The Fourier integral operators on hardy spaces associated with Herz spaces

In this paper, it is proved that the Fourier integral operators of order m, with −n < m ⩽ −(n−1)/2, are bounded from three kinds of Hardy spaces associated with Herz spaces to their corresponding Herz spaces.     

Kakeya-type sets in finite vector spaces

For a finite vector space V and a nonnegative integer r≤dim V, we estimate the smallest possible size of a subset of V, containing a translate of every r-dimensional subspace. In particular, we show that if K⊆V is the smallest subset with this property, n denotes the dimension of V, and q is the size of the underlying field, then for r bounded and r<n≤rq ^r−1, we have |V∖K|=Θ(nq ^n−r+1); this improves the previously known bounds |V∖K|=Ω(q ^n−r+1) and |V∖K|=O(n ² q ^n−r+1).     

On cardinalities of row spaces of Boolean matrices

We prove that there are exactlyn numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of row spaces ofn×n Boolean matrices. The numbers are: 2 ⁿ⁻¹+1,2 ⁿ⁻¹+2,2 ⁿ⁻¹+4, ..., 2 ⁿ⁻¹+2 ⁿ⁻², 2 ⁿ . Two consequences follow. The first is that the height of the partial order ofD-classes in the semigroup ofn×n Boolean matrices is at most 2 ⁿ⁻¹+n−1. The second is that the numbers listed above are precisely the numbers greater than 2 ⁿ⁻¹ that can serve as the cardinalities of topologies on a finite setX withn elements.     

Dynamical borel-cantelli lemma for hyperbolic spaces

We prove that almost every (resp. almost no) geodesic rays in a finite volume hyperbolic manifold of real dimensionn intersects for arbitrary large timest a decreasing family of balls of radiusr _t, provided the integral ∫ ₀ ^∞ r _t ⁿ −1 dt diverges (resp. converges).     

Potential spaces and traces of Lévy processes on <Emphasis Type="Italic">h</Emphasis>-sets

Potential spaces and Dirichlet forms associated with Lévy processes subordinate to Brownian motion in ℝ ⁿ with generator f(−Δ) are investigated. Estimates for the related Rieszand Bessel-type kernels of order s are derived which include the classical case f(r) = r ^α/2 with 0 < α < 2 corresponding to α-stable Lévy processes. For general (tame) Bernstein functions f potential representations of the trace spaces, the trace Dirichlet forms, and the trace processes on fractal h-sets are derived. Here we suppose the trace condition ∫₀¹ r ⁻⁽ⁿ⁺¹⁾ f(r ⁻²)⁻¹ h(r) dr < ∞ on f and the gauge function h. Dedicated to the 80th birthday of Klaus Krickeberg     

Potential space estimates in local Hardy spaces for Green potentials in convex domains

Let Ω be a bounded co.nvex domain in Rn(n≥3) and G(x,y) be the Green function of the Laplace operator -△ on Ω. Let hrp(Ω) = {f ∈ D'(Ω) :(E)F∈hp(Rn), s.t. F|Ω = f}, by the atom characterization of Local Hardy spaces in a bounded Lipschitz domain, the bound of f→(△)2(Gf) for every f ∈ hrp(Ω) is obtained, where n/(n 1)＜p≤1.     

T 1 theorem for Besov spaces on nonhomogeneous spaces

Suppose μ is a Radon measure on ℝ ^d , which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant C₀>0 such that for all x∈supp(μ) and r>0, μ(B(x, r))⪯C₀rⁿ, where 0<n⪯d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa’s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7].     

Integrals and Banach spaces for finite order distributions

Let B _c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B _r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A _c ⁿ to be the space of tempered distributions that are the nth distributional derivative of a unique function in B _c . Similarly with A _r ⁿ from B _r . A type of integral is defined on distributions in A _c ⁿ and A _r ⁿ . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A _c ⁿ and A _r ⁿ are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B _c and B _r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A _c ¹ is the completion of the L ¹ functions in the Alexiewicz norm. The space A _r ¹ contains all finite signed Borel measures. Many of the usual properties of integrals hold: H?lder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.     

The Dissipative Quasi-geostrophic Equation in Weak Morrey Spaces

The author studies the Cauchy problem of the dissipative quasi-geostrophic equation in weak Morrey spaces. The global well-posedness is established for any small initial data in the weak space Mp^＊,γ（R^n）, with 1〈p〈∞and A = n-（2α-1）p, and for a small external force in a time-weighted weak Morrey space.     

Dual Riemannian spaces of constant curvature on a normalized hypersurface

In this paper, the following results are obtained: 1) It is proved that, in the fourth order differential neighborhood, a regular hypersurface V _n−1 embedded into a projective-metric space K _n , n ≥ 3, intrinsically induces a dual projective-metric space $ \bar K_n $ \bar K_n . 2) An invariant analytical condition is established under which a normalization of a hypersurface V _n−1 ⊂ K _n (a tangential hypersurface $ \bar V_{n - 1} $ \bar V_{n - 1} ⊂ $ \bar K_n $ \bar K_n ) by quasitensor fields H _n ⁱ , H _i ($ \bar H_n^i $ \bar H_n^i , $ \bar H_i $ \bar H_i ) induces a Riemannian space of constant curvature. If the two conditions are fulfilled simultaneously, the spaces R _n−1 and $ \bar R_{n - 1} $ \bar R_{n - 1} are spaces of the same constant curvature $ K = - \tfrac{1} {c} $ K = - \tfrac{1} {c} . 3) Geometric interpretations of the obtained analytical conditions are given.     

On the Construction of Permutation Arrays via Mappings from Binary Vectors to Permutations

An (n, d, k)-mapping f is a mapping from binary vectors of length n to permutations of length n + k such that for all x, y {0,1}ⁿ, d_H (f(x), f(y)) ≥ d_H (x, y) + d, if d_H (x, y) ≤ (n + k) − d and d_H (f(x), f(y)) = n + k, if d_H (x, y) > (n + k) − d. In this paper, we construct an (n,3,2)-mapping for any positive integer n ≥ 6. An (n, r)-permutation array is a permutation array of length n and any two permutations of which have Hamming distance at least r. Let P(n, r) denote the maximum size of an (n, r)-permutation array and A(n, r) denote the same setting for binary codes. Applying (n,3,2)-mappings to the design of permutation array, we can construct an efficient permutation array (easy to encode and decode) with better code rate than previous results [Chang (2005). IEEE Trans inf theory 51:359–365, Chang et al. (2003). IEEE Trans Inf Theory 49:1054–1059; Huang et al. (submitted)]. More precisely, we obtain that, for n ≥ 8, P(n, r) ≥ A(n − 2, r − 3) > A(n − 1,r − 2) = A(n, r − 1) when n is even and P(n, r) ≥ A(n − 2, r − 3) = A(n − 1, r − 2) > A(n, r − 1) when n is odd. This improves the best bound A(n − 1,r − 2) so far [Huang et al. (submitted)] for n ≥ 8. The work was supported in part by the National Science Council of Taiwan under contract NSC-93-2213-E-009-117     

Minimal full polarized embeddings of dual polar spaces

Let Δ be a thick dual polar space of rank n ≥ 2 admitting a full polarized embedding e in a finite-dimensional projective space Σ, i.e., for every point x of Δ, e maps the set of points of Δ at non-maximal distance from x into a hyperplane e∗(x) of Σ. Using a result of Kasikova and Shult [11], we are able the show that there exists up to isomorphisms a unique full polarized embedding of Δ of minimal dimension. We also show that e∗ realizes a full polarized embedding of Δ into a subspace of the dual of Σ, and that e∗ is isomorphic to the minimal full polarized embedding of Δ. In the final section, we will determine the minimal full polarized embeddings of the finite dual polar spaces DQ(2n,q), DQ ⁻(2n+1,q), DH(2n−1,q ²) and DW(2n−1,q) (q odd), but the latter only for n≤ 5. We shall prove that the minimal full polarized embeddings of DQ(2n,q), DQ ⁻(2n+1,q) and DH(2n−1,q ²) are the `natural' ones, whereas this is not always the case for DW(2n−1, q).B. De Bruyn: Postdoctoral Fellow of the Research Foundation - Flanders.     

On the projection and macphail constants ofl n p spaces

We prove that the projection and Macphail constants ofl _n ^p (1≦p≦2) are asymptotically equivalent ton ^1/2 andn ^−1/2 respectively. We also obtain some relations linking certain parameters of general finite dimensional real Banach spaces. This note is a part of the author’s Ph.D. Thesis prepared at the Hebrew University of Jerusalem, under the supervision of Prof. J. Lindenstrauss, to whom the author wishes to express his thanks and appreciation.     

Weak Morrey spaces and strong solutions to the Navier-Stokes equations

We consider the Cauchy problem of Navier-Stokes equations in weak Morrey spaces. We first define a class of weak Morrey type spaces Mp*,λ(Rn) on the basis of Lorentz space Lp,∞ = Lp*(Rn)(in particular, Mp*,0(Rn) = Lp,∞, if p ＞ 1), and study some fundamental properties of them; Second,bounded linear operators on weak Morrey spaces, and establish the bilinear estimate in weak Morrey spaces. Finally, by means of Kato's method and the contraction mapping principle, we prove that the Cauchy problem of Navier-Stokes equations in weak Morrey spaces Mp*,λ(Rn) (1＜p≤n) is time-global well-posed, provided that the initial data are sufficiently small. Moreover, we also obtain the existence and uniqueness of the self-similar solution for Navier-Stokes equations in these spaces, because the weak Morrey space Mp*,n-p(Rn) can admit the singular initial data with a self-similar structure. Hence this paper generalizes Kato's results.     

Cohomogeneity one Alexandrov spaces

We obtain a structure theorem for closed, cohomogeneity one Alexandrov spaces and we classify closed, cohomogeneity one Alexandrov spaces in dimensions 3 and 4. As a corollary we obtain the classification of closed, n-dimensional, cohomogeneity one Alexandrov spaces admitting an isometric T ⁿ⁻¹ action. In contrast to the one- and two-dimensional cases, where it is known that an Alexandrov space is a topological manifold, in dimension 3 the classification contains, in addition to the known cohomogeneity one manifolds, the spherical suspension of \mathbbRP² \mathbb{R}{P^2} , which is not a manifold.     

Cutting hyperplane arrangements

We consider a collectionH ofn hyperplanes in E ^d (where the dimensiond is fixed). An ε-cutting forH is a collection of (possibly unbounded)d-dimensional simplices with disjoint interors, which cover all E ^d and such that the interior of any simplex is intersected by at mostεn hyperplanes ofH. We give a deterministic algorithm for finding a (1/r)-cutting withO(r ^d ) simplices (which is asymptotically optimal). Forr≤n ^1−σ, where δ>0 is arbitrary but fixed, the running time of this algorithm isO(n(logn) ^O(1) r ^d−1). In the plane we achieve a time boundO(nr) forr≤n ^1−δ, which is optimal if we also want to compute the collection of lines intersecting each simplex of the cutting. This improves a result of Agarwal, and gives a conceptually simpler algorithm. For ann point setX⊆E ^d and a parameterr, we can deterministically compute a (1/r)-net of sizeO(rlogr) for the range space (X, {X ϒ R; R is a simplex}), In timeO(n(logn) ^O(1) r ^d−1 +r ^O(1)). The size of the (1/r)-net matches the best known existence result. By a simple transformation, this allows us to find ε-nets for other range spaces usually encountered in computational geometry. These results have numerous applications for derandomizing algorithms in computational geometry without affecting their running time significantly. A preliminary version of this paper appeared inProceedings of the Sixth ACM Symposium on Computational Geometry, Berkeley, 1990, pp. 1–9. Work on this paper was supported by DIMACS Center.     

On webs of maximum rank

This paper contains two results on webs of maximal rank. First, we show that at each point, the web normals for a codimension-r web of maximum r-rank are (r−1)-dimensional generators of a rational normal scroll in the projectivized tangent space to the web domain, an extension of a theorem of Chern and Griffiths in the case r=2 [5]. We use this statement to deduce that webs of maximum rank are almost-Grassmannizable in the sense of Akivis [1]. Second, we show that there are exceptional (that is, maximum rank, but not algebraizable) webs W(2n, n, 2) for all n≥2. The construction relies on the properties of zero-cycles on algebraic K3 surfaces.     

Sobolev spaces associated to the harmonic oscillator

We define the Hermite-Sobolev spaces naturally associated to the harmonic oscillatorH = −δ+|x|². Structural properties, relations with the classical Sobolev spaces, boundedness of operators and almost everywhere convergence of solutions of the Schrodinger equation are also considered.     

On some characterization of probability distributions in Hilbert spaces

Summary The aim of this paper is to prove the following theorem about characterization of probability distributions in Hilbert spaces:Theorem. — Let x₁, x₂, …, x_n be n (n≥3) independent random variables in the Hilbert spaceH, having their characteristic functionals f_k(t) = E[e^i(t,x _k)], (k=1, 2, …, n): let y₁=x₁ + x_n, y₂=x₂ + x_n, …, y_n−1=x_n−1 + x_n. If the characteristic functional f(t₁, t₂, …, t_n−1) of the random variables (y₁, y₂, …, y_n−1) does not vanish, then the joint distribution of (y₁, y₂, …, y_n−1) determines all the distributions of x₁, x₂, …, x_n up to change of location.     

Hardy spaces via distribution spaces

Let ℐ(ℝⁿ) be the Schwartz class on ℝⁿ and ℐ_∞(ℝⁿ) be the collection of functions ϕ ∊ ℐ(ℝⁿ) with additional property that