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.     

On the maximal operator associated with certain rotational invariant measures

The aim of this work is to investigate the integrability properties of the maximal operator Mu,associated with a non-doubling measure μ defined on Rn. We start by establishing for a wide class of radial and increasing measures μ that Mu is bounded on all the spaces Lu^p（R^n）,P〉1.Also,we show that there is a radial and increasing measure p for which Mμ does not map Lμ^p（R^n） into weak Lμ^p（R^n）,1≤p〈∞.     

Compactness of Commutators of Riesz Potential on Morrey Spaces

In this paper, the authors give a characterization of the (L ^{p, λ} , L ^{q, λ} )-compactness for the Riesz potential commutator [b,I _α ]. More precisely, the authors prove that the commutator [b,I _α ] is a compact operator from the Morrey space L ^{p, λ} (ℝ ⁿ ) to L ^{q, λ} (ℝ ⁿ ) if and only if b ∈ VMO(ℝ ⁿ ), the BMO-closure of . The research was supported by NSF of China (Grant: 10571015, 10826046) and SRFDP of China (Grant: 20050027025).     

Hardy spaces estimates for a class of multilinear homogeneous operators

It is proved that a class of multilinear singular integral operators with homogeneous kernels are bounded operators from the product spaces to the Hardy spacesH ^r , (ℝ ⁿ ) and the weak Hardy spaceH ^r,∞ (ℝ ⁿ . As an application of this result, the L ^p ,(ℝ ⁿ ) boundedness of a class of commutator for the singular integral with homogeneous kernels is obtained. Project supparted in part by the National Natural Science Foundation of Chind (Grant No. 19131080) of China and Doctoral Programme Foundation of Institution of Higher Education (Grant No. 98002703) of China.     

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.     

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)     

The Cauchy problem for nonhomogeneous Navier-Stokes equations inL q([0,T);L p)

In this paper the Cauchy problem for a class of nonhomogeneous Navier-Stokes equations in the infinite cylinderS _T =ℝⁿ x [0,T) is considered. We construct a unique local solution inL ^q([0,T);L ^p (ℝ ⁿ )) for a class of nonhomogeneous Navier-Stokes equations provided that initial data are inL ^r (ℝ ⁿ ), wherer>1 is an exponent determined by the structure of nonlinear terms andp,q are such that 2/q=n(1/r−1/p). Meanwhile under suitable conditions we also obtain thatu(t)≠L ^q([0,∞];L ^p (ℝ ⁿ )) provided that initial data are sufficiently small. This work is supported by the National Natural Sciences Foundation of China and the Foundation of LNM Laboratory of Institute of Mechanics of the Chinese Academy of Sciences.     

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).     

L P boundedness of area function associated with operators on product spaces

Let L be the infinitesimal generator of an analytic semigroup on L2 (Rn) with suitable upper bounds on its heat kernels. Assume that L has a bounded holomorphic functional calculus on L2(Rn). In this paper,we define the Littlewood- Paley g function associated with L on Rn × Rn, denoted by GL(f)(x1, x2), and decomposition, we prove that ‖SL(f)‖p ≈‖GL(f)‖p ≈‖f‖p for 1 ＜ p ＜∞.     

Singular spherical maximal operators on a class of two step nilpotent lie groups

LetH ⁿ ≅ℝ²ⁿ ⋉ℝ be the Heisenberg group and letμ _t be the normalized surface measure for the sphere of radiust in ℝ²ⁿ . Consider the maximal function defined byM f=sup _t>0|f*μ _t |. We prove forn≥2 thatM defines an operator bounded onL ^p (H ⁿ ) provided thatp>2n/(2n−1). This improves an earlier result by Nevo and Thangavelu, and the range forL ^p boundedness is optimal. We also extend the result to a more general class of surfaces and to groups satisfying a nondegeneracy condition; these include the groups of Heisenberg type. The second author was supported in part by the National Science Foundation.     

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.     

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L ^p _a (Ω, dν _α) and for the point-evaluation derivalive functional in Besov spaces B ^p (Ω) are obtained for bounded symmetric domains Ω in ℂ ⁿ . Received October 25, 1999, Accepted December 6, 2000     

On the interpolation of some generalized function spaces of different anisotropy

In the paper, the interpolation properties of the spacesH _p ^s (v; ℝ _n ) of Sobolev-Liouville type and the spacesB _{p, q} ^s (μ ℝ _n ) of Nikol'skii-Besov type generated by functions of polynomial growth that are infinitely differentiable outside of the origin are studied. Interpolation formulas for the pairs {H(v _o ),H(v ₁)} and {B(μ₀),B(μ₁)} of spaces of the above types for which the anisotropies of the interpolated spaces do not depend on each other are proved. The investigated spaces, for certain specification of the generating functions, coincide with the classical (isotropic and anisotropic) Sobolev-Liouville and Nikol'skii-Besov spaces. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 666–672, November, 1997. Translated by A. I. Shtern     

The Helmholtz decomposition of weightedLq-spaces for Muckenhoupt weights

We consider weights of Muckenhoupt classA _q, 1<q<∞. For a bounded Lipschitz domain Ω⊂ℝⁿ we prove a compact embedding and a Poincaré inequality in weighted Sobolev spaces. These technical tools allow us to solve the weak Neumann problem for the Laplace equation in weighted spaces on ℝⁿ, ℝⁿ ₊, on bounded and on exterior domains Ω with boundary of classC ¹, which will yield the Helmholtz decomposition ofL _ω ^q(Ω)ⁿ for general ω∈A _q. This is done by transferring the method of Simader and Sohr [4] to the weighted case. Our result generalizes a result of Farwig and Sohr [2] where the Helmholtz decomposition ofL _ω ^p(Ω)ⁿ is proved for an exterior domain and weights of Muckenhoupt class without singularities or degeneracies in a neighbourhood of ϖΩ.

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Sunto In questo lavoro consideriamo dei pesi della classe di MuckenhouptA _q, 1<q<∞. Per un dominio limitato lipschitziano Ω⊂ℝⁿ, dimostriamo una immersione compatta ed una disuguaglianza di Poincaré in spazi di Sobolev con peso. Questa tecnica ci consente di risolvere il problema debole di Neumann per l’equazione di Laplace in spazi pesati in ℝⁿ, ℝⁿ ₊ in domini limitati ed in domini esterni con frontiera di classeC ¹, che conduce alla decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ per un qualsiasi ω∈A _q. Il risultato è ottenuto trasferendo il metodo di Simader e Sohr [4] al caso pesato. Quello qui presente estende un risultato di Farwig e Sohr [2] dove la decomposizione di Helmholtz diL _ω ^q(Ω)ⁿ è dimostrata per domini esterni e pesi della classe di Muckenhoupt privi di singolarità in un intorno di ϖΩ.

     

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.     

Boundedness of an oscillating multiplier on Triebel-Lizorkin spaces

Abstract In this paper, we study Triebel-Lizorkin space estimates for an oscillating multiplier mΩ,α,β. This operator was initially studied by Wainger and by Fefferman-Stein in the Lebesgue spaces. We obtain the boundedness results on the Triebel-Lizorkin space Fpα,q（R^n） for different p, q.     

Test elements for endomorphisms of free groups and algebras

LetE be a bounded Borel subset of ℝⁿ,n≥2, of positive Lebesgue measure andP _E the corresponding ‘Pompeiu transform”. We prove thatP _E is injective onL ^p(ℝⁿ) if 1≤p≤2n/(n-1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action onL ^q(ℝⁿ) for various values ofq. We also take up the question of whenP _E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.     

Integral representations of functions and embeddings of sobolev spaces on cuspidal domains

Some classes of cuspidal domainsG ⊂ ℝ ⁿ are introduced, and embeddings of the formW _p ^(l) (G)↪L_q(G),l ∈ ℕ, for sobolev spaces are established. To this end, estimates of some integral operators are needed. These operators cannot be estimated via Riesz potentials or their anisotropic analogs. Translated fromMatematicheskie Zametki, Vol. 61, No. 2, pp. 201–219, February, 1997. Translated by V. E. Nazaikinskii     

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

On the stability of solutions to quadratic programming problems

We consider the parametric programming problem (Q _p ) of minimizing the quadratic function f(x,p):=x ^T Ax+b ^T x subject to the constraint Cx≤d, where x∈ℝ ⁿ , A∈ℝ ^n×n , b∈ℝ ⁿ , C∈ℝ ^m×n , d∈ℝ ^m , and p:=(A,b,C,d) is the parameter. Here, the matrix A is not assumed to be positive semidefinite. The set of the global minimizers and the set of the local minimizers to (Q _p ) are denoted by M(p) and M ^loc (p), respectively. It is proved that if the point-to-set mapping M ^loc (·) is lower semicontinuous at p then M ^loc (p) is a nonempty set which consists of at most ? _m,n points, where ? _m,n = is the maximal cardinality of the antichains of distinct subsets of {1,2,...,m} which have at most n elements. It is proved also that the lower semicontinuity of M(·) at p implies that M(p) is a singleton. Under some regularity assumption, these necessary conditions become the sufficient ones. Received: November 5, 1997 / Accepted: September 12, 2000?Published online November 17, 2000