Every complete doubling metric space carries a doubling measure

We prove that a complete metric space carries a doubling measure if and only if is doubling and that more precisely the infima of the homogeneity exponents of the doubling measures on and of the homogeneity exponents of are equal. We also show that a closed subset of carries a measure of homogeneity exponent . These results are based on the case of compact due to Volberg and Konyagin.

     

When do connected spaces have nice connected preimages?

We prove that every connected Tychonoff space is an open monotone continuous image of a connected strictly -discrete left-separated Tychonoff space. For wide classes of connected spaces it is established that they have a finer Hausdorff strictly -discrete connected topology. Another result is that a finer Tychonoff connected strictly -discrete topology exists for any Tychonoff topology with a countable network. We show that there are Tychonoff connected spaces with countable network which are not continuous images of connected second countable spaces. It is established also that every connected Tychonoff space is an open retract of a connected homogeneous Tychonoff space, while it is not always possible to find a finer connected homogeneous topology on .

     

Continuous multiplicative mappings on

Let and be compact Hausdorff topological spaces, and let and be real Banach algebras of all real-valued continuous functions on and , respectively. The general form of continuous multiplicative mappings is given.

     

-convexity of Orlicz-Bochner spaces

A characterization of -convexity of arbitrary Banach space is given. Moreover, it is proved that the Orlicz-Bochner function space is P-convex if and only if both spaces and are -convex. In particular, the Lebesgue-Bochner space with is -convex iff is -convex.

     

Extendibility of homogeneous polynomials on Banach spaces

We study the -homogeneous polynomials on a Banach space that can be extended to any space containing . We show that there is an upper bound on the norm of the extension. We construct a predual for the space of all extendible -homogeneous polynomials on and we characterize the extendible 2-homogeneous polynomials on when is a Hilbert space, an -space or an -space.

     

Vector bundles on log terminal varieties

Let be an -dimensional variety and an ample vector bundle on of rank . We give a complete classification of pairs , with log terminal and such that is not ample. The results we obtain were conjectured by Fujita, and recently by Zhang.

     

Optimal estimation of shell thickness in Cutland's construction of Wiener measure

In Cutland's construction of Wiener measure, he used the product of Gaussian measures on , where is an infinite integer. It is mentioned by Cutland and Ng that for the product measure ,

where and with any positive infinite number. We prove here that may be replaced by with any positive infinite number. This is the optimal estimation for the shell thickness. It is also proved that . And for the *Lebesgue measure , is finite and not infinitesimal iff with finite, while for the *Lebesgue area of the sphere , should be .

     

A remark on normal derivations

Given a Hilbert space , let be operators on . Anderson has proved that if is normal and , then for all operators . Using this inequality, Du Hong-Ke has recently shown that if (instead) , then for all operators . In this note we improve the Du Hong-Ke inequality to for all operators . Indeed, we prove the equivalence of Du Hong-Ke and Anderson inequalities, and show that the Du Hong-Ke inequality holds for unitarily invariant norms.

     

Relative to any nonrecursive set

There is a countable first order structure such that for any set of integers , is not recursive if and only if there is a presentation of which is recursive in .

     

Differential forms on quotients by reductive group actions

Let be a smooth affine algebraic variety where a reductive algebraic group acts with a smooth quotient space . We show that the algebraic differential forms on which are pull-backs of forms on are exactly the -invariant horizontal differential forms on .

     

A note on norm attaining functionals

We are concerned in this paper with the density of functionals which do not attain their norms in Banach spaces. Some earlier results given for separable spaces are extended to the nonseparable case. We obtain that a Banach space is reflexive if and only if it satisfies any of the following properties: (i) admits a norm with the Mazur Intersection Property and the set of all norm attaining functionals of contains an open set, (ii) the set of all norm one elements of contains a (relative) weak* open set of the unit sphere, (iii) has and contains a (relative) weak open set of the unit sphere, (iv) is , has and contains a (relative) weak open set of the unit sphere. Finally, if is separable, then is reflexive if and only if contains a (relative) weak open set of the unit sphere.

     

Fusion in the character table

Suppose that is a Sylow -subgroup of a finite -solvable group . If , then the number of -conjugates of in can be read off from the character table of .

     

Banach spaces failing the almost isometric universal extension property

If is an infinite dimensional, separable, uniformly smooth Banach space, then there is an , a Banach space containing as a closed subspace and a norm one map from to a space which does not extend to an operator from to with .

     

A note on sequences lying in the range of a vector measure valued in the bidual

Let be a Banach space. It is unknown if every subset of lying in the range of an -valued measure is actually contained in the range of an -valued measure. In this paper we solve this problem in the case when we consider only vector measures of bounded variation.

     

Probability measures in -algebras in Hilbert spaces with conjugation

Let be a real -algebra of -real bounded operators containing no central summand of type in a complex Hilbert space with conjugation . Denote by the quantum logic of all -orthogonal projections in the von Neumann algebra . Let be a probability measure. It is shown that contains a finite central summand and there exists a normal finite trace on such that , .

     

Average curvature of convex curves in

A well-known result states that, if a curve in has geodesic curvature less than or equal to one at every point, then is embedded. The converse is obviously not true, but the embeddedness of a curve does give information about the curvature. We prove that, if is a convex embedded curve in , then the average curvature (curvature per unit length) of , denoted , satisfies . This bound on the average curvature is tight as for a horocycle.

     

The moduli of substructures of a compact complex space

We construct a space of fine moduli for the substructures of an arbitrary compact complex space . A substructure of is given by a subalgebra of the structure sheaf with the additional feature that is also a complex space; and are called equivalent if and only if and are isomorphic as subalgebras of .

Since substructures are quotients, it is only natural to start with the fine moduli space of all complex-analytic quotients of . In order to obtain a representable moduli functor of substructures, we are forced to concentrate on families of quotients which satisfy some flatness condition for relative differential modules of higher order. Considering the corresponding flatification of , we realize that its open subset consisting of all substructures turns out to be a complex space which has the required universal property.

     

Subaveraging estimate for

Let be a smooth real hypersurface of and a compact submanifold of . We generalize a result of A. Boggess and R. Dwilewicz giving, under some geometric conditions on and , an estimate of the submeanvalue on of any function on a neighbourhood of , by the norm of on a neighbourhood of in .

     

A twistor correspondence and Penrose transform for odd-dimensional hyperbolic space

For odd-dimensional hyperbolic space , we construct transforms between the cohomology of certain line bundles on (a twistor space for ) and eigenspaces of the Laplacian and of the Dirac operator on . The transforms are isomorphisms. As a corollary we obtain that every eigenfunction of or on extends as a holomorphic eigenfunction of the corresponding holomorphic operator on a certain region of the complexification of . We also obtain vanishing theorems for the cohomology of a class of line bundles on .

     

Polishness of the Wijsman topology revisited

Let be a completely metrizable space. Then the space of nonempty closed subsets of endowed with the Wijsman topology is -favorable in the strong Choquet game. As a consequence, a short proof of the Beer-Costantini Theorem on Polishness of the Wijsman topology is given.