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In this paper we deal with the following family of exponential maps $(f_\lambda:z\mapsto \lambda(e^z-1))_{\lambda\in [1,+\infty)}$. Denoting $d(\lambda)$ the hyperbolic dimension of $f_\lambda$. It is known that the function $\lambda\mapsto d(\lambda)$ is real analytic in $(1,+\infty)$, and that it is continuous in $[1,+\infty)$. In this paper we prove that this map is C$^1$ on $[1,+\infty)$, with $d'(1^+)=0$. Moreover, depending on the value of $d(1)$, we give estimates of the speed of convergence towards 0., Comment: 32 pages. A para\^itre dans Annales Academi{\ae} Scientiarum Fennic{\ae} Mathematica

LetF=(f ij )be ann×n matrix ofH∞entries, and define $$S_F = (T_{2 \otimes ln}^* \oplus T_{2 \otimes ln}^* )\left| {_{Graph T_F^* } } \right..$$ This type of operator plays an important role in Cowen-Douglas theory. We call it the imbedded operator with symbolF. In the paper [L], we have already shown thatS F is a compact pertubation ofT z⊗ * I n by BDF Theorem. In this paper, we begin to investigate the compact part ofS F . We give a practical method to calculate this compact part whenn=1 andF is any finite Blaschke product.

In this paper the constrained vector optimization problem mic C f(x), g(x) ∃ − K, is considered, where $$f:\mathbb{R}^n \to \mathbb{R}^m $$ and $$g:\mathbb{R}^n \to \mathbb{R}^p $$ are locally Lipschitz functions and $$C \subset \mathbb{R}^m $$ and $$K \subset \mathbb{R}^p $$ are closed convex cones. Several solution concepts are recalled, among them the concept of a properly efficient point (p-minimizer) and an isolated minimizer (i-minimizer). On the base of certain first-order optimalitty conditions it is shown that there is a close relation between the solutions of the constrained problem and some unconstrained problem. This consideration allows to “double” the solution concepts of the given constrained problem, calling sense II optimality concepts for the constrained problem the respective solutions of the related unconstrained problem, retaining the name of sense I concepts for the originally defined optimality solutions. The paper investigates the stability properties of thep-minimizers andi-minimizers. It is shown, that thep-minimizers are stable under perturbations of the cones, while thei-minimizers are stable under perturbations both of the cones and the functions in the data set. Further, it is shown, that sense I concepts are stable under perturbations of the objective data, while sense II concepts are stable under perturbations both of the objective and the constraints. Finally, the so called structural stability is discused.

On a compact manifold with boundary, consider the realization B of an elliptic, possibly pseudodifferential, boundary value problem having a spectral cut (a ray free of eigenvalues), say R_-. In the first part of the paper we define and discuss in detail the operator log B; its residue (generalizing the Wodzicki residue) is essentially proportional to the zeta function value at zero, zeta(B,0), and it enters in an important way in studies of composed zeta functions zeta(A,B,s)=Tr(AB^{-s}) (pursued elsewhere). There is a similar definition of the operator log_theta B, when the spectral cut is at a general angle theta. When B has spectral cuts at two angles theta < phi, one can define the sectorial projection Pi_{theta,phi}(B) whose range contains the generalized eigenspaces for eigenvalues with argument in ] theta, phi [; this is studied in the last part of the paper. The operator Pi_{theta,phi}(B) is shown to be proportional to the difference between log_theta B and log_phi B, having slightly better symbol properties than they have. We show by examples that it belongs to the Boutet de Monvel calculus in many special cases, but lies outside the calculus in general., 27 pages, minor adjustments and correction of typos. To appear in Math. Scand

Take e.g. two disjoint nondegenerate compact continua A and B in the complex plane C with connected complements and pick a simple arc γ in the complex sphere Ĉ disjoint from A ∪ B, which we call a pasting arc for A and B. Construct a covering Riemann surface Ĉγ over Ĉ by pasting two copies of $\hat{\mathbf C}\setminus\gamma$ crosswise along γ. We embed A in one sheet and B in another sheet of two sheets of Ĉγ which are copies of $\hat{\mathbf C}\setminus\gamma$ so that $\hat{\mathbf C}_{\gamma}\setminus A \cup B$ is understood as being obtained by pasting ($\hat{\mathbf C}\setminus A)\setminus \gamma$ with ($\hat{\mathbf C}\setminus B)\setminus \gamma$ crosswise along γ. In the comparison of the variational 2 capacity cup(A,$\hat{\mathbf C}_{\gamma}\setminus B$) of the compact set A considered in the open set $\hat{\mathbf C}_{\gamma}\setminus B$ with the corresponding cap(A,$\hat{\mathbf C}\setminus B$), we say that the pasting arc γ for A and B is subcritical, critical, or supercritical according as cap(A,$\hat{\mathbf C}_{\gamma}\setminus B$) is less than, equal to, or greater than cap(A,$\hat{\mathbf C}\setminus B$), respectively. We have shown in our former paper [4] the existence of pasting arc γ of any one of the above three types but that of supercritical and critical type was only shown under the additional requirment on A and B that A and B are symmetric about a common straight line simultaneously. The purpose of the present paper is to show that in the above mentioned result the additional symmetry assumption is redundant: we will show the existence of supercritical and hence of critical arc γ starting from an arbitrarily given point in $\hat{\mathbf C}\setminus A \cup B$ for any general admissible pair of A and B without any further requirment whatsoever.

The study of approximation properties of the periodic functions in Lp(p ≥ 1)-spaces, in general and in Lipschitz classes Lip α, Lip (α, p), Lip (ξ(t), p) and weighted Lipschitz class W(Lp, ω(t), β), in particular, through trigonometric Fourier series, although is an old problem and known as Fourier approximation in the existing literature, has been of a growing interests over the last four decades due to its application in filters and signals [E. Z. Psarakis and G. V. Moustakides, An L2-based method for the design of 1-D zero phase FIR digital filters, IEEE Trans. Circuits Systems I Fundam. Theory Appl., 44(7) (1997) 551–601]. The most common methods used for the determination of the degree of approximation of periodic functions are based on the minimization of the Lp-norm of f(x) - Tn(x), where Tn(x) is a trigonometric polynomial of degree n and called approximant of the function f. In this paper, we discuss the approximation properties of the periodic functions in the Lipschitz classes Lip α and W(Lp, ω(t), β), p ≥ 1 by a trigonometric polynomial generated by the product matrix means of the Fourier series associated with the function. The degree of approximation obtained in our theorems of this paper is free from p and sharper than earlier results.

This paper proves that two possible notions of Stein fillability for a contact structure are the same. A contact geometer typically says a contact manifold $(M,\xi)$(M,ξ) is Stein fillable if $M$M is the noncritical level set of a proper plurisubharmonic function $f\colon X\to [0,\infty)$f:X→[0,∞) and $\xi$ξ is the set of complex tangencies to the level set $M.$M. This is a very useful notion in contact geometry and implies strong geometric properties for the contact structure if $M$M is 3-dimensional. (In particular, the contact structure will be tight. Moreover, many tight contact structures are constructed by constructing a Stein filling of a 3-manifold.) There is another, "more intrinsic'', notion of Stein fillability. Suppose $X$X is a Stein manifold and $X$X is the interior of a closed manifold $\overline{X}$X−− such that $\overline{X}=M\cup X$X−−=M∪X and the complex structure on $X$X extends to one on $\overline{X}. $X−−. Then we can say the contact manifold $(M,\xi)$(M,ξ) is filled by $X$X, where $\xi$ξ is the set of complex tangencies to $M.$M. In this case the authors say that $M$M is the "intrinsic'' boundary of $X.$X. The main result of this paper shows the equivalence of these two notions. The first notion obviously implies the second, but the converse is non-obvious. Along the way the authors prove several results. In particular, they show that if $(M,\xi)$(M,ξ) is a 3-dimensional Stein fillable contact manifold then $M$M embeds in complex 4-space such that $\xi$ξ is the set of complex tangencies to the embedding. They also show that if $M$M is an intrinsic boundary of $X$X then $\overline{X}$X−− embeds as a domain in a larger open complex manifold with strictly pseudoconvex boundary $M.$M. However, the germ of this extension of $\overline{X}$X−− is not unique.

A classical result of A.~D. Alexandrov states that a connected compact smooth $n$-dimen\-sional manifold without boundary, embedded in $ {\mathbb R}^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane $X_{n+1}={\rm const}$ in case $M$ satisfies: for any two points $(X', X_{n+1})$, $(X', \widehat X_{n+1})$ on $M$, with $X_{n+1}>\widehat X_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$. Some variations of the Hopf Lemma are also presented. Part II [Y.Y. Li and L. Nirenberg, Chinese Ann. Math. Ser. B 27 (2006), 193--218] deals with corresponding higher dimensional problems. Several open problems for higher dimensions are described in this paper as well.

Let $z=(z_1, ..., z_n)$ and $\Delta=\sum_{i=1}^n \fr {\p^2}{\p z^2_i}$ the Laplace operator. The main goal of the paper is to show that the well-known Jacobian conjecture without any additional conditions is equivalent to the following what we call {\it vanishing conjecture}: for any homogeneous polynomial $P(z)$ of degree $d=4$, if $\Delta^m P^m(z)=0$ for all $m \geq 1$, then $\Delta^m P^{m+1}(z)=0$ when $m>>0$, or equivalently, $\Delta^m P^{m+1}(z)=0$ when $m> \fr 32 (3^{n-2}-1)$. It is also shown in this paper that the condition $\Delta^m P^m(z)=0$ ($m \geq 1$) above is equivalent to the condition that $P(z)$ is Hessian nilpotent, i.e. the Hessian matrix $\Hes P(z)=(\fr {\p^2 P}{\p z_i\p z_j})$ is nilpotent. The goal is achieved by using the recent breakthrough work of M. de Bondt, A. van den Essen \cite{BE1} and various results obtained in this paper on Hessian nilpotent polynomials. Some further results on Hessian nilpotent polynomials and the vanishing conjecture above are also derived., Comment: Latex, 34 pages

In a previous paper with Schmid (math.NT/0402382) we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound sum_{n < T} a_n exp(2 pi i n alpha) = O_\epsilon (T^{3/4+\epsilon}) uniformly for alpha real, for a_n the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L-functions., Comment: 33 pages; v2 has minor revisions; v3 is 35 pages, has a revised title, and a lengthened introduction

This paper contains four main ideas. First, it shows global existence for the steepest descent in the uniformly convex setting. Secondly, it shows existence of critical points for convex functions defined on uniformly convex spaces. Thirdly, it shows an isomorphism between the dual space of H^{1,p}[0,1] and the space H^{1,q}[0,1] where p > 2 and {1/p} + {1/q} = 1. Fourthly, it shows how the Beurling-Denny theorem can be extended to find a useful function from H^{1,p}[0,1] to L_{p}[1,0] where p > 2 and addresses the problem of using that function to establish a relationship between the ordinary and the Sobolev gradients. The paper contains some numerical experiments and two computer codes.

In this paper we give a description of hypersurfaces with AK(S) = C.Let X be an affine variety and let G(X) be the group generated by all C + -actions onX. Then AK(X) ⊂ O(X) is the subring of all regular G(X)− invariant functions onX. We give here a description of affine surfaces S with AK(S) ∼= C. We show thatif AK(S) = C then S is quasihomogeneous and so may be obtained from a smoothrational projective surface by deleting a divisor of special form, which is called a“zigzag”. We denote by A the set of all such surfaces, and by H those which haveonly three components in the zigzag. We prove that for a surface S with AK(S) ∼= Cthe following statements are equivalent: 1. S is isomorphic to a hypersurface; 2.S is isomorphic to a hypersurface S ′ = {(x,y,z) ∈ C 3 |xy = p(z)}, where p is apolynomial with simple roots only; 3. S admits a fixed-point free C + - action; 4.S ∈ H. Moreover, if S 1 ∈ H and S 2 ∈ A\H, then S 1 ×C k ≃ S 2 ×C k for any k ∈ N. 1. Introduction.In this paper we proceed with our research of the smooth surfaces with C

In this paper, we study the heat equation on a homogeneous graph, relative to the natural (nearest–neighbour) Laplacian. We find pointwise estimates for the heat and resolvent kernels, and the Lp-Lq mapping properties of the corresponding operators. In recent years, random walks on graphs have been studied as analogues of diffusions on manifolds (see, e.g., T. Coulhon [Cn], M. Kanai [K1], [K2], and N. Th. Varopoulos [V1], [V2]). More recently, I. Chavel and E. A. Feldman [CF] and M. M. H. Pang [P] considered the semigroup (e)t>0. In this paper, we consider the heat semigroup associated to the canonical Laplace operator on a homogeneous tree, which is perhaps the basic example of a graph where the cardinality of the “ball of radius r” grows exponentially in r. We study the properties of the heat and resolvent operators, and discover very close analogies with diffusions on hyperbolic spaces, in line with the philosophy of the above-mentioned authors. A homogeneous tree X of degree q + 1 is defined to be a connected graph with no loops, in which every vertex is adjacent to q + 1 other vertices. We denote by d the natural distance on X, d(x, y) being the number of edges between the vertices x and y, and by L(X) the Lebesgue space with respect to counting measure on X. On X the canonical “Laplace operator” L is defined thus: Lf(x) = f(x)− 1 q + 1 ∑ y∈X d(x,y)=1 f(y) ∀x ∈ X; it is easily seen to be bounded on L(X) for every p in [1,∞], and self–adjoint on L(X). Denote by σ2(L) the L(X) spectrum of L, and by Pλ the associated spectral resolution of the identity, such that Lf = ∫ σ2(L) λdPλf ∀f ∈ L(X), and denote inf σ2(L) by b2. For θ in [0, 1] and α with Re (α) > 0, the heat operator and the (θ, α)-resolvent of L are the operators defined by the formulae Htf = ∫ σ2(L) e−tλ dPλf ∀t ∈ R, Received by the editors October 4, 1996. 1991 Mathematics Subject Classification. Primary 43A85; Secondary 20E08, 43A90, 22E35.

A polar space S with point set P is laxly embedded in the projective space PG(d, q), d ≥ 2, if the following conditions are satisfied : (i) P is a point set of PG(d, q) which generates PG(d, q), and (ii) each line L of S is a subset of a line L′ of PG(d, q), and distinct lines L1, L2 of S define distinct lines L1, L2 of PG(d, q). In this paper we determine all polar spaces of rank at least three which are laxly embedded in PG(d, q), where d ≥ 4 if S is isomorphic to the polar space W (2m + 1, s),m ≥ 2 and s odd, arising from a symplectic polarity in PG(2m + 1, s), and where d ≥ 3 in all other cases. Laxly embedded generalized quadrangles were considered in a foregoing paper. 1991 Mathematics Subject Classification: 51A50.

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a {\sf generalized polarized variety}. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of paper by Ye-Zhang, Fujita, Andreatta-Ballico-Wisniewski. If $K_X+detE$ is nef, then by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singualarities. In particular this gives the Mukai's conjecture1 for singular varieties. We consider also the case in which $dimF = r$ for every fibers and $\pi$ is birational. Hard copies of the paper are available., Comment: 18 pages, LateX

The aim of this paper is to prove a pointwise estimate which plays a fundamental role in the analysis of both linear and nonlinear partial differential equations arising from a system of non-commuting vector fields X = {X1, . . . ,Xm}. Although our result holds in a larger setting, for the unity of presentation we confine ourselves to the specific case of smooth vector fields X1, . . .Xm satisfying Hormander’s finite rank condition [H]: Rank (Lie[X1, . . . ,Xm ])(x ) = n for every x ∈ n . Let d : n × n → + be the Carnot-Caratheodory metric associated to X1, . . . ,Xm , and for x0 ∈ n and R > 0 set B = Bd (x0,R) = {y ∈ n |d (x0, y) 0 the symbol aB will denote the concentric ball Bd (x0, aR). For a given function u we let Xu = (X1u, . . . ,Xmu) and |Xu|2 = ∑m j =1(Xj u) 2. Throughout the paper we will use the standard notation uB = 1 |B | ∫ B u(y)dy . Our main result is Theorem 1.1. Let U ⊂⊂ n , and x0 ∈ U . There exist C , R0 > 0 and a > 1 depending only on U and the vector fields X1, . . . ,Xm such that for any u ∈ C 1(aB ), R ≤ R0, and x ∈ B one has |u(x ) − uB | ≤ C ∫

A celebrated theorem in operator theory is A. Beurling's description of the invariant subspaces in $H^2$ in terms of inner functions [Acta Math. {\bf81} (1949), 239--255; MR0027954 (10,381e)]. To do the same thing for the Bergman space $L^2_a$ has been deemed virtually impossible by many analysts, in view of the fact that the lattice of invariant subspaces is so large, and that the invariant subspaces may have weird properties as viewed from the $H^2$ perspective. The size of the lattice can be appreciated from the known fact that essentially every operator on separable Hilbert space can be realized as the compression of the Bergman shift on $M\ominus N$, where $M$ and $N$ are invariant subspaces, $N\subset M$. But a Beurling-type theorem is precisely what the present paper delivers. Given an invariant subspace $M$ in $L^2_a$, consider the subspace $M\ominus TM$, where $T$ stands for multiplication by $z$. This makes sense because $TM$ is a closed subspace of $M$. In Beurling's $H^2$ case, $M\ominus TM$ is one-dimensional and spanned by an inner function. In the $L^2_a$ setting, the dimension of $M\ominus TM$ may be arbitrarily large, even infinite. However, with the correct analogous definition of inner functions in $L^2_a$, all vectors of unit norm in\break $M\ominus TM$ are $L^2_a$-inner. Following Halmos, the subspace $M\ominus TM$ is called the wandering subspace of $M$. Given an invariant subspace, a natural question is: which collections of elements generate it? In particular, one can ask for the least number of elements in a set of generators. It is known that the dimension of the wandering subspace represents a lower bound for the least number of generators. In the paper, it is shown that any orthonormal basis in the wandering subspace (which then consists of $L^2_a$-inner functions) generates $M$ as an invariant subspace. This settles the issue of the minimal number of generators. Let $P$ be the orthogonal projection $M\to M\ominus TM$, and let $L\colon M\to M$ be the operator such that $TL$ is the orthogonal projection $M\to TM$. Then, for $f\in M$, $f=Pf+TLf$. If we do the same for $Lf\in M$, we get $Lf=PLf+TL^2f$ and, inserting it into the original relation for $f$, we get $f=Pf+TPLf+T^2L^2f$. As we go on repeating this process, we get $f=Pf+TPLf+T^2PL^2f+\cdots+T^{n-1}PL^{n-1}f+T^nL^nf$. The point with this decomposition is that, apart from the last term, each term is of the form $T$ to some power times an element of $M\ominus TM$, so that $Pf+TPLf+T^2PL^2f+\cdots+T^{n-1}PL^{n-1}f$ is in\break $[M\ominus TM]$, the invariant subspace generated by $M\ominus TM$. If the operators $T^nL^n$ happened to be uniformly bounded, as they are in the case of $H^2$, $T^nL^nf$ would tend to $0$ in the weak topology, and $f$ would be in the weak closure of $[M\ominus TM]$, which by standard functional analysis coincides with $[M\ominus TM]$. However, for the Bergman space, it seems unlikely that the $T^nL^n$ are uniformly bounded for all possible invariant subspaces $M$, although no immediate counterexample comes to mind. For this reason, the authors try Abel summation instead, and consider for $0

This paper is an attempt to understand the 2-plane bundle case for the converse of the soul theorem due to J. Cheeger and D Gromoll. It is shown that there is a class of 2-plane bundles over certain 52-bundles that carry complete metrics of nonnegative sectional curvature. In particular, every 2plane bundle and every S1 -bundle over the connected sum CPnφCPn of CPn with a negative CPn carries a 2-parameter family of complete metrics with nonnegative sectional curvature. A complete noncompact Riemannian manifold with nonnegative curvature (K > 0) is diffeomorphic to the normal bundle of a closed totally geodesic submanifold of the noncompact manifold according to a fundamental theorem due to J. Cheeger and D. Gromoll [4]. They also proposed the following problem: Which vector bundles over closed manifolds with K > 0 carry complete metrics with K > 0? This paper is an attempt to understand the 2-plane bundle case. For a class of 2-plane bundles and S^-bundles, we show that there exist families of complete metrics with K > 0. Our main result is the following Theorem. Every 2-plane bundle and every S1-bundle over the connected sum CP n #CP of CP n with a negative CPn carries a 2-parameter family of complete metrics with K > 0. More generally, let (£?, h) be a Hodge manifold with positive Kahlerian curvature and let M be an associated S2bundle over B. Then, there is a two dimensional subspace H2(M), generated by two integral cohomology classes, in the cohomology group H*(M) such that every principal S1-bundle and every 2-plane bundle over M supported by an Euler class in H 2{M) carries a 2-parameter family of complete metrics with K>0.

Let $d\geq 2$ be an integer and let $2d/(d-1) < q \leq \infty$. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality \begin{equation*} \big\|\widehat{f\sigma}\big\|_{L^q_{{\rm rad}}L^2_{{\rm ang}}(\mathbb{R}^d)} \leq {\bf C}_{d,q}\, \|f\|_{L^2(\mathbb{S}^{d-1},{\rm d}\sigma)}, \end{equation*} established by L. Vega in 1988. Letting $\mathcal{A}_d \subset (2d/(d-1), \infty]$ be the set of exponents for which the constant functions on $\mathbb{S}^{d-1}$ are the unique extremizers of this inequality, we show that: (i) $\mathcal{A}_d$ contains the even integers and $\infty$; (ii) $\mathcal{A}_d$ is an open set in the extended topology; (iii) $\mathcal{A}_d$ contains a neighborhood of infinity $(q_0(d), \infty]$ with $q_0(d) \leq \left(\tfrac{1}{2} + o(1)\right) d\log d$. In low dimensions we show that $q_0(2) \leq 6.76\,;\,q_0(3) \leq 5.45 \,;\, q_0(4) \leq 5.53 \,;\, q_0(5) \leq 6.07$. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions., Comment: 21 pages

Let ${\cal S}{\cal U}(r, L_0)$ denote the moduli space of semi stable vector bundles of rank $r$ and fixed determinant $L_0$ of degree $d$ on a smooth curve $C$ of genus $g \geq 3$. In this paper we describe the group of automorphisms of $ {\cal S}{\cal U}(r, L_0) $. The analogue of this result is carried out for the space ${\cal U}(r,d) $ of semi stable vector bundles of rark $r$ and degree $d$. As an application of the technics we use, we give in the appendix at the end of the paper a proof of the Torelli theorem for the moduli spaces ${\cal S}{\cal U}(r, L_0) $ for any rank $r$ and degree $d$., 48 pages, LaTeX

Let Φ be an irreducible root system (sometimes we denote Φ by Φ(X) to indicate its type X). Choose a simple root system Π in Φ. Let Φ+ (resp. Φ~) be the corresponding positive (resp. negative) root system of Φ. By a subsystem Φ' of Φ (resp. of Φ + ), we mean that Φ' is a subset of Φ (resp. of Φ+) which itself forms a root system (resp. a positive root system). We refer the readers to Bourbaki's book for the detailed information about root systems. Among all subsystems of Φ, the subsystems of Φ of rank 2 and of type Φ A\ x A\ are of particular importance in the theory of Weyl groups and affine Weyl groups (see the papers by Jian-yi Shi). In the present paper, we shall compute the number of such subsystems of Φ for an irreducible root system Φ of any type. Some interesting properties of Φ are also obtained. 1. The number h(a). Let ( , ) be an inner product of the euclidean space E spanned by Φ. For any a e Φ, we denote by |α| the length of α, by α v the dual root 2α/(α, α) of a and by sa the reflection in E which sends any vector v e E to sa(υ) = v - (v, α v )α. For a, β G Φ, we write a (α)nΦ+ and D~(a) = D(α) n Φ" . Let d(a) be the cardinality of the set Z>+(α). Also, we denote by ht(α) the height of α, i.e. ht(α) = Σβenaβ = Σβenaββ with aβ e z

w 1. Introduction In this paper we show that the Bergman projection operator for certain smooth bounded pseudoconvex domains does not preserve smoothness as measured by Sobolev norms. Let Q be a smooth bounded pseudoconvex domain in C" and let p~t) denote the orthogonal projection from L2(f~) onto the Bergman subspace B(ff~)=L2(~) f3 r with respect to the weighted norm I{fll~ J'olf(z)l 2 e-t{lzll2dV)l/2. Let wk(t)) denote the Sobolev space consisting of functions whose derivatives of order ~ -to(k, g2). On the other hand, there is a large collection of results implying that for certain types of domains the unweighted Bergman projection p=pr preserves W k for all k~0. (See [FK] for the strictly pseudoconvex case; for results on weakly pseudoconvex domains the reader may consult [Ko2], [Ca], [Si] and the recent [BStl], [Ch] as well as the references cited therein. Most of these results are focused on the a-Neumann operator rather than P; see [BSt2] for the connection. Also, except for [Kol], positive results in this area are typically valid for any choice of smooth positive weight function on 0.) The question of whether or not P is similarly well-behaved for all weakly pseudoconvex domains has remained open for many years. In this paper we show that this is not the case; in fact when Q is the so-called "worm domain" of Diederich and Forna~ss then P does not map W* to W k when k>~zr/(total amount of winding). This latter quantity is explained in section 4 below, where the construction of the worm is reviewed and the main result is proved. The proof depends on computations for a piecewise Levi-flat model domain depending in turn on certain one-dimensional computations; these are treated in sections 3 and 2, respectively. Section 5 contains additional remarks and questions.

In this paper we have studied the Chebyshev and interpolation errors for functions in $\Cal C(E)$, the normed algebra of analytic functions on a compact set $E$ of positive transfinite diameter. The $(p,q)$-order and generalized $(p,q)$-type have been characterized in terms of these approximation errors. Finally, we have obtained a saturation theorem for $f\in\Cal C(E)$ which can be extended to an entire function of $(p,q)$-order 0 or 1 and for entire functions of minimal generalized $(p,q)$-type.

In this paper, we study general properties of $\alpha$-localized wavelets and multiresolution analyses, when $\frac12 < \alpha\leq\infty$. Related to the latter, we improve a well-known result of A. Cohen by showing that the correspondence $m\mapsto{\widehat\varphi}=\prod_1^\infty m(2^{-j}\cdot)$, between low-pass filters in $H^{\alpha}(\Bbb T)$ and Fourier transforms of $\alpha$-localized scaling functions (in $H^{\alpha}(\Bbb R)$), is actually a homeomorphism of topological spaces. We also show that the space of such filters can be regarded as a connected infinite dimensional manifold, extending a theorem of A. Bonami, S. Durand and G. Weiss, in which only the case $\alpha=\infty$ is treated. These two properties, together with a careful study of the "phases" that give rise to a wavelet from the MRA, will allow us to prove that the space ${\Cal W}_\alpha$, of $\alpha$-localized wavelets, is arcwise connected with the topology of $L^2((1+|x|^2)^{\alpha}\,dx)$ (modulo homotopy classes). This last result is new even for the case $\alpha=\infty$, as well as the considerations about the "homotopy degree" of a wavelet.

The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by "regular mappings" (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called $(s,t)$-regular mappings. These mappings are the same as ordinary regular mappings when $s = t$, but otherwise they behave somewhat like projections. In particular, they can map sets with Hausdorff dimension $s$ to sets of Hausdorff dimension $t$. We mostly consider the case of mappings between Euclidean spaces, and show in particular that if $f\colon {\mathbf R}^s\to {\mathbf R}^n$ is an $(s,t)$-regular mapping, then for each ball $B$ in ${\mathbf R}^s$ there is a linear mapping $\lambda \colon {\mathbf R}^s\to {\mathbf R}^{s-t}$ and a subset $E$ of $B$ of substantial measure such that the pair $(f,\lambda )$ is bilipschitz on $E$. We also compare these mappings in comparison with "nonlinear quotient mappings" from [6].

In this paper we illustrate the explicit implementation of a method for computing limit cycles which bifurcate from a 2-dimensional isochronous set contained in ℝ3, when we perturb it inside a class of differential systems. This method is based in the averaging theory. As far as we know all applications of this method have been made perturbing noncompact surfaces, as for instance a plane or a cylinder in ℝ3. Here we consider polynomial perturbations of degree d of an isochronous torus. We prove that, up to first order in the perturbation, at most 2(d+1) limit cycles can bifurcate from a such torus and that there exist polynomial perturbations of degree d of the torus such that exactly ν limit cycles bifurcate from such a torus for every ν ∈ {2, 4,...,2(d + 1)}.

In this paper we introduce the one-sided weighted spaces ${\mathcal L}^{-}_w (\beta)$, $-1 < \beta < 1$. The purpose of this definition is to obtain an extension of the Weyl fractional integral operator $I_{\alpha}^+$ from $L^p_w$ into a suitable weighted space. Under certain condition on the weight $w$, we have that ${\mathcal L}^{-}_w (0)$ coincides with the dual of the Hardy space $H_{-}^1(w)$. We prove for $0

In this paper we study free boundary regularity in a parabolic two-phase problem below the continuous threshold. We consider unbounded domains $\Omega\subset\mathsf{R}^{n+1}$ assuming that $\partial\Omega$ separates $\mathsf{R}^{n+1}$ into two connected components $\Omega^1=\Omega$ and $\Omega^2=\mathsf{R}^{n+1}\setminus\overline\Omega$. We furthermore assume that both $\Omega^1$ and $\Omega^2$ are parabolic NTA-domains, that $\partial\Omega$ is Ahlfors regular and for $i\in\{1,2\}$ we define $\omega^i(\hat{X}^i,\hat{t}^i,\cdot)$ to be the caloric measure at $(\hat{X}^i,\hat{t}^i)\in \Omega^i$ defined with respect to $\Omega^i$. In the paper we make the additional assumption that $\omega^i(\hat{X}^i,\hat{t}^i,\cdot)$, for $i\in\{1,2\}$, is absolutely continuous with respect to an appropriate surface measure $\sigma$ on $\partial\Omega$ and that the Poisson kernels $k^i(\hat{X}^i,\hat{t}^i,\cdot)=d\omega^i(\hat{X}^i,\hat{t}^i,\cdot)/d\sigma$ are such that $\log k^i(\hat{X}^i,\hat{t}^i,\cdot)\in \mathrm{VMO}(d\sigma)$. Our main result (Theorem 1) states that, under these assumptions, $C_r(X,t)\cap\partial\Omega$ is Reifenberg flat with vanishing constant whenever $(X,t)\in\partial\Omega$ and $\min\{\hat{t}^1,\hat{t}^2\}>t+4r^2$. This result has an important consequence (Theorem 3) stating that if the two-phase condition on the Poisson kernels is fulfilled, $\Omega^1$ and $\Omega^2$ are parabolic NTA-domains and $\partial\Omega$ is Ahlfors regular then if $\Omega$ is close to being a chord arc domain with vanishing constant we can in fact conclude that $\Omega$ is a chord arc domain with vanishing constant.

In this paper, we classify all solutions ofwith the finite conditionsHere c is a positive number and β > −1.

In this paper, we study the existence of entropy solutions for some nonlinear $p(x)-$elliptic equation of the type $$Au - \mbox{div }\phi(u) + H(x,u,\nabla u) = \mu,$$ where $A$ is an operator of Leray-Lions type acting from $W_{0}^{1,p(x)}(\Omega)$ into its dual, the strongly nonlinear term $H$ is assumed only to satisfy some nonstandard growth condition with respect to $|\nabla u|,$ here $\>\phi(\cdot)\in C^{0}(I\!\!R,I\!\!R^{N})\>$ and $\mu$ belongs to ${\mathcal{M}}_{0}^{b}(\Omega)$.

In this paper, the authors study a complete moment convergence result for Sung?s type weighted sums of ?*-mixing random variables. This result extends and improves the corresponding theorem of Sung [S.H. Sung, Complete convergence for weighted sums of ?*-mixing random variables, Discrete Dyn. Nat. Soc. 2010 (2010), Article ID 630608, 13 pages].

In this paper, we prove the local converse theorem for $${\mathrm {Sp}}_{2r}(F)$$ over a p-adic field F. More precisely, given two irreducible supercuspidal representations of $${\mathrm {Sp}}_{2r}(F)$$ with the same central character such that they are generic with the same additive character and they have the same gamma factors when twisted with generic irreducible representations of $${\mathrm {GL}}_n(F)$$ for all $$1\le n\le r$$ , then these two representations must be isomorphic. Our proof is based on the local analysis of the local integrals which define local gamma factors. A key ingredient of the proof is certain partial Bessel function property developed by Cogdell–Shahidi–Tsai recently. The same method can give the local converse theorem for $${\mathrm {U}}(r,r)$$ .

Let $K$ be the Calderon-Zygmund convolution kernel on $\mathbb{R}^d~(d\geq2)$. Christ and Journe defined the commutator associated with $K$ and $a\in~L^\infty(\mathbb{R}^d)$ by which is an extension of the classical Calderon commutator.In this paper, we show that$T_a$ is weighted weak type $(1,1)$ bounded with $A_1$ weight for $d\geq2$.

A space $$ Y $$ is called an extension of a space $$ X $$ if $$ Y $$ contains $$ X $$ as a dense subspace. An extension $$ Y $$ of $$ X $$ is called a one-point extension of $$ X $$ if $$ Y\backslash X $$ is a singleton. In present paper, we extend the well-known and important fact “any locally compact Hausdorff non-compact space $$ X $$ has a one-point compact Hausdorff extension”, which was proved by Alexandroff, to more general topological space.

Chow and Lou in 2003 had shown that the analogue of the Hamilton Ricci flow on surfaces in the combinatorial setting converges to the Thruston’s circle packing metric. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper of Chow and Lou was that the weights are nonnegative. We show that the same results on convergence of Ricci flow can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities. As a consequence we obtain theorem of existence of Thurston’s circle packing metric for a wider range of weights.

In this paper we prove that the Hurwitz space $$\mathcal {H}_{9,8}$$ , which parameterizes 8-sheeted covers of $${\mathbb P }^1$$ by curves of genus 9, is unirational. Our construction leads to an explicit Macaulay2 code, which will randomly produce a nodal curve of degree 8 of geometric genus 9 with 12 double points and together with a pencil of degree 8.

Let F be an NWUE distribution with mean 1 and G be the stationary renewal distribution of F. We would expect G to converge in distribution to the unit exponential distribution as its mean goes to 1. In this paper, we derive sharp bounds for the Kolmogorov distance between G and the unit exponential distribution, as well as between G and an exponential distribution with the same mean as G. We apply the bounds to geometric convolutions and to first passage times.

For fixed real numbers $$c>0,$$ $$\alpha >-\frac{1}{2},$$ the finite Hankel transform operator, denoted by $$\mathcal {H}_c^{\alpha }$$ is given by the integral operator defined on $$L^2(0,1)$$ with kernel $$K_{\alpha }(x,y)= \sqrt{c xy} J_{\alpha }(cxy).$$ To the operator $$\mathcal {H}_c^{\alpha },$$ we associate a positive, self-adjoint compact integral operator $$\mathcal Q_c^{\alpha }=c\, \mathcal {H}_c^{\alpha }\, \mathcal {H}_c^{\alpha }.$$ Note that the integral operators $$\mathcal {H}_c^{\alpha }$$ and $$\mathcal Q_c^{\alpha }$$ commute with a Sturm-Liouville differential operator $$\mathcal D_c^{\alpha }.$$ In this paper, we first give some useful estimates and bounds of the eigenfunctions $$\varphi ^{(\alpha )}_{n,c}$$ of $$\mathcal H_c^{\alpha }$$ or $$\mathcal Q_c^{\alpha }.$$ These estimates and bounds are obtained by using some special techniques from the theory of Sturm-Liouville operators, that we apply to the differential operator $$\mathcal D_c^{\alpha }.$$ If $$(\mu _{n,\alpha }(c))_n$$ and $$\lambda _{n,\alpha }(c)=c\, |\mu _{n,\alpha }(c)|^2$$ denote the infinite and countable sequence of the eigenvalues of the operators $$\mathcal {H}_c^{(\alpha )}$$ and $$\mathcal Q_c^{\alpha },$$ arranged in the decreasing order of their magnitude, then we show an unexpected result that for a given integer $$n\ge 0,$$ $$\lambda _{n,\alpha }(c)$$ is decreasing with respect to the parameter $$\alpha .$$ As a consequence, we show that for $$\alpha \ge \frac{1}{2},$$ the $$\lambda _{n,\alpha }(c)$$ and the $$\mu _{n,\alpha }(c)$$ have a super-exponential decay rate. Also, we give a lower decay rate of these eigenvalues. As it will be seen, the previous results are essential tools for the analysis of a spectral approximation scheme based on the eigenfunctions of the finite Hankel transform operator. Some numerical examples will be provided to illustrate the results of this work.

This paper deals with an initial-boundary value problem for the chemotaxis-(Navier–)Stokes system $$\begin{aligned} \left\{ \begin{array}{lcll} n_t + u\cdot \nabla n &{}=&{} \Delta n - \nabla \cdot (n\nabla c), \qquad &{}\quad x\in \Omega , \ t>0, \\ c_t + u\cdot \nabla c &{}=&{}\Delta c - nc, \qquad &{}\quad x\in \Omega , \ t>0, \\ u_t + \kappa (u\cdot \nabla )u &{}=&{} \Delta u - \nabla P + n\nabla \phi , \qquad &{}\quad x\in \Omega , \ t>0, \\ \nabla \cdot u =0, &{} &{} \qquad &{}\quad x\in \Omega , t>0, \end{array} \right. \end{aligned}$$ in a bounded convex domain $$\Omega \subset \mathbb {R}^2$$ with smooth boundary, with $$\kappa \in \mathbb {R}$$ and a given smooth potential $$\phi :\Omega \rightarrow \mathbb {R}$$ . It is known that for each $$\kappa \in \mathbb {R}$$ and all sufficiently smooth initial data this problem possesses a unique global classical solution $$(n^{(\kappa )},c^{(\kappa )},u^{(\kappa }))$$ . The present work asserts that these solutions stabilize to $$(n^{(0)},c^{(0)},u^{(0)})$$ uniformly with respect to the time variable. More precisely, it is shown that there exist $$\mu >0$$ and $$C>0$$ such that whenever $$\kappa \in (-1,1)$$ , $$\begin{aligned}&\Big \Vert n^{(\kappa )}(\cdot ,t)-n^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} + \Big \Vert c^{(\kappa )}(\cdot ,t)-c^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \\&\quad +\, \Big \Vert u^{(\kappa )}(\cdot ,t)-u^{(0)}(\cdot ,t)\Big \Vert _{L^\infty (\Omega )} \le C |\kappa | e^{-\mu t} \end{aligned}$$ for all $$t>0$$ . This result thereby provides an example for a rigorous quantification of stability properties in the Stokes limit process, as frequently considered in the literature on chemotaxis-fluid systems in application contexts involving low Reynolds numbers.

A function f meromorphic on ℂ\ℝ is said to be in the generalized Nevanlinna class N κ (κ ϵ ℤ+), if f is symmetric with respect to ℝ and the kernel $$ {\mathbf{N}}_{\omega }(z)\coloneq \frac{f(z)-\overline{f\left(\omega \right)}}{z-\overline{\omega}} $$ has κ negative squares on ℂ+. The generalized Stieltjes class $$ {\mathbf{N}}_{\kappa}^k\left(\kappa, k\in {\mathrm{\mathbb{Z}}}_{+}\right) $$ is defined as the set of functions f ϵ N κ such that z f ϵ N k . The full indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ consists in the following: Given κ, k ϵ ℤ+, and a sequence $$ \mathbf{s}={\left\{{s}_i\right\}}_{i=0}^{\infty } $$ of real numbers, to describe the set of functions $$ f\in {\mathbf{N}}_{\kappa}^k $$ , which satisfy the asymptotic expansion $$ f(z)=-\frac{s_0}{z}-\cdots -\frac{s_2n}{z^{2n+1}}+o\left(\frac{1}{z^{2n+1}}\right)\kern1em \left(z=-y\in {\mathrm{\mathbb{R}}}_{-},y\uparrow \infty \right) $$ for all n big enough. In the present paper, we will solve the indefinite Stieltjes moment problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ within the M. G. Krein theory of u-resolvent matrices applied to a Pontryagin space symmetric operator A [0;N] generated by $$ {\mathfrak{J}}_{\left[0;N\right]} $$ . The u-resolvent matrices of the operator A [0;N] are calculated in terms of generalized Stieltjes polynomials, by using the boundary triple’s technique. Some criteria for the problem $$ {MP}_{\kappa}^k\left(\mathbf{s}\right) $$ to be solvable and indeterminate are found. Explicit formulae for Pade approximants for the generalized Stieltjes fraction in terms of generalized Stieltjes polynomials are also presented.

In this paper, the authors introduce a generalized Sakaguchi type spiral-likefunction class S(\lambda, \beta, s, t) and obtain sharp upper bound to thesecond Hankel determinant |H_{2}(1)| for the function f in the above class.Relevances of the main result are also briefly indicated.

In this paper we complete the proof of the fact that the Hausdorff dimensions of graphs of density continuous functions vary continuously between one and two. This result was announced in our previous paper, but the proof there contained a gap and the construction given there should also be slightly modified. This correction is done in this paper.

In this paper we introduce the backward operator is $\nabla$ and study the notion of $\nabla-$ statistical convergence and $\nabla-$ statistical Cauchy sequence using by almost $\left(\lambda_{m}\mu_{n}\right)$ convergence in $\chi^{2}-$Riesz space and also some inclusion theorems are discussed.

We consider a one-parameter family of closed, embedded hypersurfaces moving with normal velocity $$G_\kappa = \big ( \sum _{i < j} \frac{1}{\lambda _i+\lambda _j-2\kappa } \big )^{-1}$$ , where $$\lambda _1 \le \cdots \le \lambda _n$$ denote the curvature eigenvalues and $$\kappa $$ is a nonnegative constant. This defines a fully nonlinear parabolic equation, provided that $$\lambda _1+\lambda _2>2\kappa $$ . In contrast to mean curvature flow, this flow preserves the condition $$\lambda _1+\lambda _2>2\kappa $$ in a general ambient manifold. Our main goal in this paper is to extend the surgery algorithm of Huisken–Sinestrari to this fully nonlinear flow. This is the first construction of this kind for a fully nonlinear flow. As a corollary, we show that a compact Riemannian manifold satisfying $$\overline{R}_{1313}+\overline{R}_{2323} \ge -2\kappa ^2$$ with non-empty boundary satisfying $$\lambda _1+\lambda _2 > 2\kappa $$ is diffeomorphic to a 1-handlebody. The main technical advance is the pointwise curvature derivative estimate. The proof of this estimate requires a new argument, as the existing techniques for mean curvature flow due to Huisken–Sinestrari, Haslhofer–Kleiner, and Brian White cannot be generalized to the fully nonlinear setting. To establish this estimate, we employ an induction-on-scales argument; this relies on a combination of several ingredients, including the almost convexity estimate, the inscribed radius estimate, as well as a regularity result for radial graphs. We expect that this technique will be useful in other situations as well.

For positive integers N and M, the general hypergeometric Cauchy polynomials c M,N,n (z) (M, N ≥ 1; n ≥ 0) are defined by $$\frac{1}{(1+t)^z} \frac{1}{{}_2F_1(M,N;N+1;-t)}=\sum_{n=0}^\infty c_{M,N,n}(z)\, \frac{t^n}{n!}\,, $$ where $${{}_2 F_1(a,b;c;z)}$$ is the Gauss hypergeometric function. When M = N = 1, c n = c 1,1,n are the classical Cauchy numbers. In 1875, Glaisher gave several interesting determinant expressions of numbers, including Bernoulli, Cauchy and Euler numbers. In the aspect of determinant expressions, hypergeometric Cauchy numbers are the natural extension of the classical Cauchy numbers, though many kinds of generalizations of the Cauchy numbers have been considered by many authors. In this paper, we show some interesting expressions of generalized hypergeometric Cauchy numbers. We also give a convolution identity for generalized hypergeometric Cauchy polynomials.

Let $$G=\mathrm{GL}_{2n}$$ over a totally real number field F and $$n\ge 2$$ . Let $$\Pi $$ be a cuspidal automorphic representation of $$G(\mathbb {A})$$ , which is cohomological and a functorial lift from SO $$(2n+1)$$ . The latter condition can be equivalently reformulated that the exterior square L-function of $$\Pi $$ has a pole at $$s=1$$ . In this paper, we prove a rationality result for the residue of the exterior square L-function at $$s=1$$ and also for the holomorphic value of the symmetric square L-function at $$s=1$$ attached to $$\Pi $$ . As an application of the latter, we also obtain a period-free relation between certain quotients of twisted symmetric square L-functions and a product of Gaus sums of Hecke characters.

The purpose of this paper is to give necessary and sufficient conditions for the $$L^p$$ -well-posedness (resp. $$B_{p,q}^s$$ -well-posedness) for the second order degenerate differential equation with finite delay: $$(Mu)''(t)=Au(t)+Gu'_t+Fu_t+f(t),(t\in [0,2\pi ])$$ with periodic boundary conditions $$Mu(0)=Mu(2\pi )$$ , $$(Mu)'(0)=(Mu)'(2\pi )$$ , where A, M are closed linear operators in a Banach space X satisfying $$D(A)\subset D(M)$$ , F and G are delay operators on $$L^p([-2\pi ,0];X)$$ (resp. $$B_{p,q}^s([-2\pi ,0];X)$$ ).

In this paper, we study the existence of positive solutions to the following Schrodinger system: $$\left\{ {\begin{array}{*{20}{c}} { - \Delta u + {V_1}\left( x \right)u = {\mu _1}\left( x \right){u^3} + \beta \left( x \right){v^2}u,x \in {\mathbb{R}^N},} \\ { - \Delta v + {V_2}\left( x \right)v = {\mu _2}\left( x \right){v^3} + \beta \left( x \right){u^2}v,x \in {\mathbb{R}^N},} \\ {u,v \in {H^1}\left( {{\mathbb{R}^N}} \right),} \end{array}} \right.$$ where N = 1; 2; 3; V1(x) and V2(x) are positive and continuous, but may not be well-shaped; and µ1(x), µ2(x) and β(x) are continuous, but may not be positive or anti-well-shaped. We prove that the system has a positive solution when the coefficients Vi(x), µi(x) (i = 1; 2) and β(x) satisfy some additional conditions.

This paper deals with two topics mentioned in the title. First, it is proved that function f in L p (∂D a ) can be decomposed into a sum g + h, where D a is an angular domain in the complex plane, g and h are the non-tangential limits of functions in H p (D a ) and $${H^p}\left( {\overline D _a^c} \right)$$ in the sense of L p (D a ), respectively. Second, the sufficient and necessary conditions between boundary values of holomorphic functions and distributions in n-dimensional complex space are obtained.