Jackson-type inequalities and exact values of widths of classes of functions in the spaces <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>, 1 ≤ <Emphasis Type="Italic">p</Emphasis> < ∞

In the spaces S^p, 1 p < , introduced by Stepanets, we obtain exact Jackson-type inequalities and compute the exact values of widths of classes of functions determined by averaged moduli of continuity of order m.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 595–605, May, 2004.     

Square functions and <Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> calculus on subspaces of <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and on Hardy spaces

Let X be a (closed) subspace of L^p with 1≤p<∞, and let A be any sectorial operator on X. We consider associated square functions on X, of the form and we show that if A admits a bounded H^∞ functional calculus on X, then these square functions are equivalent to the original norm of X. Then we deduce a similar result when X=H¹(ℝ^N) is the usual Hardy space, for an appropriate choice of || ||_F. For example if N=1, the right choice is the sum for h ∈ H¹(ℝ), where H denotes the Hilbert transform.     

The fractional Hausdorff operators on the Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>)

In this paper, we study the high-dimensional fractional Hausdorff operators and establish their boundedness on the real Hardy spaces H ^p (? ⁿ ) for 0 < p < 1.     

<Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness of Calderón-Zygmund operators on product spaces

In this paper, we prove the product H^p boundedness of Calderón- Zygmund operators which were considered by Fefferman and Stein. The methods used in this paper are new even for the classical H^p boundedness of Calderón- Zygmund operators, namely, using some subtle estimates together with the H^p–L^p boundedness of product vector valued Calderón-Zygmund operators.This project was supported by the NNSF (No. 10271015 & No. 10310201047) of China and the second (corresponding) author was also supported by the RFDP (No. 20020027004) of China.Mathematics Subject Classification (2000):42B20, 42B30, 42B25     

Bases of exponents in weighted spaces <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">−π, π</Emphasis>)

The system of exponents $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} $ \left\{ {e^{i\lambda _n t} } \right\}_{n \in \mathbb{Z}} is considered. A sufficient condition for a Riesz-property basis in the weighted space L ^p (−π, π) is obtained.     

An extremal problem in the Hardy space <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for 0&lt;<Emphasis Type="Italic">p</Emphasis>&lt;∞

We prove that if the function determining a linear functional over the Hardy space is analytic on the disk of radius greater than 1 then the extremal function of this functional is analytic on the same disk.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Best orthogonal trigonometric approximations of the classes <Emphasis Type="Italic">B</Emphasis><Superscript>Ω</Superscript><Subscript><Emphasis Type="Italic">p</Emphasis>,θ</Subscript> of periodic functions of many variables

We obtain exact-order estimates for the best orthogonal trigonometric approximations of the classes B ^Ω _p,θ of periodic functions of many variables in the space L _q .     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

On the regularity of scalar <Emphasis Type="Italic">p</Emphasis>-Harmonic functions in <Emphasis Type="Italic">n</Emphasis> dimensions with 2 ≤ <Emphasis Type="Italic">p</Emphasis> < <Emphasis Type="Italic">n</Emphasis>

Using Tilli’s technique [Cal Var 25(3):395–401, 2006], we shall give a new proof of the regularity of the local minima of the functional

$J\left( u\right) =\int\limits_{\Omega } \left\vert \partial u\right\vert^{p}\,dx$

with Ω a domain of class C ^{0, 1} in \({\mathbb{R}^{n}}\) and 2 ≤ p < n.

     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> and <Emphasis Type="Italic">L</Emphasis><Superscript>∞</Superscript> norm estimates of the cost of the controllability for heat equations

This paper is concerned with the bound of the cost of approximate controllability and null controllability of heat equations, i.e., the minimal Lp norm and L∞ norm of a control needed to control the system approximately or a control needed to steer the state of the system to zero. The methods we use combine observability inequalities, energy estimates for heat equations and the dual theory.     

Cycles in <Emphasis Type="Italic">G</Emphasis>-orbits in <Emphasis Type="Italic">G</Emphasis><Superscript><Emphasis Type="Italic">ℂ</Emphasis></Superscript>-flag manifolds

There is a natural duality between orbits of a real form G of a complex semisimple group G on a homogeneous rational manifold Z=G /P and those of the complexification K of any of its maximal compact subgroups K: (,) is a dual pair if is a K-orbit. The cycle space C() is defined to be the connected component containing the identity of the interior of {g:g() is non-empty and compact}. Using methods which were recently developed for the case of open G-orbits, geometric properties of cycles are proved, and it is shown that C() is contained in a domain defined by incidence geometry. In the non-Hermitian case this is a key ingredient for proving that C() is a certain explicitly computable universal domain.Research of the first author partially supported by Schwerpunkt Global methods in complex geometry and SFB-237 of the Deutsche Forschungsgemeinschaft.The second author was supported by a stipend of the Deutsche Akademische Austauschdienst.     

Lower semicontinuity of mappings with bounded (<Emphasis Type="Italic">θ</Emphasis>, 1)-weighted (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-distortion

We prove that, under some extra conditions, the locally uniform limit of mappings with bounded (θ, 1)-weighted (p, q)-distortion is a mapping of bounded (θ, 1)-weighted (p, q)-distortion too. Moreover, we obtain the lower semicontinuity of the distortion coefficients.     

On existence of a universal function for <Emphasis Type="Italic">L</Emphasis> <Superscript> <Emphasis Type="Italic">p</Emphasis> </Superscript>[0, 1] with <Emphasis Type="Italic">p</Emphasis>∈(0, 1)

We show that, for every number p ∈ (0, 1), there is g ∈ L¹[0, 1] (a universal function) that has monotone coefficients c_k(g) and the Fourier–Walsh series convergent to g (in the norm of L¹[0, 1]) such that, for every f ∈ L^p[0, 1], there are numbers δ_k = ±1, 0 and an increasing sequence of positive integers N_q such that the series ∑ _k=0^+∞δ_kc_k(g)W_k (with {W_k} theWalsh system) and the subsequence \(\sigma _{{N_q}}^{\left( \alpha \right)}\), α ∈ (?1, 0), of its Cesáro means converge to f in the metric of L^p[0, 1].     

Regular<Emphasis Type="Italic">n</Emphasis>-simplices in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> with vertices in ℤ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper theI andII regularn-simplices are introduced. We prove that the sufficient and necessary conditions for existence of anI regularn-simplex in ℝ ⁿ are that ifn is even thenn = 4m(m + 1), and ifn is odd thenn = 4m + 1 with thatn + 1 can be expressed as a sum of two integral squares orn = 4m - 1, and that the sufficient and necessary condition for existence of aII regularn-simplex in ℝ ⁿ isn = 2m ² - 1 orn = 4m(m + 1)(m ∈ ℕ). The connection between regularn-simplex in ℝ ⁿ and combinational design is given.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the Schrödinger type operators

Let L _k = (?Δ) ^k + V ^k be a Schrödinger type operator, where k ≥ 1 is a positive integer and V is a nonnegative polynomial. We obtain the L ^p estimates for the operators ?^2k L _k ^?1 and ? ^k L _k ^?1/2 .     

Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling <Emphasis Type="Italic">ν</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, ·) → + ∞ AS <Emphasis Type="Italic">p</Emphasis> → + ∞

Over a large range of the pressure, one cannot ignore the fact that the viscosity grows significantly (even exponentially) with increasing pressure. This paper concerns long-time and large-data existence results for a generalization of the Navier-Stokes fluid whose viscosity depends on the shear rate and the pressure. The novelty of this result stems from the fact that we allow the viscosity to be an unbounded function of pressure as it becomes infinite. In order to include a large class of viscosities and in order to explain the main idea in as simple a manner as possible, we restrict ourselves to a discussion of the spatially periodic problem. K. R. Rajagopal thanks the National Science Foundation for its support.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>-Poincaré inequality for general symmetric forms

Lp Poincare inequalities for general symmetric forms are established by new Cheeger＇s isoperimetric constants. Lp super-Poincare inequalities are introduced to describe the equivalent conditions for the Lp compact embedding, and the criteria via the new Cheeger＇s constants for those inequalities are presented. Finally, the concentration or the volume growth of measures for these inequalities are studied.     

Hochschild Cohomology of Deformation Quantizations over ℤ/<Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>ℤ

Let A ₁ be an Azumaya algebra over a smooth affine symplectic variety X over Spec F _p , where p is an odd prime. Let A be a deformation quantization of A ₁ over the p-adic integers. In this note we show that for all n ≥ 1, the Hochschild cohomology of A/p ⁿ A is isomorphic to the de Rham-Witt complex \(W_{n}{\Omega }^{\ast }_{X}\) of X over \(\mathbb {Z}/p^{n}\mathbb {Z}\). We also compute the center of deformations of certain affine Poisson varieties over F _p .