Behaviour of holomorphic automorphisms on equicontinuous subsets of the space

Consider a compact Hausdorff topological space , a -triple and , the -triple of all continuous -valued functions with the pointwise operations and the norm of the supremum. Let be the group of all holomorphic automorphisms of the unit ball of that map every equicontinuous subset lying strictly inside into another such a set. The real Banach-Lie group and its Lie algebra are investigated. The identity connected component of is identified when has the strong Banach-Stone property. This extends to the infinite dimensional setting a well known result concerning the case .

     

A factorization theorem for the derivative of a function in

We show that a function is the derivative of a function in the Hardy space of the unit disk for if and only if where and . Here, can be chosen to be non-vanishing, , and . As an application, we characterize positive measures on the unit disk such that the operator is bounded from the tent space to , where .

     

Open covers and partition relations

An open cover of a topological space is said to be an -cover if there is for each finite subset of the space a member of the cover which contains the finite set, but the space itself is not a member of the cover. We prove theorems which imply that a set of real numbers has Rothberger's property if, and only if, for each positive integer , for each -cover of , and for each function from the two-element subsets of , there is a subset of such that is constant on , and each element of belongs to infinitely many elements of (Theorem 1). A similar characterization is given of Menger's property for sets of real numbers (Theorem 6).

     

Gradient estimates for positive solutions of the Laplacian with drift

Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where If is a vector field such that and on for some nonnegative constants and then we show that any positive solution of the equation satisfies the estimate

on , for all In particular, for the case when this estimate is advantageous for small values of and when it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).

     

An extension of H. Cartan's theorem

In this article we prove that if , , is a bounded pseudoconvex domain with real analytic boundary, then for each , there exists a fixed open neighborhood of and an open neighborhood of in such that any can be extended holomorphically to , and that the action defined by

is real analytic in joint variables. This extends H. Cartan's theorem beyond the boundary. Some applications are also discussed here.

     

Gaussian estimates and regularized groups

We show that if a bounded analytic semigroup on satisfies a Gaussian estimate of order and is the generator of its consistent semigroup on , then generates a -regularized group on where . We obtain the estimate of () and the -independence of , and give applications to Schrödinger operators and elliptic operators of higher order.

     

Toeplitz -algebras on ordered groups and their ideals of finite elements

Let be a discrete abelian group and an ordered group. Denote by the minimal quasily ordered group containing . In this paper, we show that the ideal of finite elements is exactly the kernel of the natural morphism between these two Toeplitz -algebras. When is countable, we show that if the direct sum of -groups , then .

     

On the product of two generalized derivations

Two elements and in a ring determine a generalized derivation on by setting for any in . We characterize when the product is a generalized derivation in the cases when the ring is the algebra of all bounded operators on a Banach space , and when is a -algebra . We use these characterizations to compute the commutant of the range of .

     

A Liouville-type theorem on halfspaces for the Kohn Laplacian

Let be the Kohn Laplacian on the Heisenberg group and let be a halfspace of whose boundary is parallel to the center of . In this paper we prove that if is a non-negative -superharmonic function such that

then in .

     

A bound for

Let denote the largest irreducible character degree of a finite group , and let be a prime. Two results are obtained. First, we show that, if is a -solvable group and if , then . Next, we restrict attention to solvable groups and show that, if and if is a Sylow -subgroup of , then .

     

Interpolation in inflated Hilbert spaces

The interpolation problem for a reflexive algebra Alg is this: Given two operators and , under what conditions can we be sure that there will exist an operator in Alg such that ? There are simple necessary conditions that have been investigated in several earlier papers. Here we present an example to show that the conditions are not, in general, sufficient. We also suggest a strengthened set of conditions which are necessary and are ``almost' sufficient, in the sense that they will ensure that lies in the weak-operator closure of the set {:Alg}.

     

A mean value theorem on bounded symmetric domains

Let be a Cartan domain of rank and genus and , , the Berezin transform on ; the number can be interpreted as a certain invariant-mean-value of a function around . We show that a Lebesgue integrable function satisfying , , must be -harmonic. In a sense, this result is reminiscent of Delsarte's two-radius mean-value theorem for ordinary harmonic functions on the complex -space , but with the role of radius played by the quantity .

     

Multinomial coefficients modulo a prime

We say that the multinomial coefficient (m.c.) has order and power . Let be the number of m.c. that are not divisible by and have order with powers which are not larger than . If and

then for any integer

     

On covering multiplicity

Let be a system of arithmetic sequences which forms an -cover of (i.e. every integer belongs at least to members of ). In this paper we show the following surprising properties of : (a) For each there exist at least subsets of with such that . (b) If forms a minimal -cover of , then for any there is an such that for every there exists an for which and

     

Periodic solutions of damped differential systems with repulsive singular forces

We consider the periodic boundary value problem for the singular differential system: where , , and . The singular potential is of repulsive type in the sense that as . Under Habets-Sanchez's strong force condition on at the origin, the existence results, obtained by coincidence degree in this paper, have no restriction on the damping forces . Meanwhile, some quadratic growth of the restoring potentials at infinity is allowed.

     

Small values of polynomials and potentials with normalization

For a polynomial of degree , normalized by the condition

we show that has at most , where is explicitly given and sharp for each . Similar estimates are given for other normalizations, such as , and for planar measure, and for generalized polynomials and potentials, thereby extending work of Cuyt, Driver and the author for . The relation to Remez inequalities is briefly discussed.

     

Extremal points of a functional on the set of convex functions

We investigate the extremal points of a functional , for a convex or concave function . The admissible functions are convex themselves and satisfy a condition . We show that the extremal points are exactly and if these functions are convex and coincide on the boundary . No explicit regularity condition is imposed on , , or . Subsequently we discuss a number of extensions, such as the case when or are non-convex or do not coincide on the boundary, when the function also depends on , etc.

     

Complete positivity of elementary operators

In this paper, we prove that if is an -dimensional subspace of , then is -reflexive, where denotes the greatest integer not larger than . By the result, we show that if is an elementary operator on a -algebra , then is completely positive if and only if is -positive.

     

type free resolutions of monomial ideals

Let be a monomial ideal of . Bayer-Peeva-Sturmfels studied a subcomplex of the Taylor resolution, defined by a simplicial complex . They proved that if is generic (i.e., no variable appears with the same non-zero exponent in two distinct monomials which are minimal generators), then is the minimal free resolution of , where is the Scarf complex of . In this paper, we prove the following: for a generic (in the above sense) monomial ideal and each integer , there is an embedded prime of . Thus a generic monomial ideal with no embedded primes is Cohen-Macaulay (in this case, is shellable). We also study a non-generic monomial ideal whose minimal free resolution is for some . In particular, we prove that if all associated primes of have the same height, then is Cohen-Macaulay and is pure and strongly connected.

     

On a problem of Dynkin

Consider an -superdiffusion on , where is an uniformly elliptic differential operator in , and . The -polar sets for are subsets of which have no intersection with the graph of , and they are related to the removable singularities for a corresponding nonlinear parabolic partial differential equation. Dynkin characterized the -polarity of a general analytic set in term of the Bessel capacity of , and Sheu in term of the restricted Hausdorff dimension. In this paper we study in particular the -polarity of sets of the form , where and are two Borel subsets of and respectively. We establish a relationship between the restricted Hausdorff dimension of and the usual Hausdorff dimensions of and . As an application, we obtain a criterion for -polarity of in terms of the Hausdorff dimensions of and , which also gives an answer to a problem proposed by Dynkin in the 1991 Wald Memorial Lectures.