A new Leray formula for smooth functions on bounded domains in Cn

By means of a new technique of integral representations in C ⁿ given by the authors, we establish a new abstract formula with a vector function W for smooth functions on bounded domains in C ⁿ , which is different from the well-known Leray formula. This new formula eliminates the term that contains the parameter A from the classical Leray formula, and especially on some domains the uniform estimates for the are very simple. From the new Leray formula, we can obtain correspondingly many new formulas for smooth functions on many domains in C ⁿ , which are different from the classical ones, when we properly select the vector function W.     

Traces of monotone Sobolev functions

In this paper we prove that ifu: ${\mathbb{B}}^n \to {\mathbb{R}}$ , where ${\mathbb{B}}^n $ is the unit ball in ? ⁿ , is a monotone function in the Sobolev space W^1·p ( ${\mathbb{B}}^n $ ), andn ? 1 <p ≤n, thenu has nontangential limits at all the points of $\partial {\mathbb{B}}^n $ except possibly on a set ofp-capacity zero. The key ingredient in the proof is an extension of a classical theorem of Lindelöf to monotone functions in W^1·p ( ${\mathbb{B}}^n $ ),n ? 1 <p ≤n.     

On a class of smooth Frechet subalgebras of C * -algebras

The paper contributes to understanding the differential structure in a C ^*-algebra. Refining the Banach $(D_p^*)$ -algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra C ^p [a,b] of p-times continuously differentiable functions, we investigate a Frechet $(D_\infty^*)$ -subalgebra $\ensuremath{{\mathcal B}}$ of a C ^*-algebra as a noncommutative analogue of the algebra C ^?∞?[a,b] of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\ensuremath{{\mathcal B}}$ as an inverse limit over n of Banach $(D^*_n)$ -algebras. Several examples of such smooth algebras are exhibited.     

A gradient-projective basis of compactly supported wavelets in dimension n > 1

A given set W = {W _X } of n-variable class C ¹ functions is a gradient-projective basis if for every tempered distribution f whose gradient is square-integrable, the sum $\sum\limits_\chi {(\int_{\mathbb{R}^n } {\nabla f \cdot } \nabla W_\chi ^* )} W_\chi $ converges to f with respect to the norm $\left\| {\nabla ( \cdot )} \right\|_{L^2 (\mathbb{R}^n )} $ . The set is not necessarily an orthonormal set; the orthonormal expansion formula is just an element of the convex set of valid expansions of the given function f over W. We construct a gradient-projective basis W = {W _x } of compactly supported class C ^2?? functions on ? ⁿ such that where X has the structure $\chi = (\tilde \chi ,\nu )$ , ν ∈ ?. W is a wavelet set in the sense that the functions indexed by $\tilde \chi $ are generated by an averaging of lattice translations with wave propagations, and there are two additional discrete parameters associated with the latter. These functions indexed by $\tilde \chi $ are the unit-scale wavelets. The support volumes of our unit-scale wavelets are not uniformly bounded, however. While the practical value of this construction is doubtful, our motivation is theoretical. The point is that a gradient-orthonormal basis of compactly supported wavelets has never been constructed in dimension n > 1. (In one dimension the construction of such a basis is easy — just anti-differentiate the Haar functions.)     

Exponential laws for ultrametric partially differentiable functions and applications

We establish exponential laws for certain spaces of differentiable functions over a valued field $\mathbb{K}$ . For example, we show that $$C^{(\alpha ,\beta )} \left( {U \times V,E} \right) \cong C^\alpha \left( {U,C^\beta \left( {V,E} \right)} \right)$$ if α ∈ (?₀ ∪ {∞}) ⁿ , β ∈ (?₀ ∪ {∞}) ^m , $U \subseteq \mathbb{K}^n$ and $V \subseteq \mathbb{K}^m$ are open (or suitable more general) subsets, and E is a topological vector space. As a first application, we study the density of locally polynomial functions in spaces of partially differentiable functions over an ultrametric field (thus solving an open problem by Enno Nagel), and also global approximations by polynomial functions. As a second application, we obtain a new proof for the characterization of C ^r -functions on (? _p ) ⁿ in terms of the decay of their Mahler expansions. In both applications, the exponential laws enable simple inductive proofs via a reduction to the one-dimensional, vector-valued case.     

Jackson-Bernstein theorem in strictly pseudoconvex domains in Cn

Let ω be aC ²-smooth strictly pseudoconvex domain in Cn, letA(ω) denote the class of functions holomorphic in the interior \(\mathop \Omega \limits^ \circ\) and continuous in ω, letP(ω) be the closure of holomorphic polynomials in the uniformC(ω)-norm, and let \(\Lambda ^\alpha (\Omega ) \subset A(\Omega )\) be the Hölder class of holomorphic functions. With the assumptionP(ω)=A(ω) a theorem concerning Λ^α(Ω) analogous to the classical Jackson-Bernstein theorem is proved.     

Lyashko–Looijenga morphisms and submaximal factorizations of a Coxeter element

When W is a finite reflection group, the noncrossing partition lattice $\operatorname{NC}(W)$ of type W is a rich combinatorial object, extending the notion of noncrossing partitions of an n-gon. A formula (for which the only known proofs are case-by-case) expresses the number of multichains of a given length in $\operatorname{NC}(W)$ as a generalized Fu?–Catalan number, depending on the invariant degrees of W. We describe how to understand some specifications of this formula in a case-free way, using an interpretation of the chains of $\operatorname{NC}(W)$ as fibers of a Lyashko–Looijenga covering ( $\operatorname{LL}$ ), constructed from the geometry of the discriminant hypersurface of W. We study algebraically the map $\operatorname{LL}$ , describing the factorizations of its discriminant and its Jacobian. As byproducts, we generalize a formula stated by K. Saito for real reflection groups, and we deduce new enumeration formulas for certain factorizations of a Coxeter element of W.     

Reconstructing vector bundles on curves from their direct image on symmetric powers

Let C be an irreducible smooth complex projective curve, and let E be an algebraic vector bundle of rank r on C. Associated to E, there are vector bundles ${{\mathcal F}_n(E)}$ of rank nr on S ⁿ (C), where S ⁿ (C) is the n-th symmetric power of C. We prove the following: Let E ₁ and E ₂ be two semistable vector bundles on C, with genus ${(C)\, \geq\, 2}$ . If ${{\mathcal F}_n(E_1)\,\simeq \, {\mathcal F}_n(E_2)}$ for a fixed n, then ${E_1 \,\simeq\, E_2}$ .     

Global Existence and Blow-Up Results for a Classical Semilinear Parabolic Equation

The author studies the boundary value problem of the classical semilinear parabolic equations ut-△u = |u|p-1u inΩ×(0, T), and u = 0 on the boundary × [0, T) and u = φ at t = 0, where Rnis a compact C1domain, 1 < p ≤ p S is a fixed constant, and φ∈ C1 0(Ω) is a given smooth function. Introducing a new idea, it is shown that there are two sets W and Z, such that for φ∈ W, there is a global positive solution u(t) ∈ W with H1omega limit 0 and for φ∈ Z, the solution blows up at finite time.     

Long time small solutions to nonlinear parabolic equations

A sharp result on global small solutions to the Cauchy problem $$u_t = \Delta u + f\left( {u,Du,D^2 u,u_t } \right)\left( {t > 0} \right),u\left( 0 \right) = u_0 $$ In Rⁿ is obtained under the the assumption thatf is C^1+r forr>2/n and ‖u ₀‖C²(R ⁿ ) +‖u ₀‖W ₁ ² (R ⁿ ) is small. This implies that the assumption thatf is smooth and ‖u ₀ ‖W ₁ ^k (R ⁿ )+‖u ₀‖W ₂ ^k (R ⁿ ) is small fork large enough, made in earlier work, is unnecessary.     

The computation and perturbation analysis for weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29(1):39–48, 2007) has defined a weighted group inverse of rectangular matrices. Let A∈C ^m×n ,W∈C ^n×m , if X∈C ^m×n satisfies the system of matrix equations $$(W_{1})\quad AWXWA=A,\quad\quad (W_{2})\quad XWAWX=X,\quad\quad (W_{3})\quad AWX=XWA$$ X is called the weighted group inverse of A with W, and denoted by A _W ^# . In this paper, we will study the algebra perturbation and analytical perturbation of this kind weighted group inverse A _W ^# . Under some conditions, we give a decomposition of B _W ^# ?A _W ^# . As a results, norm estimate of ‖B _W ^# ?A _W ^# ‖ is presented (where B=A+E). An expression of algebra of perturbation is also obtained. In order to compute this weighted group inverse with ease, we give a new representation of this inverse base on Gauss-elimination, then we can calculate this weighted inverse by Gauss-elimination. In the end, we use a numerical example to show our results.     

A contribution to stability theory for nonlinear Neumann problem

In this paper we study the stability of the solutions of some nonlinear Neumann problems, under perturbations of the domains in the Hausdorff complementary topology. We consider the problem $${{\left\{\begin{array}{c}-\text{ div}\;\left(a\left( x,\nabla u_{\Omega}\right)\right)=0 \;\text{in}\; \Omega \\ {a\left( x, \nabla u_{\Omega}\right) \cdot \nu=0\; \text{on}\; \partial\Omega}\end{array}\right.}}$$ where ${{\mathbf{R}^n \times \mathbf{R}^n \rightarrow \mathbf{R}^n}}$ is a Caratheodory function satisfying the standard monotonicity and growth conditions of order p, 1?<?p?<???. If ?? _h is a uniformly bounded sequence of connected open sets in R ⁿ , n ??? 2, we prove that if ${{\Omega_{h}^{c} \rightarrow \Omega^{c}}}$ in the Hausdorff metric, ${|\Omega_{h}| \rightarrow |\Omega|}$ and the geodetic distances satisfy the inequality ${d_{\Omega}\left( x,y\right) \leq \liminf_{h} d_{\Omega_{h}} \left( x,y\right)}$ for every ${x, y \in \Omega,}$ then ${\nabla u_{\Omega_h} \rightarrow\nabla u_{\Omega}}$ strongly in L ^p , provided that W ^{1, ??}(??) is dense in the space L ^{1, p} (??) of all functions whose gradient belongs to L ^p (??, R ⁿ ).     

On the perturbation of weighted group inverse of rectangular matrices

Cen (Math. Numer. Sin. 29:39–48, 2007) defined a weighted group inverse of rectangular matrices. For given matrices A∈C ^m×n and W∈C ^n×m , if X∈C ^m×n satisfies $$( W_{1} )\ AWXWA=A, \qquad ( W_{2} ) \ XWAWX=X,\qquad ( W_{3} )\ AWX=XWA $$ then X is called the W-weighted group inverse, which is denoted by $A_{W}^{\#}$ . In this paper, for given rectangular matrices A and E and B=A+E, we investigate the perturbation of the weighted group inverse $A_{W}^{\#}$ and present the upper bounds for $\|B_{W}^{\#} \|$ .     

The \bar \partial -problem on q-pseudoconvex domains with applications

For a q-pseudoconvex domain Ω in ? ⁿ , 1 ≤ q ≤ n, with Lipschitz boundary, we solve the $\bar \partial $ -problem with exact support in Ω. Moreover, we solve the $\bar \partial $ -problem with solutions smooth up to the boundary over Ω provided that it has smooth boundary. Applications are given to the solvability of the tangential Cauchy-Riemann equations on the boundary.     

The explicit solutions ofC_{p,q}^{k + \alpha } -form for the\bar \partial -equations on pseudoconvex open sets

In this paper, we study the integral solution operators for the $\bar \partial $ -equations on pseudoconvex domains. As a generalization of [1] for the $\bar \partial $ -equations on pseudoconvex domains with boundary of classC ^∞, we obtain the explicit integral operator solutions of $C_{p,q}^{k + \alpha } $ -form for the $\bar \partial $ -equations on pseudoconvex open sets with boundary ofC ^k (k≥0) and the sup-norm estimates of which solutions have similar as that [1] in form.     

Integration of H?lder forms and currents in snowflake spaces

For an oriented n-dimensional Lipschitz manifold M we give meaning to the integral ${\int_M f \, dg_1 \wedge \cdots \wedge dg_n}$ in case the functions ${f, g_1, \ldots, g_n}$ are merely H?lder continuous of a certain order by extending the construction of the Riemann?CStieltjes integral to higher dimensions. More generally, we show that for ${\alpha \in (\tfrac{n}{n+1},1]}$ the n-dimensional locally normal currents in a locally compact metric space (X, d) represent a subspace of the n-dimensional currents in (X, d ^?? ). On the other hand, for ${n \geq 1}$ and ${\alpha \leq \tfrac{n}{n+1}}$ the vector space of n-dimensional currents in (X, d ^?? ) is zero.     

Comparison of Different Definitions of Traces for a Class of Ramified Domains with Self-Similar Fractal Boundaries

We consider a class of ramified bidimensional domains Ω with a self-similar fractal boundary Γ^?∞?, which is supplied with a probability measure μ called the self-similar measure. Emphasis is put on the case when the domain is not a ε???δ domain as defined by Jones and the fractal set is not totally disconnected. We compare two notions of trace on Γ^?∞? for functions in W ^1,q (Ω): the classical one, see for instance the book by Jonnson and Wallin, 1984, using the strict definition of a function at a point of $\overline{\Omega}$ , and another one proposed in 2007 and heavily relying on self-similarity. We prove that the two traces coincide μ-almost everywhere on Γ^?∞?. As a corollary, we characterize the critical number $\bar q$ for which for all $q<\bar q$ (resp. $q > \bar q$ ) there is a (resp. no) continuous extension operator from W ^1,q (Ω) to W ^1,q (?²).     

Symmetry breaking bifurcation from solutions concentrating on the equator of \mathbb{S}^N

We are concerned with the elliptic problem $${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$ , where ${\Delta _{{S^n}}}$ is the Laplace-Beltrami operator on $\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)$ , and p ? 2. We construct a smooth branch C of solutions concentrating on the equator S ⁿ ∩ {x _{n+1 = 0}}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.     

Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We study the principal Dirichlet eigenvalue of the operator \({L_A=\Delta^{\alpha/2}+Ab(x)\cdot\nabla}\) , on a bounded C ^1,1 regular domain D. Here \({\alpha\in(1,2)}\) , \({\Delta^{\alpha/2}}\) is the fractional Laplacian, \({A\in\mathbb{R}}\) , and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W ^1,2d/(d+α)(D). We prove that the eigenvalue remains bounded, as A→ + ∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of \({\Delta^{\alpha/2}}\) for the Dirichlet condition.     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.