Lax Projective Embeddings of Polar Spaces

Let be a non-degenerate polar space of rank n 3 where all of its lines have at least three points. We prove that, if admits a lax embedding e : in a projective space defined over a skewfield K, then is a classical and defined over a sub-skewfield K₀ of K. Accordingly, admits a full embedding e₀ in a K₀-projective space ₀. We also prove that, under suitable hypotheses on e and e₀, there exists an embedding such that and preserves dimensions.Received: March, 2004     

Exceptional Extended Dual Polar Spaces

We address the classification problem of flag-transitive geometries with diagrams of the form where the leftmost edge symbolizes the geometry of vertices and edges of a complete graph ons + 2 vertices and the residue of an element of the leftmost type is a finite thick classical dual polar space. These geometries are known asextended dual polar spaces. An extended dual polar space is called affine if it possesses a flag-transitive automorphism group which contains a normal subgroup acting regularly on the set of elements of the leftmost type. For a dual polar space D with three points per line there exists a unique 2-simply connected affine extension A(D) of D. We show that a flag-transitive extended dual polar space is either a quotient of A(D) for some D or isomorphic to one of 19 exceptional geometries whose full automorphism groups are isomorphic respectively to Sym₈,U₄(2).2,Sp₆(2) × 2,Sp₆(2), 3 · U4(3).2²,U₄(3).2²,U₅(2).2,McL.2,HS.2,Suz.2,Sp₈(2), 3 · Fi₂₂.2,Fi₂₂.2,Co₂ × 2,Co₂,Fi₂₄(s = 4,t = 2),Fi₂₄(s = t = 3),F₁andFi₂₃.     

Embeddings of Generalised Morrey Smoothness Spaces

We study embeddings between generalised Triebel–Lizorkin–Morrey spaces ε^s_?,p,q（R^d） and within the scales of further generalised Morrey smoothness spaces like N^s_?,p,q（R^d）,B_p,q^s,?（R^d） and F_p,q^s,?（R^d）.The latter have been investigated in a recent paper by the first two authors （2023）,while the embeddings of the scale N^s_?,p,q（R^d） were mainly obtained in a paper of the first and last two authors（2022）.Now we concentrate on the characterisation of the spaces ε^s_?,p,q（R^d）.Our approach requires a wavelet characterisation of those spaces which we establish for the system of Daubechies’wavelets.Then we prove necessary and sufficient conditions for the embedding ■.We can also provide some almost final answer to the question when ε^s_?,p,q（R^d） is embedded into C（R^d）,complementing our recent findings in case of N^s_?,p,q（R^d）.     

Absolutely Continuous Embeddings of Rearrangement-Invariant Spaces

Compactness of embeddings between rearrangement invariant spaces is closely related to absolute continuity of these embeddings. We study absolutely continuous embeddings between rearrangement invariant spaces. In particular it is shown that an absolutely continuous embedding is never optimal. We give sufficient (and under additional hypotheses necessary) conditions for absolute continuity of these embeddings. We also provide quantitative estimates of absolutely continuous embeddings.     

Compact Embeddings of Besov Spaces in Exponential Orlicz Spaces

Let 1 < p < , 0 < v < p', let be a bounded domainin Rⁿ, and denote by id the limiting compact embedding of theBesov space (Rⁿ) into the exponentialOrlicz space L_exp(t^v)(), mapping a function f onto its restrictionf|. In 1993 Triebel established, among others, two-sided estimatesfor the entropy numbers of id, which are even asymptoticallyoptimal for ‘small’ . The aim of the paper is toimprove the upper bounds in the case of ‘large’, where Triebel's estimates are not yet sharp, thus making afurther step towards the conjectured correct asymptotic behaviour.     

Strict Embeddings of Rearrangement Invariant Spaces

A Banach space E of measurable functions on [0,1] is called rearrangement invariant if E is a Banach lattice and equimeasurable functions have identical norms. The canonical inclusion E ? F of two rearrangement invariant spaces is said to be strict if functions from the unit ball of E have absolutely equicontinuous norms in F. Necessary and sufficient conditions for the strictness of canonical inclusion for Orlicz, Lorentz, and Marcinkiewicz spaces are obtained, and the relations of this concept to the disjoint strict singularity are studied.     

On the Generation of Dual Polar Spaces of Unitary Type Over Finite Fields

It is demonstrated that the generating rank of the dual polar space of typeU_2n(q²) is when q > 2. It is also shown that this is equal to the embedding rank of this geometry.     

On Embeddings of Tori in Euclidean Spaces

Using the relation between the set of embeddings of tori into Euclidean spaces modulo ambient isotopies and the homotopy groups of Stiefel manifolds, we prove new results on embeddings of tori into Euclidean spaces.     

Interpolation and Embeddings of Weighted Tent Spaces

Given a metric measure space X, we consider a scale of function spaces \(T^{p,q}_s(X)\), called the weighted tent space scale. This is an extension of the tent space scale of Coifman, Meyer, and Stein. Under various geometric assumptions on X we identify some associated interpolation spaces, in particular certain real interpolation spaces. These are identified with a new scale of function spaces, which we call Z -spaces, that have recently appeared in the work of Barton and Mayboroda on elliptic boundary value problems with boundary data in Besov spaces. We also prove Hardy–Littlewood–Sobolev-type embeddings between weighted tent spaces.     

Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

Polar Foliations of Symmetric Spaces

We prove that a polar foliation of codimension at least three in an irreducible compact symmetric space is hyperpolar, unless the symmetric space has rank one. For reducible symmetric spaces of compact type, we derive decomposition results for polar foliations.     

Entropy Numbers of Embeddings of Weighted Besov Spaces

We investigate the asymptotic behavior of the entropy numbers of the compact embedding $$ B^{s_1}_{p_1,q_1} \!\!(\mbox{\footnotesize\bf R}^d, \alpha) \hookrightarrow B^{s_2}_{p_2,q_2} \!\!({\xxR}). $$ Here $B^s_{p,q} \!({\mbox{\footnotesize\bf R}^d}, \alpha)$ denotes a weighted Besov space, where the weight is given by $w_\alpha (x) = (1+| x |^2)^{\alpha/2}$, and $B^{s_2}_{p_2,q_2} \!({\mbox{\footnotesize\bf R}^d})$ denotes the unweighted Besov space, respectively. We shall concentrate on the so-called limiting situation given by the following constellation of parameters: $s_2 < s_1$, $0 < p_1,p_2 \le \infty$, and $$ \alpha = s_1 - \frac{d}{p_1} - s_2 + \frac{d}{p_2} > d \, \max \Big(0, \frac{1}{p_2}-\frac{1}{p_1}\Big). $$ In almost all cases we give a sharp two-sided estimate.     

Extension and Boundedness of Operators on Morrey Spaces from Extrapolation Techniques and Embeddings

We prove that operators satisfying the hypotheses of the extrapolation theorem for Muckenhoupt weights are bounded on weighted Morrey spaces. As a consequence, we obtain at once a number of results that have been proved individually for many operators. On the other hand, our theorems provide a variety of new results even for the unweighted case because we do not use any representation formula or pointwise bound of the operator as was assumed by previous authors. To extend the operators to Morrey spaces we show different (continuous) embeddings of (weighted) Morrey spaces into appropriate Muckenhoupt \(A_1\) weighted \(L_p\) spaces, which enable us to define the operators on the considered Morrey spaces by restriction. In this way, we can avoid the delicate problem of the definition of the operator, often ignored by the authors. In dealing with the extension problem through the embeddings (instead of using duality), one is neither restricted in the parameter range of the p’s (in particular \(p=1\) is admissible and applies to weak-type inequalities) nor the operator has to be linear. Another remarkable consequence of our results is that vector-valued inequalities in Morrey spaces are automatically deduced. On the other hand, we also obtain \(A_\infty \)-weighted inequalities with Morrey quasinorms.     

Osculating Hyperplanes and Asymptotic Directions of Codimension Two Submanifolds of Euclidean Spaces

We define the concepts of binormal and asymptotic directions for submanifolds embedded with codimension 2 into Euclidean spaces and obtain necessary conditions, in terms of the existence of such directions, for the convexity and the sphericity of these submanifolds.     

Embeddings of Rearrangement Invariant Spaces that are not Strictly Singular

We give partial answers to the following conjecture: the natural embedding of a rearrangement invariant space E into L ₁([0,1]) is strictly singular if and only if G does not embed into E continuously, where G is the closure of the simple functions in the Orlicz space L with (x) = exp(x²)-1.     

Uniform Embeddings of Free Products into Uniformly Convex Banach Spaces

Let A and B be countable discrete groups, and let = A * B betheir free product. We show that if A and B are uniformly embeddableinto a uniformly convex Banach space, then so is . 2000 MathematicsSubject Classification 46L89, 20F65.