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Stein’s higher Riesz transforms are translation invariant operators on$L$^{2}($R$^{$n$}) built from multipliers whose restrictions to the unit sphere are eigenfunctions of the Laplace–Beltrami operators. In this article, generalizing Stein’s higher Riesz transforms, we construct a family of translation invariant operators by using discrete series representations for hyperboloids associated to the indefinite quadratic form of signature ($p$,$q$). We prove that these operators extend to$L$^{$r$}-bounded operators for 1<$r$<∞ if the parameter of the discrete series representations is generic.

We compute local Gromov–Witten invariants of cubic surfaces at all genera. We use a deformation a of cubic surface to a nef toric surface and the deformation invariance of Gromov–Witten invariants.

A recent approach of D. S. Lubinsky yields universality in random matrix theory and fine zero spacing of orthogonal polynomials under very mild hypothesis on the weight function, provided the support of the generating measure μ is [-1,1]. This paper provides a method with which analogous results can be proven on general compact subsets of$R$. Both universality and fine zero spacing involves the equilibrium measure of the support of μ. The method is based on taking polynomial inverse images, by which results can be transferred from [-1,1] to a system of intervals, and then to general sets.

We are studying the Diophantine exponent μ_{$n$,$l$}defined for integers 1≤$l$<$n$and a vector α∈ℝ^{$n$}by letting $$\mu_{n,l}=\sup\{\mu\geq0: 0 < \Vert\underline{x}\cdot\alpha\Vert<H(\underline{x})^{-\mu}\ \text{for infinitely many}\ \underline{x}\in\mathcal{C}_{n,l}\cap\mathbb{Z}^n\},$$ where $\cdot$ is the scalar product, $\|\cdot\|$ denotes the distance to the nearest integer and $\mathcal{C}_{n,l}$ is the generalised cone consisting of all vectors with the height attained among the first$l$coordinates. We show that the exponent takes all values in the interval [$l$+1,∞), with the value$n$attained for almost all α. We calculate the Hausdorff dimension of the set of vectors α with μ_{$n$,$l$}(α)=μ for μ≥$n$. Finally, letting$w$_{$n$}denote the exponent obtained by removing the restrictions on $\underline{x}$ , we show that there are vectors α for which the gaps in the increasing sequence μ_{$n$,1}(α)≤...≤μ_{$n$,$n$-1}(α)≤$w$_{$n$}(α) can be chosen to be arbitrary.

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We prove continuity and Schatten–von Neumann properties of such operators when acting on$L$^{2}.

So-called exceptional isomorphisms of low-dimensional spinor groups are directly worked out for the terminal two cases of rank five and six (in the sense of quadratic modules) at the level of special Clifford groups with norm characters as group schemes over a ring. Explicit formulas of identifications are available on big cells, restrictions to which being justified by scheme theory.

We present new results concerning the solvability, or lack of thereof, in the Cauchy problem for the $\bar\partial$ operator with initial values assigned on a weakly pseudoconvex hypersurface, and provide illustrative examples.

We give several characterizations of those sequences of holomorphic self-maps {φ_{$n$}}_{$n$≥1}of the unit disk for which there exists a function$F$in the unit ball $\mathcal{B}=\{f\in H^{\infty}: \|f\|_\infty\leq1\}$ of$H$^{∞}such that the orbit {$F$∘φ_{$n$}:$n$∈ℕ} is locally uniformly dense in $\mathcal{B}$ . Such a function$F$is said to be a $\mathcal{B}$ -universal function. One of our conditions is stated in terms of the hyperbolic derivatives of the functions φ_{$n$}. As a consequence we will see that if φ_{$n$}is the$n$th iterate of a map φ of $\mathbb{D}$ into $\mathbb{D}$ , then {φ_{$n$}}_{$n$≥1}admits a $\mathcal{B}$ -universal function if and only if φ is a parabolic or hyperbolic automorphism of $\mathbb{D}$ . We show that whenever there exists a $\mathcal{B}$ -universal function, then this function can be chosen to be a Blaschke product. Further, if there is a $\mathcal{B}$ -universal function, we show that there exist uniformly closed subspaces consisting entirely of universal functions.

We introduce a basis of the Orlik–Solomon algebra labeled by chambers, the so called chamber basis. We consider structure constants of the Orlik–Solomon algebra with respect to the chamber basis and prove that these structure constants recover D. Cohen’s minimal complex from the Aomoto complex.

In the present paper we prove, that in the real normed space$X$, having at least three dimensions, the Nordlander’s conjecture about the modulus of convexity of the space$X$is true, i.e. from the validity of Day’s inequality for a fixed real number from the interval (0,2), follows that$X$is an inner product space.