Littlewood–Paley theorem on spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)</Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

We point out that if the Hardy–Littlewood maximal operator is bounded on the space L ^p(t)(ℝ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ, then the well-known characterization of the spaces L ^p (ℝ), 1 < p < ∞, by the Littlewood–Paley theory extends to the space L ^p(t)(ℝ). We show that, for n > 1 , the Littlewood–Paley operator is bounded on L ^p(t) (ℝ ⁿ ), 1 < a ≤ p(t) ≤ b < ∞, t ∈ ℝ ⁿ , if and only if p(t) = const. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 12, pp. 1709–1715, December, 2008.     

Properties of <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-smooth mappings with one-dimensional gradient range

We find necessary and sufficient conditions for a curve in ℝ ^m×n to be the gradient range of a C ¹-smooth function υ: Ω ⊂ ℝ ⁿ → ℝ ^m . We show that this curve has tangents in a weak sense; these tangents are rank 1 matrices and their directions constitute a function of bounded variation. We prove also that in this case v satisfies an analog of Sard’s theorem, while the level sets of the gradient mapping ▿υ: Ω → ℝ ^m×n are hyperplanes.     

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.     

Some remarks on the Hardy-Littlewood maximal function on variable <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

We show that any pointwise multiplier for BMO(ℝⁿ) generates a function p from the class (ℝⁿ) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable L^p space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions which nevertheless belong to (ℝⁿ).     

S<Superscript>1</Superscript>-Valued Sobolev maps

We describe the structure of the space W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) , where 0 < s < ∞ and 1 ≤ p < ∞. According to the values of s, p, and n, maps in W^s,p( \mathbbSⁿ;\mathbbS¹ ) {W^{s,p}}\left( {{\mathbb{S}^n};{\mathbb{S}^1}} \right) can either be characterised by their phases or by a couple (singular set, phase).     

Atomic hardy-type spaces between <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript> on metric spaces with non-doubling measures

Let ( Y,d,dl )\left( {\mathcal{Y},d,d\lambda } \right) be (ℝ ⁿ , |·|, μ), where |·| is the Euclidean distance, μ is a nonnegative Radon measure on ℝ ⁿ satisfying the polynomial growth condition, or the Gauss measure metric space (ℝ ⁿ , |·|, d _λ ), or the space (S, d, ρ), where S ≡ ℝ ⁿ ⋉ ℝ⁺ is the (ax + b)-group, d is the left-invariant Riemannian metric and ρ is the right Haar measure on S with exponential growth. In this paper, the authors introduce and establish some properties of the atomic Hardy-type spaces { X_s ( Y ) }_{0 < s \leqslant ￥}\left\{ {X_s \left( \mathcal{Y} \right)} \right\}_{0 < s \leqslant \infty } and the BMO-type spaces { BMO( Y, s ) }_{0 < s \leqslant ￥}\left\{ {BMO\left( {\mathcal{Y}, s} \right)} \right\}_{0 < s \leqslant \infty }. Let H ¹ ( Y )\left( \mathcal{Y} \right) be the known atomic Hardy space and L ₀¹ ( Y )\left( \mathcal{Y} \right) the subspace of f ∈ L ¹ ( Y )\left( \mathcal{Y} \right) with integral 0. The authors prove that the dual space of X _s ( Y )\left( \mathcal{Y} \right) is BMO( Y,s )BMO\left( {\mathcal{Y},s} \right) when s ∈ (0,∞), X _s ( Y )\left( \mathcal{Y} \right) = H ¹ ( Y )\left( \mathcal{Y} \right) when s ∈ (0, 1], and X _∞ ( Y )\left( \mathcal{Y} \right) = L ₀¹ ( Y )\left( \mathcal{Y} \right) (or L ¹ ( Y )\left( \mathcal{Y} \right)). As applications, the authors show that if T is a linear operator bounded from H ¹ ( Y )\left( \mathcal{Y} \right) to L ¹ ( Y )\left( \mathcal{Y} \right) and from L ¹ ( Y )\left( \mathcal{Y} \right) to L ^1,∞ ( Y )\left( \mathcal{Y} \right), then for all r ∈ (1,∞) and s ∈ (r,∞], T is bounded from X _r ( Y )\left( \mathcal{Y} \right) to the Lorentz space L ^1,s ( Y )\left( \mathcal{Y} \right), which applies to the Calderón-Zygmund operator on (ℝ ⁿ , |·|, μ), the imaginary powers of the Ornstein-Uhlenbeck operator on (ℝ ⁿ , |·|, d _γ ) and the spectral operator associated with the spectral multiplier on (S, d, ρ). All these results generalize the corresponding results of Sweezy, Abu-Shammala and Torchinsky on Euclidean spaces.     

A direct proof for lower semicontinuity of polyconvex functionals

Lower semicontinuity for polyconvex functionals of the form ∫_Ω g(detDu)dx with respect to sequences of functions fromW ^1,n (Ω;ℝ ⁿ ) which converge inL ¹ (Ωℝ ⁿ ) and are uniformly bounded inW ^1,n−1 (Ω;ℝ ⁿ ), is proved. This was first established in [5] using results from [1] on Cartesian Currents. We give a simple direct proof which does not involve currents. We also show how the method extends to prove natural, essentially optimal, generalizations of these results. Supported by MURST, Gruppo Nazionale 40% Partially supported by Australian Research Council     

Multi-parameter singular radon transforms I: The <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript> theory

The purpose of this paper is to study the L ² boundedness of operators of the form f ↦ ψ(x) ∫ f (γ _t (x))K(t)dt, where γ _t (x) is a C ^∞ function defined on a neighborhood of the origin in (t, x) ∈ ℝ ^N × ℝ ⁿ , satisfying γ ₀(x) ≡ x, ψ is a C ^∞ cut-off function supported on a small neighborhood of 0 ∈ ℝ ⁿ , and K is a “multi-parameter singular kernel” supported on a small neighborhood of 0 ∈ ℝ ^N . The goal is, given an appropriate class of kernels K, to give conditions on γ such that every operator of the above form is bounded on L ². The case when K is a Calderón-Zygmund kernel was studied by Christ, Nagel, Stein, and Wainger; we generalize their conditions to the case when K has a “multi-parameter” structure. For example, when K is given by a “product kernel.” Even when K is a Calderón- Zygmund kernel, our methods yield some new results. This is the first paper in a three part series, the later two of which are joint with E. M. Stein. The second paper deals with the related question of L ^p boundedness, while the third paper deals with the special case when γ is real analytic.     

On Transference of Multipliers on Matrix Weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Spaces

We consider a periodic matrix weight W defined on ℝ ^d and taking values in the N×N positive-definite matrices. For such weights, we prove transference results between multiplier operators on L _p (ℝ ^d ;W) and L_p(\mathbb T^d;W)L_{p}(\mathbb {T}^{d};W), 1<p<∞, respectively. As a specific application, we study transference results for homogeneous multipliers of degree zero.     

Fundamental groups of manifolds with <Emphasis Type="Italic">S</Emphasis><Superscript>1</Superscript>-category 2

A closed topological n-manifold M ⁿ is of S ¹-category 2 if it can be covered by two open subsets W ₁,W ₂ such that the inclusions W _i → M ⁿ factor homotopically through maps W _i → S ¹ → M ⁿ . We show that the fundamental group of such an n-manifold is a cyclic group or a free product of two cyclic groups with nontrivial amalgamation. In particular, if n = 3, the fundamental group is cyclic.      

Integral self-affine tiles in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> part II: Lattice tilings

Let A be an expanding n×n integer matrix with |det(A)|=m. Astandard digit set D for A is any complete set of coset representatives forℤ ⁿ /A(ℤ ⁿ ). Associated to a given D is a setT (A, D), which is the attractor of an affine iterated function system, satisfyingT=∪ _d∈D (T+d). It is known thatT (A, D) tilesℝ ⁿ by some subset ofℤ ⁿ . This paper proves that every standard digit set D gives a setT (A, D) that tilesℝ ⁿ with a lattice tiling.     

Approximation by K-finite functions in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> spaces

Let Γ ⊂ ℝⁿ, n ≥ 2, be the boundary of a bounded domain. We prove that the translates by elements of Γ of functions which transform according to a fixed irreducible representation of the orthogonal group form a dense class in L ^p (ℝⁿ) for . A similar problem for noncompact symmetric spaces of rank one is also considered. We also study the connection of the above problem with the injectivity sets for weighted spherical mean operators. The first author was supported in part by a grant from UGC via DSA-SAP Phase IV.     

Exact order of relative widths of classes <Emphasis Type="Italic">W</Emphasis><Stack><Subscript>1</Subscript><Superscript>r</Superscript></Stack> in the space <Emphasis Type="Italic">L</Emphasis><Subscript>1</Subscript>

We determine the exact order of relative widths of classes W ₁ ^r of periodic functions in the space L ₁ as n → ∞ under restrictions on higher derivatives of approximating functions. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 10, pp. 1409–1417, October, 2005.     

Realizability of the <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">k</Emphasis></Subscript>-distance functions by homology classes of path spaces

In previous papers, we have constructed and studied mappings d _k : M × M → ℝ called the H _k -distance functions. The main result of this paper is a theorem on realizability of the generalized distances d _k (υ, w), υ, w ∈ M, by critical values of the length functional L: Ω(M, υ, w) → ℝ generated by nontrivial homology classes of the space Ω(M, υ, w) of paths joining the points υ and w.     

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.     

Harnack estimate for the <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">k</Emphasis></Superscript>-flow

We obtain the Harnack estimate of the solution to Hk-flow in Euclidean space Rn 1,for k ＞ 0.By using this estimate,we get some corollaries about the translation soliton.     

SOME NEW PROPERTIES ON L^p（Ω, H^n）

Let L^p（Ω, H^n） indicate the L^P-space of the maps for Heisenberg group target. In this paper some new properties are obtained for the space L^p（Ω, H^n）     

On a characterization of orthogonality with respect to particular sequences of random variables in <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>

This note deals with the orthogonality between sequences of random variables. The main idea of the note is to apply the results on equidistant systems of points in a Hilbert space to the case of the space L ²(Ω, F, ℙ) of real square integrable random variables. The main result gives a necessary and sufficient condition for a particular sequence of random variables (elements of which are taken from sets of equidistant elements of L ²(Ω, F, ℙ) to be orthogonal to some other sequence in L ²(Ω, F, ℙ). The result obtained is interesting from the point of view of the time series analysis, since it can be applied to a class of sequences random variables that exhibit a monotonically increasing variance. An application to ergodic theorem is also provided.     

Best approximation by ridge functions in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We study the approximation of the classes of functions by the manifold R _n formed by all possible linear combinations of n ridge functions of the form r(a · x)): It is proved that, for any 1 ≤ q ≤ p ≤ ∞, the deviation of the Sobolev class W ^r _p from the set R _n of ridge functions in the space L _q (B ^d ) satisfies the sharp order n ^-r/(d-1).     

Von Neumann-Jordan Constants of Absolute Normalized Norms on C^n

In this note, we give some estimations of the Von Neumann-Jordan constant C _{N J} (∥·∥_ψ) of Banach space (ℂ ⁿ , ∥·∥_ψ), where ∥·∥_ψ is the absolute normalized norm on ℂ ⁿ given by function ψ. In the case where ψ and φ are comparable, n=2 and C _{N J} (∥·∥_ψ)=1, we obtain a formula of computing C _{N J} (∥·∥_ψ). Our results generalize some results due to Saito and others. Received May 11, 2002, Accepted November 20, 2002 This work is partly supported by NNSF of China (No. 19771056)