of a compact simple Lie group G such that the convolution algebra of G with the coproduct twisted by \({\mathcal F}\) is *-isomorphic to the convolution algebra of the q-deformation G _q of G, while the coboundary of \({\mathcal F}^{-1}\) coincides with Drinfeld’s KZ-associator defined via monodromy of the Knizhnik–Zamolodchikov equations.     

Unlösbarkeit der Gleichung $\alpha x^n + \beta y^n = z^n$ und Gleichverteilung von $\root n \of{\alpha x^n + \beta y^n}$

We show that the unsolvability of the Diophantine equation is equivalent to a good uniform distribution of the set . The proof depends on the asymptotic evaluation of the Gauss sum .     

A Generalization of Bochner’s Extension Theorem to Rough Tubes

This paper generalizes Bochner’s extension theorem to tubes X+iℝ ^m where the set X⊂ℝ ^m is not necessarily a manifold.     

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.     

A Simple Proof of Perelman’s Collapsing Theorem for 3-manifolds

We will simplify earlier proofs of Perelman’s collapsing theorem for 3-manifolds given by Shioya–Yamaguchi (J. Differ. Geom. 56:1–66, 2000; Math. Ann. 333: 131–155, 2005) and Morgan–Tian ( [math.DG], 2008). A version of Perelman’s collapsing theorem states: “Let {M³_i}\{M^{3}_{i}\} be a sequence of compact Riemannian 3-manifolds with curvature bounded from below by (−1) and $\mathrm{diam}(M^{3}_{i})\ge c_{0}>0$\mathrm{diam}(M^{3}_{i})\ge c_{0}>0 . Suppose that all unit metric balls in M³_iM^{3}_{i} have very small volume, at most v _i →0 as i→∞, and suppose that either M³_iM^{3}_{i} is closed or has possibly convex incompressible toral boundary. Then M³_iM^{3}_{i} must be a graph manifold for sufficiently large i”. This result can be viewed as an extension of the implicit function theorem. Among other things, we apply Perelman’s critical point theory (i.e., multiple conic singularity theory and his fibration theory) to Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3-manifolds.     

A generalization of Schmidt’s Theorem on groups with all subgroups nilpotent

According to Schmidt’s Theorem a finite group whose proper subgroups are all nilpotent (or a finite group without non-nilpotent proper subgroups) is solvable. In this paper we prove that every finite group with less than 22 non-nilpotent subgroups is solvable and that this estimate is sharp.     

The Diederich–Fornaess index and the global regularity of the \bar{\partial }-Neumann problem

We describe along the guidelines of Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), the constant ${\mathcal {E}}_s$ which is needed to control the commutator of a totally real vector field $T_{{\mathcal {E}}}$ with $\bar{\partial }^*$ in order to have $H^s$ a-priori estimates for the Bergman projection $B_k, k\ge q-1$ , on a smooth $q$ -convex domain $D\subset \subset {\mathbb {C}}^{n}$ . This statement, not explicit in Kohn (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999), yields regularity of $B_k$ in specific Sobolev degree $s$ . Next, we refine the pseudodifferential calculus at the boundary in order to relate, for a defining function $r$ of $D$ , the operators $(T^+)^{-\frac{\delta }{2}}$ and $(-r)^{\frac{\delta }{2}}$ . We are thus able to extend to general degree $k\ge 0$ of $B_k$ , the conclusion of (Quantitative Estimates for Global Regularity. Analysis and Geometry in Several Complex Variables, pp. 97–128. Trend Math. Birkhäuser, Boston, 1999) which only holds for $q=1$ and $k=0$ : if for the Diederich–Fornaess index $\delta $ of $D$ , we have $(1-\delta )^{\frac{1}{2}}\le {\mathcal {E}}_s$ , then $B_k$ is $H^s$ -regular.     

A primal-dual homotopy algorithm for $$\ell _{1}$$-minimization with $$\ell _{\infty }$$ℓ∞-constraints

In this paper we propose a primal-dual homotopy method for \(\ell _1\)-minimization problems with infinity norm constraints in the context of sparse reconstruction. The natural homotopy parameter is the value of the bound for the constraints and we show that there exists a piecewise linear solution path with finitely many break points for the primal problem and a respective piecewise constant path for the dual problem. We show that by solving a small linear program, one can jump to the next primal break point and then, solving another small linear program, a new optimal dual solution is calculated which enables the next such jump in the subsequent iteration. Using a theorem of the alternative, we show that the method never gets stuck and indeed calculates the whole path in a finite number of steps. Numerical experiments demonstrate the effectiveness of our algorithm. In many cases, our method significantly outperforms commercial LP solvers; this is possible since our approach employs a sequence of considerably simpler auxiliary linear programs that can be solved efficiently with specialized active-set strategies.     

Littlewood-Paley Formulas and Carleson Measures for Weighted Fock Spaces Induced by A ∞ $A_{\infty }$ -Type Weights

Potential Analysis - We obtain Littlewood-Paley formulas for Fock spaces ${\mathcal {F}}_{\beta ,\omega }^{q}$ induced by weights $\omega \in {A}_{\infty }^{restricted} = \cup _{1 \le p <...     

A Note on the Pader "A Short Proof of Liapounoff''''s Convexity Theorem" of J. Lindenstrauss

J.Lindenstrauss once gave a short proof of Liapounoff＇s Convexity Theorem by using induction [1]. Now we give a more direct way to prove the theorem other than using induction. Here is the Liapounoff＇i theorem: Theorem Let μ_1,μ_2,…,μ_n,be finite positive non-atomic measures on some measure space X.Then M={(μ_1(A),μ_2(A),…,μ(A))|,measurable}is a closed and convex subset of R~n.     

A Characterization of Sobolev Spaces on the Sphere and an Extension of Stolarsky’s Invariance Principle to Arbitrary Smoothness

In this paper, we study reproducing kernel Hilbert spaces of arbitrary smoothness on the sphere $\mathbb{S}^{d} \subset\mathbb{R}^{d+1}$ . The reproducing kernel is given by an integral representation using the truncated power function $(\mathbf{x} \cdot\mathbf{z} - t)_{+}^{\beta-1}$ supported on spherical caps centered at z of height t, which reduces to an integral over indicator functions of open spherical caps if β=1, as studied in Brauchart and Dick (Proc. Am. Math. Soc. 141(6):2085–2096, 2013). This is analogous to a generalization of the reproducing kernel to arbitrary smoothness on the unit cube by Temlyakov (J. Complex. 19(3):352–391, 2003). We show that the reproducing kernel is a sum of the Euclidean distance ∥x?y∥ of the arguments of the kernel raised to the power of 2β?1 and an adjustment in the form of a Kampé de Fériet function that ensures positivity of the kernel if 2β?1 is not an even integer; otherwise, a limit process introduces logarithmic terms in the distance. For $\beta\in\mathbb{N}$ , the Kampé de Fériet function reduces to a polynomial, giving a simple closed form expression for the reproducing kernel. Stolarsky’s invariance principle states that the sum of all mutual distances among N points plus a certain multiple of the spherical cap $\mathbb{L}_{2}$ -discrepancy of these points remains constant regardless of the choice of the points. Rearranged differently, it provides a reinterpretation of the spherical cap $\mathbb{L}_{2}$ -discrepancy as the worst-case error of equal-weight numerical integration rules in the Sobolev space over $\mathbb{S}^{d}$ of smoothness (d+1)/2 provided with the reproducing kernel 1?C _d ∥x?y∥ for some constant C _d . Using the new function spaces, we establish an invariance principle for a generalized discrepancy extending the spherical cap $\mathbb{L}_{2}$ -discrepancy and give a reinterpretation as the worst-case error in the Sobolev space over $\mathbb{S}^{d}$ of arbitrary smoothness s=β?1/2+d/2. Previously, Warnock’s formula, which is the analog to Stolarsky’s invariance principle for the unit cube [0,1] ^s , has been generalized using similar techniques in Dick (Ann. Mat. Pura Appl. (4) 187(3):385–403, 2008).     

On L 2-Estimates for \bar{\partial} on a Pseudoconvex Domain in a Complete Kähler Manifold with Positive Holomorphic Bisectional Curvature

Let Ω be a relatively compact pseudoconvex domain in a complete Kähler manifold X with positive holomorphic bisectional curvature. If Ω has positive inner reach and is defined by a plurisubharmonic function of class \(\mathcal{C}^{1}\) , we generalize the existence of the Diederich–Fornaess exponent for the distance function to the boundary ?Ω. This property allows us to prove L ² estimates for the \(\bar{\partial}\) operator and regularity properties for the \(\bar{\partial}\) -Neumann operator.