A relation is described between Arnold's strange duality and a polar duality between the Newton polytopes which is mostly due to M. Kobayashi. It is shown that this relation continues to hold for the extension of Arnold's strange duality found by C. T. C. Wall and the author. By a method of Ehlers–Varchenko, the characteristic polynomial of the monodromy of a hypersurface singularity can be computed from the Newton diagram. This method is generalized to the isolated complete intersection singularities embraced in the extended duality. This is used to explain the duality of characteristic polynomials of the monodromy discovered by K. Saito for Arnold's original strange duality and extended by the author to the other cases.

It is proved that the dual of a Banach space with the Mazur intersection property is almost weak* Asplund. Analogously, the predual of a dual space with the weak* Mazur intersection property is almost Asplund. Through the use of these arguments, it is found that, in particular, almost all (in the Baire sense) equivalent norms on [lscr ]1(Γ) and [lscr ]∞(Γ) are Fréchet differentiable on a dense Gδ subset. Necessary conditions for Mazur intersection properties in terms of convex sets satisfying a Krein–Milman type condition are also discussed. It is also shown that, if a Banach space has the Mazur intersection property, then every subspace of countable codimension can be equivalently renormed to satisfy this property.

Le but de cet article est de démontrer une version locale de certains résultats classiques concernant l'espace de Hardy H1 et l'action sur cet espace des opérateurs de Calderón–Zygmund. Plus précisément, voici deux exemples de questions étudiées.

La première concerne l'invariance de l'espace H1 relativement aux fonctions maximales utilisées pour le définir. On sait que, si une fonction f est telle qu'une certaine fonction maximale raisonnable M1f soit intégrable dans ℝn, alors toute autre fonction maximale raisonnable M2f est également intégrable dans ℝn. Si on suppose seulement que M1f est intégrable sur une boule de ℝn, que peut-on dire de l'intégrabilitéde M2f?

La seconde concerne l'action sur l'espace H1 des opérateurs de Calderón–Zygmund. Un résultat classique de cette théorie affirme que, si T est un opérateur de Calderón–Zygmund (respectivement un opérateur de Calderón–Zygmund vérifiant la condition T*() = 0) et si f ∈ H1, alors T(f) ∈ L1 (respectivement T(f) ∈ H1). Que peut-on dire de T(f), si on suppose seulement qu'une certaine fonction maximale associée à f est intégrable sur une boule de ℝn?

La deuxième question s'est naturellement posée au cours de notre travail [3], dans lequel nous avons eu besoin de certains résultats de la Section 4 du présent article. A notre connaissance, ces résultats ne se trouvent pas dans la littérature et pour les obtenir, on doit “localiser” convenablement les méthodes utilisées classiquement pour établir les résultats globaux correspondants; nous en rappelons les grandes lignes, qui fixent le plan de notre article. La description de l'action des opérateurs de Calderón–Zygmund sur H1 est basée sur la décomposition atomique de cet espace; de façon un peu similaire, notre réponse à la deuxième question passe par le résultat de localisation de la Section 3 qui est, sans être aussi précis, dans l'esprit des décompositions atomiques. A son tour, la décomposition atomique de H1 est une conséquence du fait que cet espace puisse être défini au moyen de “grandes” fonctions maximales; parallèlement, notre résultat de localisation de la Section 3 utilise une version locale de ce résultat, qui fait l'objet de la Section 2.

A reflection principle is given for temperature functions on a rectangle in the plane with much weaker conditions than the classical continuity to zero at the boundary, which improves a continuous version of Widder.

As an application of this result a uniqueness theorem is given for solutions of the Cauchy problem of the heat equation on a semi-infinite rod.

The purpose of the paper is to introduce a new linearization method for local well-posedness of nonlinear evolution equations. This approach is based on an implicit function theorem of Nash–Moser type. The technique is illustrated by an application to a general class of fully nonlinear parabolic partial differential equations of arbitrary order on ℝn. The estimates required by the Nash–Moser technique are derived for the higher order Sobolev norms of the solutions of the linearized parabolic equation using semigroup theory and elliptic theory. In particular, a priori estimates, resolvent estimates and commutator estimates are involved. The general method based on a combination of Nash–Moser techniques with semigroup theory is applicable to other problems and has already been used to prove short-time solvability for some nonlinear Schrödinger type equation. This approach might be useful in other situations as well since it compensates for a loss of derivatives in the estimates of the solutions of the linearized equation.

The weak type 1 for the Mehler maximal function is studied via a precise estimate for the ‘maximal kernel’. This, in turn, allows the geometry involved in this setting to be described.

Let M be a manifold with conical ends. (For precise definitions see the next section; we only mention here that the cross-section [Kscr ] can have a nonempty boundary.) We study the scattering for the Laplace operator on M. The first question that we are interested in is the structure of the absolute scattering matrix [Sscr ](s). If M is a compact perturbation of ℝn, then it is well-known that [Sscr ](s) is a smooth perturbation of the antipodal map on a sphere, that is,

formula here

On the other hand, if M is a manifold with a scattering metric (see [8] for the exact definition), it has been proved in [9] that [Sscr ](s) is a Fourier integral operator on [Kscr ], of order 0, associated to the canonical diffeomorphism given by the geodesic flow at distance π. In our case it is possible to prove that [Sscr ](s) is in fact equal to the wave operator at a time t = π plus C∞ terms. See Theorem 3.1 for the precise formulation. This result is not too difficult and is obtained using only the separation of variables and the asymptotics of the Bessel functions.

Our second result is deeper and concerns the scattering phase p(s) (the logarithm of the determinant of the (relative) scattering matrix).

It is proved that Thom spectra of generalized braid groups are the wedges of suspensions over the Eilenberg–MacLane spectrum for ℤ/2. The precise structure of the Thom spectra of the generalized braid groups of the types C and D is obtained. For the generalized braid groups of type C the natural pairing analogous to the pairing of the classical braids is studied. This pairing generates the multiplicative structure of the Thom spectrum such that the corresponding bordism theory has the coefficient ring isomorphic to the polynomial ring over ℤ/2 on one generator of dimension 1[ratio ]ℤ/2[s].

Some of the arguments and techniques developed by the authors in a previous paper are applied to the study of the boundedness of certain operators associated with the Ornstein–Uhlenbeck semigroup. In particular, a simple proof is given of the weak type 1 with respect to the Gaussian measure of the Riesz transforms of order 1 and the Littlewood–Paley g-function which then is extended to show the same property for the Riesz transforms of order 2. For Riesz transforms of higher order boundedness is shown in appropriate spaces close to L1.

The ideas of Friedman and Qin are extended to find the wall-crossing formulae for the Donaldson invariants of algebraic surfaces with pg = 0, q > 0 and anticanonical divisor −K effective, for any wall ζ with lζ = ¼(ζ2 − p1) being 0 or 1.

The Fourier series of the elements in the generalized Bergman spaces bp, q of harmonic functions over D and over [Copf ] (as well as those of holomorphic functions) is analysed. It is shown that the trigonometric system Ω = {r[mid ]k[mid ]eikϕ}k∈ℤ is never a basis of b1, 1 and b∞, 0 for any weighted L1-norm and L∞-norm over D. The same result holds in the special case of Bargmann–Fock space over [Copf ] (with respect to the weighted L1-norms and L∞-norms) which answers a question of Garling and Wojtaszczyk. On the other hand examples are given of weighted L1-norms and L∞-norms over [Copf ] where Ω is indeed a basis of b1, 1 and b∞, 0. Moreover, using similar methods, a weight is constructed on D where b∞, ∞ is not isomorphic to l∞ which shows that there are weighted spaces whose Banach space classifications differ completely from those which have been characterized so far.

The essential norm of weighted composition operators on weighted Banach spaces of analytic functions is computed in terms of the weights and the inducing symbols. As a consequence the boundedness and compactness of these operators is characterized. As another consequence the essential norm of composition operators on weighted Bloch spaces is obtained and, consequently, the boundedness and compactness of composition operators on these spaces is also characterized. Particular instances of weighted Bloch spaces are the Lipschitz spaces. The method used allows a unified treatment of the problem of boundedness and compactness on these spaces.

McMichael proved that the convolution with the (euclidean) arclength measure supported on the curve t [xrarr ] (t, t2, …, tn), 0 < t < 1, maps Lp(ℝn) boundedly into Lp′(ℝn) if and only if 2n(n+1)/(n2+n+2) [les ] p [les ] 2. In proving this, a uniform estimate on damping oscillatory integrals with polynomial phase was crucial. In this paper, a remarkably simple proof of the same estimate on oscillatory integrals is presented. In addition, it is shown that the convolution operator with the affine arclength measure on any polynomial curve in ℝn maps Lp(ℝn) boundedly into Lp′(ℝn) if p = 2n(n+1)/(n2+n+2).

The spectrum and essential spectrum of a self-adjoint operator in a real Hilbert space are characterized in terms of Palais–Smale conditions on its quadratic form and Rayleigh quotient respectively.

Let G be a discrete group and (G, G+) be a quasi-ordered group. Set G0+ = G+∩(G+)−1 and G1 = (G+[setmn ]G0+)∪{e}. Let [Fscr ]G1(G) and [Fscr ]G+(G) be the corresponding Toeplitz algebras. In the paper, a necessary and sufficient condition for a representation of [Fscr ]G+(G) to be faithful is given. It is proved that when G is abelian, there exists a natural C*-algebra morphism from [Fscr ]G1(G) to [Fscr ]G+(G). As an application, it is shown that when G = ℤ2 and G+ = ℤ+ × ℤ, the K-groups K0([Fscr ]G1(G)) ≅ ℤ2, K1([Fscr ]G1(G)) ≅ ℤ and all Fredholm operators in [Fscr ]G1(G) are of index zero.

A classical result of Kadison concerning the extension, via the Hahn–Banach theorem, of extremal states on unital self-adjoint linear manifolds (that is, operator systems) in C*-algebras is generalised to the setting of noncommutative convexity, where one has matrix states (that is, unital completely positive linear maps) and matrix convexity. It is shown that if ϕ is a matrix extreme point of the matrix state space of an operator system R in a unital C*-algebra A, then ϕ has a completely positive extension to a matrix extreme point Φ of the matrix state space of A. This result leads to a characterisation of extremal matrix states as pure completely positive maps and to a new proof of a decomposition of C*-extreme points.

Let ϕ be a hyperbolic map. Cocycle equations of the form f = uϕ·g·αu−1 are considered, with f, g, u taking values in a compact connected Lie group G, α being an automorphism of G and f, g being Hölder continuous. When the eigenvalues of the derivative of α have modulus 1, it is proved that any measurable solution u has a Hölder continuous version. This condition on α is optimal. When f, g are Ck then u may be taken to be Ck−1+ε for any ε ∈ (0, 1).

Every subalgebra of [Lscr ]([Hscr ]) which is isometrically weak*-homeomorphic with H∞(Ω) is reflexive, when Ω is a finitely connected region in the complex plane.

When S is a discrete subsemigroup of a discrete group G such that G = S−1S, it is possible to extend circle-valued multipliers from S to G, to dilate (projective) isometric representations of S to (projective) unitary representations of G, and to dilate/extend actions of S by injective endomorphisms of a C*-algebra to actions of G by automorphisms of a larger C*-algebra. These dilations are unique provided they satisfy a minimality condition. The (twisted) semigroup crossed product corresponding to an action of S is isomorphic to a full corner in the (twisted) crossed product by the dilated action of G. This shows that crossed products by semigroup actions are Morita equivalent to crossed products by group actions, making powerful tools available to study their ideal structure and representation theory. The dilation of the system giving the Bost–Connes Hecke C*-algebra from number theory is constructed explicitly as an application: it is the crossed product C0([Aopf ]f)[rtimes ]ℚ*+, corresponding to the multiplicative action of the positive rationals on the additive group [Aopf ]f of finite adeles.

Consider a discrete-time regenerative phenomenon with associated renewal sequence un. General results for the supremum of um+1, …, um+n are developed for those renewal sequences {un} for which the first m + 1 elements match those of a fixed renewal sequence {vn}, that is, u0 = v0, …, um = vm. A series of associated lemmas are developed in the process.