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The aim of the present paper is the computation of Green's functions for the powers Δm of the invariant Laplace operator on rank-one Hermitian symmetric spaces. Starting with the noncompact case, the unit ball in ℂd, we obtain a complete result for m= 1, 2 in all dimensions. For m≥ 3 the formulas grow quite complicated so we restrict ourselves to the case of the unit disc (d= 1) where we develop a method, possibly applicable also in other situations, for reducing the number of integrations by half, and use it to give a description of the boundary behaviour of these Green functions and to obtain their (multi-valued) analytic continuation to the entire complex plane. Next we discuss the type of special functions that turn up (hyperlogarithms of Kummer). Finally we treat also the compact case of the complex projective space ℙd (for d= 1, the Riemann sphere) and, as an application of our results, use eigenfunction expansions to obtain some new identities involving sums of Legendre (d= 1) or Jacobi (d > 1) polynomials and the polylogarithm function. The case of Green's functions of powers of weighted (no longer invariant, but only covariant) Laplacians is also briefly discussed.

Given function Ω on ℝn , we define the fractional maximal operator and the fractional integral operator by and respectively, where 0 < α < n. In this paper we study the weighted norm inequalities of MΩα and TΩα for appropriate α, s and A(p, q) weights in the case that Ω∈ Ls(Sn-1)(s> 1), homogeneous of degree zero.

This paper is a study of summability methods that are based on Dirichlet convolution. If f(n) is a function on positive integers and x is a sequence such that then x is said to be Af-summable to L. The necessary and sufficient condition for the matrix Af to preserve bounded variation of sequences is established. Also, the matrix Af is investigated as ℓ − ℓ and G − G mappings. The strength of the Af-matrix is also discussed.

In dealing with the spectral synthesis property for a plane curve with nonzero curvature, a key step is to have a uniform L∞ estimate for some smoothing operators related to the curve. In this paper, we will show that the same L∞ estimate holds true for a plane curve that may have zero curvature.

In this paper, we solve the-Neumann problem on (0, q) forms, 0 ≤ q ≤ n, in the strictly pseudoconvex non-isotropic Siegel domain:where aj> 0 for j = 1,2, . . . , n. The metric we use is invariant under the action of the Heisenberg group on the domain. The fundamental solution of the related differential equation is derived via the Laguerre calculus. We obtain an explicit formula for the kernel of the Neumann operator. We also construct the solution of the corresponding heat equation and the fundamental solution of the Laplacian operator on the Heisenberg group.

In this paper we introduce the concepts of hyperbolic and elliptic areas and prove uncountably many new geometric isoperimetric inequalities on the surfaces of constant curvature.

In this paper, we define the η-invariant for a cusped hyperbolic 3-manifold and discuss some of its applications. Such an invariant detects the chirality of a hyperbolic knot or link and can be used to distinguish many links with homeomorphic complements.

In this paper, we study the local bifurcations of critical periods in the neighborhood of a nondegenerate centre of the reduced Kukles system. We find at the same time the isochronous systems. We show that at most three local critical periods bifurcate from the Christopher-Lloyd centres of finite order, at most two from the linear isochrone and at most one critical period from the nonlinear isochrone. Moreover, in all cases, there exist perturbations which lead to the maximum number of critical periods. We determine the isochrones, using the method of Darboux: the linearizing transformation of an isochrone is derived from the expression of the first integral. Our approach is a combination of computational algebraic techniques (Gröbner bases, theory of the resultant, Sturm’s algorithm), the theory of ideals of noetherian rings and the transversality theory of algebraic curves.

For ameromorphic (or harmonic) function ƒ, let us call the dilation of ƒ at z the ratio of the (spherical)metric at ƒ(z) and the (hyperbolic)metric at z. Inequalities are knownwhich estimate the sup norm of the dilation in terms of its Lp norm, for p > 2, while capitalizing on the symmetries of ƒ. In the present paper we weaken the hypothesis by showing that such estimates persist even if the Lp norms are taken only over the set of z on which ƒ takes values in a fixed spherical disk. Naturally, the bigger the disk, the better the estimate. Also, We give estimates for holomorphic functions without zeros and for harmonic functions in the case that p = 2.

The aim of this paper is to derive the expansion of the following class of multi-dimensional integralswith respect to the large parameter λ when Ω is a subset of ℝn, a > 0, w is a strictly positive and bounded function on Σ and fp means an integration in the finite part sense of Hadamard (see Section 2). This is performed for weak assumptions bearing on pseudofunction K and by extending to higher dimensional cases the tools developed in the one-dimensional context. The range of applications of the proposed results is outlined by the exhibition of several examples.

We study the question of nonexistence of extremal functions for perturbations of some sharp inequalities such as those of Moser-Trudinger (1971) and Chang- Marshall (1985). We shall show that for each critically sharp (in a sense that will be precisely defined) inequality of the formwhere is a collection of measurable functions on a finite measure space (I, μ) and O a nonnegative continuous function on [0, ∞), we have a continuous Ψ on [0, ∞) with 0 ≤ Ψ ≤ Φ, but withnot being attained even if the supremum in (1) is attained. We then apply our results to the Moser-Trudinger and Chang-Marshall inequalities. Our result is to be contrasted with the fact shown by Matheson and Pruss (1994) that if Ψ(t) = o(Φ(t) as t —> ∞ then the supremum in (2) is attained. In the present paper, we also give a converse to that fact.

In 1975, L. Fejes Toth conjectured that in Ed, d ≥ 5, the sausage arrangement is denser than any other packing of n unit balls. This has been known if the convex hull Cn of the centers has low dimension. In this paper, we settle the case when the inner m-radius of Cn is at least O(ln d/m). In addition, we consider the extremal properties of finite ballpackings with respect to various intrinsic volumes.

In the context of a locally compact abelian group, we establish maximal theorem counterparts for weak type (1,1) multipliers of the classical de Leeuw theorems for individual strong multipliers. Special methods are developed to handle the weak type (1,1) estimates involved since standard linearization methods such as Lorentz space duality do not apply to this case. In particular, our central result is a maximal theorem for convolutions with weak type (1,1) multipliers which opens avenues of approximation. These results complete a recent series of papers by the authors which extend the de Leeuw theorems to a full range of strong type and weak type maximal multiplier estimates in the abstract setting.

Let M2n+1 be a compact contact manifold and 𝓐 the set of associated metrics. Using the scalar curvature R and the *-scalar curvature R*, in [5] we defined the "total scalar curvature", by and showed that the critical points of I(g) on 𝓐 are the K-contact metrics, i.e. metrics for which the characteristic vector field is Killing. In this paper we compute the second variation of I(g) and prove that the index of I(g) and of —I(g) are both positive at each critical point. As an application we show that the classical total scalar curvature A(g) = ∫M R dVg restricted to 𝓐 cannot have a local minimum at any Sasakian metric.

In an earlier paper we provided a counterexample to an old conjecture of Hopf. In this note we show that the "strong sweeping out property" obtains for the Hopf operators (Tt) both when t —> +∞ and when t —> 0+, that is a.e. convergence fails in the worst possible way.

Let M be a compact, connected, orientable, irreducible 3-manifold with a torus boundary. It is known that if two Dehn fillings on M along the boundary produce a reducible manifold and a manifold containing a Klein bottle, then the distance between the filling slopes is at most three. This paper gives a remarkably short proof of this result.

In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we make some remarks on connections between the mentioned property of a projective variety and its adjunction properties., 7 pages, Plain TeX Version 3.141

The main purpose of this paper is to investigate isotropic immersions with low codimensions into a real space form.

The paper is concerned with the comparison of two solutions for a one-dimensional stochastic partial differential equation. Noting that support compactness of solutions propagates with passage of time, we define the SCP property and show that the SCP property and the strong positivity are two contrasting properties of solutions for one-dimensional SPDEs, which are due to degeneracy of the noise-term coefficient

Let (M, °T″) be a compact strongly pseudo-convex CR-manifold with trivial canonical line bundle. Then, in [A-M2], a weak version of the Bogomolov type theorem for deformations of CR-structures was shown by an analogy of the Tian- Todorov method. In this paper, we show that: in the very strict sense, there is a counterexample.

In this paper, extending the results in [ 1 ], we establish a necessary and sufficient condition for oscillation in a large class of singular (i.e., the difference operator is nonatomic) neutral equations.

In this paper we study the local bifurcation of critical periods of periodic orbits in the neighborhood of a nondegenerate centre of a vector field with a homogeneous nonlinearity of the third degree. We show that at most three local critical periods bifurcate from a weak linear centre of finite order or from the linear isochrone and at most two local critical periods from the nonlinear isochrone. Moreover, in both cases, there are perturbations with the maximum number of critical periods.

A geometric characterization is given for a collection of points in ℝd to produce a singular l1 -distance matrix. Some quantitative results are established in terms of "characteristic matrices". The results in this paper generalize those of Dyn, Light and Cheney and have application to ridge function interpolation.

In this paper, we consider the oscillatory behavior of the second order neutral delay differential equationwhere t ≥ t0,T and σ are positive constants, a,p, q € C(t0, ∞), R),f ∊ C[R, R]. Some sufficient conditions are established such that the above equation is oscillatory. The obtained oscillation criteria generalize and improve a number of known results about both neutral and delay differential equations.

In this paper, we are concerned with the basic problem defined in [9]. Formulas for δV(0)and δ∞V(0),respectively the generalized and asymptotic gradient of the value function at zero, corresponding to an L2 -additive perturbation of dynamics are given. Under the normality condition, δV(0)turns out to be a compact subset of L2, formed entirely of arcs, and V is locally finite and Lipschitz at 0. Moreover, estimations of the generalized directional derivative and Dini's derivative of V at 0 are derived. Supplementary conditions imply that Dini's derivative of V at 0 exists, and V is actually strictly differentiate at this point.

In this paper we use a particular kind of weighted Sobolev space and pseudo-differential operators to study H3s propagation of singularities for the solution u ∊ Hs of the equations with second order.

The purpose of this paper is to characterize the weight functions for which the Hardy operator , with non-decreasing function ƒ, is bounded between various weighted Lp-spaces for a wide range of indices. Our characterizations complement for the most part those of E. T. Sawyer [11] and V. D. Stepanov [15] for the Hardy operator of non-increasing function.

Chern and Hamilton considered the integral of the Webster scalar curvature as a functional on the set of CR-structures on a compact 3-dimensional contact manifold. Critical points of this functional can be viewed as Riemannian metrics associated to the contact structure for which the characteristic vector field generates a 1-parameter group of isometries i.e. K-contact metrics. Tanno defined a higher dimensional generalization of the Webster scalar curvature, computed the critical point condition of the corresponding integral functional and found that it is not the K-contact condition. In this paper two other generalizations are given and the critical point conditions of the corresponding integral functionals are found. For the second of these, this is the K-contact condition, suggesting that it may be the proper generalization of the Webster scalar curvature.

In this paper, we develop a new approach to obtain the compactness of the fluctuation processes for Boltzmann dynamics. Our method is applicable to Kac's model, already studied by Uchiyama, but it covers many other cases. A novelty worth mentioning is the use of the weak topology of a Hilbert space.

Let 0 < wbe a smooth function on a complete Riemannian manifold Mn, and define L = — Δ — ▽ (log w) and Rw =Ric - w -1 Hess w.In this paper we show that if Rw ≥ —nK, (K ≥0), then the positive solutions of (L + ∂/∂t)u —0 satisfy a gradient estimate of the same form as that obtained by Li and Yau ([LY]) when Lis the Laplacian. This is used to obtain a parabolic Harnack inequality, which in turn, yields upper and lower Gaussian estimates for the heat kernel of L.The results obtained are applied to study the LPmapping properties of t→ e-tL μfor measures μ which are α-dimensional in a sense that generalises the local uniform α-dimensionality introduced by R. S. Strichartz ([St2], [St3]).

In this paper it is shown that the computation of higher dimensional harmonic volume, defined in [1], can be reduced to Harris' computation in the onedimensional case (See [3]), so that higher dimensional harmonic volume may be computed essentially as an iterated integral. We then use this formula to produce a specific smooth curve , namely a specific double cover of the Fermat quartic, so that the image of the second symmetric product of in its Jacobian via the Abel-Jacobi map is algebraically inequivalent to the image of under the group involution on the Jacobian.

In this paper, under certain hypothesis, we characterize generalized hyperbolic cylinders as the only null 2-type hypersurfaces in a Lorentz space.

In this paper we examine the examples of isospectral but non-isometric Riemannian manifolds given by Milnor, Ikeda, and Vignéras. Of these, only Milnor's example is accounted for by Sunada's method of constructing isospectral manifolds, and even then only as an "unnatural" construction.

This paper gives another construction of (0, p)-forms on a complex analytic space and of the operator. This construction is independent of the one in [1] and apart from the general result of Section 1 of [1], it can be read independently. As in [1], the hypotheses on S are the following: S has normal singularities, its singular locus X is smooth, the exceptional divisor in a desingularization of S is irreducible.

In [6] and [7], Krasnosel'skiĭ proved several fundamental fixed point principles for operators leaving invariant a cone in a Banach space. In [9], Nussbaum extended one of the results, the theorem about compression and expansion of a cone, to k-setcontraction maps, k < 1. Other versions for completely continuous maps were given by Fournier-Peitgen [2] and G. Fournier [1].The purpose of this paper is to generalise some of these results to upper semi continuous multivalued maps which are K-set contractions, k < 1, and differentiable at the origin and infinity.

The main topic of this paper is the study of the fundamental solution of the heat equation for the symmetric spaces of positive definite matrices, Pos(n,R).Our first step is to develop a “False Abel Inverse Transform” which transforms functions of compact support on an euclidean space into integrable functions on the symmetric space. The transform is shown to satisfy the relation is the usual Laplacian with a constant drift).Using this transform, we find explicit formulas for the heat kernel in the cases n = 2 and n = 3. These formulas allow us to give the asymptotic development for the heat kernel as t tends to infinity. Finally, we give an upper and lower bound of the same type for the heat kernel.

This paper deals with the Stefan-type problem with a zone of coexistence of both phases. We formulate the problem in the enthalpy form and show that the interfaces between the liquid and the mushy, the mushy and the solid phase are smooth. Our approach is to study the structures of the level sets of the solution via Sard's Lemma and the implicit function theorem.

The object of this paper is to study the local properties of light harmonic mappings.

This paper is devoted to the machinery necessary to apply the general theory of monotone dynamical systems to neutral functional differential equations. We introduce an ordering structure for the phase space, investigate its compatibility with the usual uniform convergence topology, and develop several sufficient conditions of strong monotonicity of the solution semiflows to neutral equations. By applying some general results due to Hirsch and Matano for monotone dynamical systems to neutral equations, we establish several (generic) convergence results and an equivalence theorem of the order stability and convergence of precompact orbits. These results are applied to show that each orbit of a closed biological compartmental system is convergent to a single equilibrium.

In this paper we shall develop the scattering theory introduced by Lax and Phillips [5] for the following general wave equation; where Ω is an exterior domain Rn(n ≥ 3) with the smooth boundary δΩ and B is either a Dirichlet boundary condition or of the form Bu = Vi(x)aij(x)δju+σ(x)u with the unit outer normal vector v(x) = (v1 , … , vn) at x ∈ δ Ω. The precise assumptions on α(x), aij(x),q(x), σ(x) are denoted below. If Ω is an inhomogeneous medium with the density ρ (x), the propagation of waves is described by (1.1) with a(x) = a(x)2 ρ(x), aij(x) = ρ-l(x) δij and q(x) = 0 with the velocity a(x).

For h € ℝ and ϕ : ℝ2 → ℝ define Lhϕ: ℝ2 → ℝ by (Lhϕ)(x, y) = ϕ (x + h, y) + ϕ (x — h,y) — ϕ (x,y + h) — ϕ (x,y — h) for all (x,y) € ℝ2. The aim of the paper is to establish the following "stability" theorem concerning the functional equationif δ > 0, f : ℝ2 → ℝ andthen there exists ɛ > 0 ami ϕ : ℝ2 → ℝ such that (Lhϕ)(x, y ) = 0 for all x, y,h ∊ ℝ and

In this paper we will study the Fubini theorem for stochastic integrals with respect to semimartingales in Hilbert space.Let (Ω, , P) he a probability space, (X, , μ) a measure space, H and G two Hilbert spaces, L(H, G) the space of bounded linear operators from H into G, Z an H-valued semimartingale relative to a given filtration, and φ: X × R+ × Ω → L(H, G) a function such that for each t ∈ R+ the iterated integrals are well-defined (the integrals with respect to μ are Bochner integrals). It is often necessary to have sufficient conditions for the process Y1 to be a version of the process Y2 (e.g. [1], proof of Theorem 2.11).

The main result of this paper establishes the existence and uniqueness of integral representations of KMS functionals on nuclear *- algebras. Our first result is about representations of *-algebras by means of operators having a common dense domain in a Hilbert space. We show, under certain regularity conditions, that (Powers) self-adjoint representations of a nuclear *-algebra, which admit a direct integral decomposition, disintegrate into representations which are almost all self-adjoint. We then define and study the class of self-derivative algebras. All algebras with an identity are in this class and many other examples are given. We show that if is a self-derivative algebra with an equicontinuous approximate identity, the cone of all positive forms on is isomorphic to the cone of all positive invariant kernels on These in turn correspond bijectively to the invariant Hilbert subspaces of the dual space This shows that if is a nuclear -space, the positive cone of has bounded order intervals, which implies that each positive form on has an integral representation in terms of the extreme generators of the cone. Given a continuous exponentially bounded one-parameter group of *-automorphisms of we can define the subcone of all invariant positive forms satisfying the KMS condition. Central functionals can be viewed as KMS functionals with respect to a trivial group action. Assuming that is a self-derivative algebra with an equicontinuous approximate identity, we show that the face generated by a self-adjoint KMS functional is a lattice. If is moreover a nuclear *-algebra the previous results together imply that each self-adjoint KMS functional has a unique integral representation by means of extreme KMS functionals almost all of which are self-adjoint.

Pandey and Chaudhary [13] recently developed the theory of Hilbert transform of Schwartz distribution space (DLp)',p > 1 in one dimension using Parseval's types of relations for one dimensional Hilbert transform [17] and noted that their theory coincides with the corresponding theory for the Hilbert transform developed by Schwartz [16] by using the technique of convolution in one dimension.The corresponding theory for the Hilbert transform in n-dimension is considerably harder and will be successfully accomplished in this paper.

This paper deals with the differential equation: ẍ + μF(ẋ) + x = ƒ( X, ẋ, t/Tμ) for μ ≫ 1 where F is a piecewise linear function and f is a periodic function of period μT, where T is to be chosen. It is established that periodic forced vibrations exist in an annular domain R(μ) constructed for the free vibration (ƒ ≡ 0), provided ƒ is not of higher order than Subsequently with ƒ = A cos (2πt/μT*), an asymptotic treatment of the forced vibration problem is carried out, by finding the proper initial conditions and the proper period μT* of f. Finally it is concluded that μT* is close to the period of the free vibration.

Let I be an interval, and let for 1 ≦ j ≦ I < ∞ be abutted subintervals such that . Let τ j be a linear differential expression defined on I j . In this paper we study densely defined operators associated with (0.1)

In this paper we investigate the zeros of power series1for some functions of coefficients A. In particular, we derive upper and lower bounds for the number of zeros of f in its domain of analyticity.

For certain classes of singular symmetric differential operators L of order 2n, this paper considers the problem of determining sufficient conditions for L to be of limit point type or of limit circle type. The operator discussed here is defined by

We name a functional equation with restricted argument one in which at least one of the variables is restricted to a certain discrete subset of the domain of the other variable(s). In particular, the subset may consist of a single element.The purpose of this paper is to present a functional equation satisfied only by cosine functions.

The main result of this paper is the following regularity result:THEOREM. Let D ⊂ CNbe a bounded, strongly pseudoconvex domain with bD of class Ck, k ≧ 3. Let Σ ⊂ bD be an N-dimensional totally real submanifold, and let f ∊ A(D) satisfy |f| = 1 on Σ, |f| < 1 on. If Σ is of class Cr, 3 ≦ r < k, then the restriction fΣ = f|Σ of f to Σ is of class Cr − 0, and if Σ is of class Ck, then fΣ is of class Ck − 1.Here, of course, A(D) denotes the usual space of functions continuous on , holomorphic on D, and we shall denote by Ak(D), k = 1, 2, …, the space of functions holomorphic on D whose derivatives or order k lie in A(D).