Amenability of compact hypergroup algebras

We show that for a compact hypergroup K, the hypergroup algebra is amenable as a Banach algebra if the set of hyperdimensions of irreducible representations of K is bounded above. Conversely if is amenable, the set of ratios of the hyperdimension to the dimension of irreducible representations of K is bounded above. These are equivalent for compact commutative hypergroups.     

‐subgroups in the space Cremona group

We prove that if X is a rationally connected threefold and G is a p‐subgroup in the group of birational selfmaps of X, then G is an abelian group generated by at most 3 elements provided that . We also prove a similar result for under an assumption that G acts on a (Gorenstein) G‐Fano threefold, and show that the same holds for under an assumption that G acts on a G‐Mori fiber space.     

On the existence of RUC systems in rearrangement invariant spaces

We characterize rearrangement invariant spaces X on [0, 1] with the property that each orthonormal system in X which is uniformly bounded in some Marcinkiewicz space , for equivalent to , , is a system of Random Unconditional Convergence (RUC system).     

On a necessary aspect for the Riesz basis property for indefinite Sturm‐Liouville problems

In 1996, H. Volkmer observed that the inequality is satisfied with some positive constant for a certain class of functions f on [ ? 1, 1] if the eigenfunctions of the problem form a Riesz basis of the Hilbert space . Here the weight is assumed to satisfy a.e. on ( ? 1, 1). We present two criteria in terms of Weyl–Titchmarsh m‐functions for the Volkmer inequality to be valid. Note that one of these criteria is new even for the classical HELP inequality. Using these results we improve the result of Volkmer by showing that this inequality is valid if the operator associated with the spectral problem satisfies the linear resolvent growth condition. In particular, we show that the Riesz basis property of eigenfunctions is equivalent to the linear resolvent growth if r is odd.     

Cohomological characterisation of monads

Let X be an n‐dimensional smooth projective variety with an n‐block collection , with , of coherent sheaves on X that generate the bounded derived category . We give a cohomological characterisation of torsion‐free sheaves on X that are the cohomology of monads of the form where . We apply the result to get a cohomological characterisation when X is the projective space, the smooth hyperquadric or the Fano threefold V₅. We construct a family of monads on a Segre variety and apply our main result to this family.     

Critical point equation on four‐dimensional compact manifolds

The aim of this article is to study the space of metrics with constant scalar curvature of volume 1 that satisfies the critical point equation for simplicity CPE metrics. It has been conjectured that every CPE metric must be Einstein. Here, we shall focus our attention for 4‐dimensional half conformally flat manifolds M⁴. In fact, we shall show that for a nontrivial must be isometric to a sphere and f is some height function on      

Nefness of adjoint bundles for ample vector bundles of corank 3

Let be an ample vector bundle of rank on a smooth complex projective variety X of dimension n. The aim of this paper is to describe the structure of pairs as above whose adjoint bundles are not nef for . Furthermore, we give some immediate consequences of this result in adjunction theory.     

The second CR Yamabe invariant

Let be a closed, connected, strictly pseudoconvex CR manifold with dimension . We define the second CR Yamabe invariant in terms of the second eigenvalue of the Yamabe operator and the volume of M over the pseudo‐convex pseudo‐hermitian structures conformal to θ. Then we study when it is attained and classify the CR‐sphere by its second CR Yamabe invariant. This work is motivated by the work of B. Ammann and E. Humbert 1 on the Riemannian context.     

Note on Nikodym maximal in the variable Lebesgue spaces

In this paper we consider the k‐plane Nikodym maximal estimates in the variable Lebesgue spaces . We first formulate the problem about the boundedness of the k‐plane Nikodym maximal and show that the maximal estimate in is equivalent to that in for . So, the optimal Nikodym maximal estimate in follows from Cordoba's estimate.     

Riemann boundary value problem for harmonic functions in Clifford analysis

In this article, we mainly deal with the boundary value problem for harmonic function with values in Clifford algebra: where is a Liapunov surface in , the Dirac operator , are unknown functions with values in a universal Clifford algebra Under some assumptions, we show that the boundary value problem is solvable.     

The estimates of Bessel integrals with oscillatory factors

In this paper, we establish some sharp estimates of Bessel integrals with oscillatory factors . As an application, we obtain the boundedness of the oscillatory singular integral operators with variable kernels.     

Quantification of Pełczyński's property (V)

A Banach space X has Pełczyński's property (V) if for every Banach space Y every unconditionally converging operator is weakly compact. In 1962, Aleksander Pełczyński showed that spaces for a compact Hausdorff space K enjoy the property (V), and some generalizations of this theorem have been proved since then. We introduce several possibilities of quantifying the property (V). We prove some characterizations of the introduced quantitative versions of this property, which allow us to prove a quantitative version of Pelczynski's result about spaces and generalize it. Finally, we study the relationship of several properties of operators including weak compactness and unconditional convergence, and using the results obtained we establish a relation between quantitative versions of the property (V) and quantitative versions of other well known properties of Banach spaces.     

Stability of integral operators on a space of homogeneous type

In this paper, we consider integral operators T on compact spaces of homogeneous type with finite diameter, whose kernels have certain Hölder regularity and mild singularity near the diagonal. We show that given any , the ‐stability of the operator is equivalent for different , where I stands for the identity operator.     

Embeddings of weighted Morrey spaces

We consider embedding theorems within the scale of weighted Morrey spaces , where the weight w belongs to the Muckenhoupt class and . This includes, in particular, the classical setting of weighted Lebesgue spaces. We study some typical examples for the weight like , but also deal with quite general assumptions.     

Global existence of solutions for a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source

This paper deals with a fully parabolic chemotaxis system with consumption of chemoattractant and logistic source under homogeneous Neumann boundary conditions in a smooth bounded domain . The functions χ and f are assumed to generalize the chemotactic sensitivity function and logistic source respectively. Under some conditions, we obtain that the corresponding initial‐boundary value problem possesses a unique global classical solution that is uniformly bounded.     

Almost minimizers for semilinear free boundary problems with variable coefficients

We study regularity results for almost minimizers of the functional where is a matrix with Hölder continuous coefficients. In the case we show that an almost minimizer belongs to , where the exponent η is related with the competition between the Hölder continuity of the matrix , the parameter of almost minimization and γ. In some sense, this regularity is optimal. As far as the case is concerned, our results show that an almost minimizer is locally in each phase and , improving in some sense a recent result of David & Toro.     

The essential spectrum of a singular Sturm–Liouville operator

In this paper we study the essential spectrum of the operator where is a positive absolutely continuous function on (0, 1) that resembles for some . We prove that the essential spectrum of coincides with the essential spectrum of the operator .     

On the structure of sets with positive reach

We give a complete characterization of compact sets with positive reach (proximally C ¹ sets) in the plane and of one‐dimensional sets with positive reach in . Further, we prove that if is a set of positive reach of topological dimension , then A has its “k‐dimensional regular part” which is a k‐dimensional “uniform” C ^{1, 1} manifold open in A and can be locally covered by finitely many ‐dimensional DC surfaces. We also show that if has positive reach, then can be locally covered by finitely many semiconcave hypersurfaces.     

Dimension of images of subspaces under mappings in Triebel‐Lizorkin spaces

Let and let be a ‐quasicontinuous representative of a mapping in the Triebel‐Lizorkin space . We find an optimal value of such that for a.e. the Hausdorff dimension of is at most α. We construct examples to show that the value of β is optimal and we show that it does not increase once p goes below the critical value α.