SOME MULTIPLIER THEOREMS FOR ANISOTROPIC HARDY SPACES ——In Memory of Professor Yongsheng Sun

Let A be a symmetric expansive matrix and Hp(Rn) be the anisotropic Hardy space associated with A. For a function m in L∞(Rn), an appropriately chosen function η in Cc∞(Rn) and j ∈ Z define mj(ξ) = m(Ajξ)η(ξ). The authors show that if 0 ＜ p ＜ 1 and (m)j belongs to the anisotropic nonhomogeneous Herz space K11/p-1,p(Rn), then m is a Fourier multiplier from Hp(Rn) to Lp(Rn). For p = 1, a similar result is obtained if the space K10,1(Rn) is replaced by a slightly smaller space K(w).Moreover, the authors show that if 0 ＜ p ≤ 1 and if the sequence {(mj)V} belongs to a certain mixednorm space, depending on p, then m is also a Fourier multiplier from Hp(Rn) to Lp(Rn).     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

Boundedness of Marcinkiewicz integrals and their commutators in H1 (?n × ?m)

The authors establish the boundedness of Marcinkiewicz integrals from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ¹(ℝ ⁿ × ℝ ^m ) and their commutators with Lipschitz functions from the Hardy space H ¹ (ℝ ⁿ × ℝ ^m ) to the Lebesgue space L ^q (ℝ ⁿ × ℝ ^m ) for some q > 1.     

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

Riesz transformations of distributions and a generalized Hardy space

LeiE(ℝⁿ) be the space of all functions on ℝⁿ which can continue to the entire holomorphic functions on ℂⁿ. We define Riesz transformation R_j of distributions as a linear transformation of the quotient spaceD′(ℝⁿ)/E(ℝⁿ) to itself, j=1,2,..., n. These generalized Riesz transformations share the same properties with the classical ones, such as . As applications we generalize further a theorem of F. & M. Riesz generalized by Stein and Weiss, and then define a generalized Hardy space, of which some properties are studied.     

Characterization for commutators of n -dimensional fractional Hardy operators

In this paper, it was proved that the commutator generated by an n-dimensional fractional Hardy operator and a locally integrable function b is bounded from L ^p1 (ℝ ⁿ ) to L ^p2 (ℝ ⁿ ) if and only if b is a CṀO(ℝ ⁿ ) function, where 1/p ₁ − 1/p ₂ = β/n, 1 < p ₁ < ∞, 0 ⩽ β < n. Furthermore, the characterization of on the homogenous Herz space (ℝ ⁿ ) was obtained. This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 10571014, 10371080) and the Doctoral Programme Foundation of Institute of Higher Education of China (Grant No. 20040027001)     

Hardy Spaces on ? n with Pointwise Variable Anisotropy

In this work we develop highly geometric Hardy spaces, for the full range 0<p≤1. These spaces are constructed over multi-level ellipsoid covers of ℝ ⁿ that are highly anisotropic in the sense that the ellipsoids can change shape rapidly from point to point and from level to level. This generalizes previous work on anisotropic Hardy spaces where the geometry of the space was ‘fixed’ over ℝ ⁿ and extends Hardy spaces over spaces of homogeneous type, where the theory holds for p values that are ‘close’ to 1.     

Pseudodifferential Operators on Local Hardy Spaces

In this work, we study the continuity of pseudodifferential operators on local Hardy spaces h ^p (ℝ ⁿ ) and generalize the results due to Goldberg and Taylor by showing that operators with symbols in S _1,δ ⁰(ℝ ⁿ ), 0≤δ<1, and in some subclasses of S _1,1⁰(ℝ ⁿ ) are bounded on h ^p (ℝ ⁿ ) (0<p≤1). As an application, we study the local solvability of the planar vector field L=∂ _t +ib(x,t)∂ _x , b(x,t)≥0, in spaces of mixed norm involving Hardy spaces. Work supported in part by CNPq, FINEP, and FAPESP.     

Convergence of Sobolev spaces on varying manifolds

We deal with variational problems on varying manifolds in ℝⁿ. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝⁿ. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μ_h -a.e., being {μ_h} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs (P_{μ h} H(μ_h)), where P_{μ h} are the projectors onto the tangent spaces T_{μ h}. When μ_h belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary.     

Decompositions of the Hardy space Hz^1（Ω）

We give a decomposition of the Hardy space Hz^1（Ω） into ＂div-curl＂ quantities for Lipschitz domains in R^n. We also prove a decomposition of Hz^1（Ω） into Jacobians det Du, u ∈ W0^1,2 （Ω,R^2） for Ω in R^2. This partially answers a well-known open problem.     

Estimates of Jacobians by subdeterminants

Let ƒ: Ω → ℝⁿ be a mapping in the Sobolev space W^1,n−1(Ω,ℝⁿ), n ≥ 2. We assume that the determinant of the differential matrix Dƒ (x) is nonnegative, while the cofactor matrix D^#ƒ satisfies , where L^p(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L _loc ¹ (Ω). Estimates above and below L _loc ¹ (Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions on the differential matrix.     

Large classes of minimally supported frequency wavelets of<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ) and<Emphasis Type="Italic">H</Emphasis><Superscript>2</Superscript>(ℝ)

We introduce a new method to construct large classes of minimally supported frequency (MSF) wavelets of the Hardy space H ² (ℝ)and symmetric MSF wavelets of L ² (ℝ),and discuss the classification of such wavelets. As an application, we show that there are uncountably many such wavelet sets of L ² (ℝ)and H ² (ℝ).We also enumerate some of the symmetric wavelet sets of L ² (ℝ)and all wavelet sets of H ² (ℝ)consisting of three intervals. Finally, we construct families of MSF wavelets of L ₂ (ℝ)with Fourier transform even and not vanishing in any neighborhood of the origin.     

The maximal (C, α, β) operator of two-parameter walsh-fourier series

The two-parameter dyadic martingale Hardy spacesH _p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from H_p to L_p (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H ₁ ^# , L₁), where the Hardy space H ₁ ^# is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H ₁ ^# converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on H_p whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈H_p, the (C, α, β) means converge to f in H_p norm. The same results are proved for the conjugate (C, α, β) means, too.     

Geometry of real grassmann manifolds. Parts I, II

The paper deals with the properties of the exterior algebra ℝ(Λⁿ) related to the Euclidean structure on ℝ(Λⁿ) induced by the scalar product in ℝ(Λⁿ). A geometric interpretation of inner multiplication for simple p-vectors is given. An invariant form of the Cartan criterion for the simplicity of a p-vector is given. The Plücker model realizing the real Grassmann manifold as a submanifold of the Euclidean space ℝ(Λⁿ), and an isometry of this submanifold onto the classical Grassmann manifold with SO(n)-invariant metric are described. A canonical decomposition of bivectors is given. Bibliography: 12 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 246, 1997, pp. 84–107. Translated by N. Yu: Netsvetaev.     

The classification of complete orientable 2-dimensional area-minimizing mod 2 surfaces in ℝn

Let S ⊂ ℝⁿ be a complete 2-dimensional areaminimizing mod 2 surface. Then S = x₁ (M₁) ∪ … ∪ x_r (M_r) where each M_j is connected, x_j: M_j → V_j is a classical minimal immersion into an affine subspace V_j of ℝⁿ, and the subspaces V₁,…, V_r are pairwise orthogonal. Here we prove that if M_j is orientable, then x_j (M_j) is either aflat plane or, in suitable coordinates, a generalized complex hyperbola.     

Some remarks on Legendrian rectifiable currents

A rectifiable current of dimension n−1 in the sphere bundle Sℝⁿ≃ℝⁿ×S ^{n −1} for euclidean space is Legendrian if it annihilates the contact 1-form α (i.e. T(α∧φ)=0 for all forms φ of degree n−2). Such a current may be naturally associated to any convex set or to any singular real analytic variety, and induces the curvature measures of such a set. We prove that the projection to ℝⁿ of a carrier of a general such T is C ²-rectifiable in the sense of Anzellotti–Serapioni. We deduce that the boundary of a set with positive reach, as well as its singular skeleta, are C ²-rectifiable. In case ∂T= 0 we prove also that the curvature measures associated to T satisfy the analogues of the classical variational formulas for curvature integrals. It follows that such formulas are valid for the curvature measures of subsets of space forms. Received: 3 December 1997/ Revised version: 25 May 1998     

Discrete characterization of the Paley-Wiener space with several variables

In this paper, we obtain a characterization of the Paley-Wiener space with several variables, which is denoted byB _{π, p} (R ⁿ ), 1≤p<∞, i.e., for 1<p<∞,B _{π, p} (R ⁿ ) is isomorphic tol _p (Z ⁿ ), and forp=1,B _{π, 1} (R ⁿ ) is isomorphic to the discrete Hardy space with several variables, which is denoted byH(Z ⁿ ). This project is supported by the National Natural Science Foundation of China (19671012) and Doctoral Programme Institution of Higher Education Foundation of Chinese Educational Committee and supported by Youth Foundation of Sichuan.     

The Dichotomy Between Traces on d-sets F in R^n and the Density of D（R^n／Г） in Function Spaces

A space Apq^s （R^n） with A ： B or A = F and s ∈R, 0 〈 p, q 〈 ∞ either has a trace in Lp（Г）, where Г is a compact d-set in R^n with 0 〈 d 〈 n, or D（R^n／Г） is dense in it. Related dichotomy numbers are introduced and calculated.     

Totally real minimal 2-spheres in quaternionic projective space

Let HPn be the quaternionic projective space with constant quaternionic sectional curvature 4. Then locally there exists a tripe {I, J, K} of complex structures on HPn satisfying U = -JI = K,JK = -KJ = /, KI = -IK = J. A surface M(？) HPn is called totally real, if at each point p ∈M the tangent plane TPM is perpendicular to I(TPM), J(TPM) and K(TPM). It is known that any surface M(？)RPn(？) HPn is totally real, where RPn (?) HPn is the standard embedding of real projective space in HPn induced by the inclusion R in H, and that there are totally real surfaces in HPn which don't come from this way. In this paper we show that any totally real minimal 2-sphere in HPn is isometric to a full minimal 2-sphere in Rp2m (？) RPn(？) HPn with 2m≤n. As a consequence we show that the Veronese sequences in KP2m (m≥1) are the only totally real minimal 2-spheres with constant curvature in the quaternionic projective space.