GLOBAL WELL-POSEDNESS FOR FRACTIONAL NAVIER-STOKES EQUATIONS IN VARIABLE EXPONENT FOURIER-BESOV-MORREY SPACES

In this paper we study the Cauchy problem of the incompressible fractional Navier-Stokes equations in critical variable exponent Fourier-Besov-Morrey space F N^(s(·))p(·),h(·),q(R^(3))with s(·)=4-2α-3/p(·).We prove global well-posedness result with small initial data belonging to FN^(4-2α-3/p(·))p(·),h(·)q(R^(3)).The result of this paper extends some recent work.

MIXED LIPSCHITZ SPACES AND THEIR APPLICATIONS

The purpose of this paper is to introduce bi-parameter mixed Lipschitz spaces and characterize them via the Littlewood-Paley theory.As an application,we derive a boundedness criterion for singular integral operators in a mixed Journéclass on mixed Lipschitz spaces.Key elements of the paper are the development of the Littlewood-Paley theory for a special mixed Besov spaces,and a density argument for the mixed Lipschitz spaces in the weak sense.

Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution

Here we consider the following strongly singular integral TΩ,γ,α,βf（x,t）=∫R^ne^i｜y｜^-βΩ（y/｜y｜）/｜y｜^n＋af（x-y,t-γ（｜y｜））dy, where Ω∈L^p（S^n-1）,p〉1,n〉1,α〉0 and γis convex on （0,∞）.We prove that there exists A（p,n） 〉 0 such that if β 〉 A（p,n） （1 ＋α）, then TΩ,γ,α,β is bounded from L^2 （R^n＋1） to itself and the constant is independent of γ Furthermore,when Ω∈ C^∞ （S^n-1 ）, we will show that TΩ,γ,α,β is bounded from L^2 （R^n＋l） to itself only if β〉 2α and the constant is independent of γ.

Hausdorff Operators on Function Spaces

The authors mainly study the Hausdorff operators on Euclidean space Rn. They establish boundedness of the Hausdorff operators in various function spaces, such as Lebesgue spaces, Hardy spaces, local Hardy spaces and Herz type spaces. The results reveal that the Hausdorff operators have better performance on the Herz type Hardy spaces HKP（Rn） than their performance on the Hardy spaces Hv（Rn） when 0 〈 p 〈 1. Also, the authors obtain some new results and reprove or generalize some known results for the high dimensional Hardy operator and adjoint Hardy operator.

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator H_? on the real line R. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space L^p(R)and the Hardy space H^1(R). The key idea is to reformulate H_? as a Calder′on-Zygmund convolution operator,from which its boundedness is proved by verifying the Hrmander condition of the convolution kernel. Secondly,to prove the boundedness on the Hardy space H^p(R) with 0 < p < 1, we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of H^p(R) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H^1(R).

The Cauchy Integral Operator on Weighted Hardy Space

The authors show that the Cauchy integral operator is bounded from Hωp(R1) to hωp(R1) (the weighted local Hardy space). To prove the results, a kind of generalized atoms is introduced and a variant of weighted "Tb theorem" is considered.

Oscillatory hyper Hilbert transforms along general curves

We consider the oscillatory hyper Hilbert transform Hγ,α,βf（x） = ∫0^∞ f（x - Г（t））eit-βt-（1＋α）dt, where Г（t） = （t, γ（t）） in R^2 is a general curve. When γ is convex, we give a simple condition on γ such that Hγ,α,βis bounded on L2 when β ≥ 3α, β 〉 0. As a corollary, under this condition, we obtain the LP-boundedness of Hγ,α,β when 2β/（2β - 3α） 〈 p 〈 2β/（3α）. When F is a general nonconvex curve, we give some more complicated conditions on γ such that Hγ,α,βis bounded on L2. As an application, we construct a class of strictly convex curves along which Hγ,α,β is bounded on L2 only if β 〉 2α 〉 0.

A class of iterative greedy algorithms related to Blaschke product

Mobius transforms,Blaschke products and starlike functions as typical conformal mappings of one complex variable give rise to nonlinear phases with non-negative phase derivatives with the latter being de ned by instantaneous frequencies of signals they represent.The positive analytic phase derivative has been a widely interested subject among signal analysts(see Gabor(1946)).Research results of the positive analytic frequency and applications appears in the literature since the middle of the 20th century.Of the positive frequency study a directly related topic is positive frequency decomposition of signals.The mainly focused methods of such decompositions include the maximal selection method and the Blaschke product unwinding method,and joint use of the mentioned methods.In this paper,we propose a class of iterative greedy algorithms based on the Blaschke product and adaptive Fourier decomposition.It generalizes the Blaschke product unwinding method by subtracting constants other than the averages of the remaining functions,aiming at larger winding numbers,and subtracting n-Blaschke forms of the remaining functions,aiming at generating larger numbers of zero-crossings,to fast reduce energy of the remaining terms.Furthermore,we give a comprehensive and rigorous proof of the converging rate in terms of the zeros of the remainders.Finite Blaschke product methods are proposed to avoid the in nite phase derivative dilemma,and to avoid the computational diculties.

Convolutions, Tensor Products and Multipliers of the Orlicz-Lorentz Spaces

In this paper, the authors first give the properties of the convolutions of Orlicz- Lorentz spaces Aφ1,w and Aφ2,w on the locally compact abelian group. Secondly, the authors obtain the concrete representation as function spaces for the tensor products of Orlicz-Lorentz spaces Aφ1,w and Aφ2,w, and get the space of multipliers from the space Aφ1,w to the space Mφ2.w. Finally, the authors discuss the homogeneous properties for the Orlicz-Lorentz space Aφ,w^p,q.

Stability of nonlinear Schrodinger equations on modulation spaces

We consider the Cauchy problem for a family of SchrSdinger equations with initial data in modulation spaces Mp,1^s. We develop the existence, uniqueness, blowup criterion, stability of regularity, scattering theory, and stability theory.

Weighted L^p Estimates for Maximal Commutators of Multilinear Singular Integrals

This paper is concerned with the pointwise estimates for the sharp function of two kinds of maximal commutators of multilinear singular integral operators T∑b^＊ and TПb^＊ which are generalized by a weighted BMO function b and a multilinear singular integral operator T, respectively. As applications, some commutator theorems are established.

Oscillatory hyper Hilbert transforms along variable curves

For n = 2 or 3 and x ∈ Rn, we study the oscillatory hyper Hilbert transform Tα,β f(x)=∫Rf(x-Г(t,x))e^-i|t|-β|t|^-1-α dt along an appropriate variable curveГ(t, x) in R^n (namely,Г(t, x) is a curve in R^n for each fixed x), where α〉β〉0. We obtain some LP boundedness theorems of Tα,β, under some suitable conditions on α and β. These results are extensions of some earlier theorems. However, Tα,β f(x) is not a convolution in general. Thus, we only can partially employ the Plancherel theorem, and we mainly use the orthogonality principle to prove our main theorems.

Endpoint Estimates for Generalized Multilinear Fractional Integrals on the Non-homogeneous Metric Spaces

In this paper, some endpoint estimates for the generalized multilinear fractional integrals Ia,m on the non-homogeneous metric spaces are established.

Transference on Some Non-convolution Operators from Euclidean Spaces to Torus

The authors prove the certain de Leeuw type theorems on some non-convolution operators,and give some applications on certain known results.