BOUNDEDNESS OF CALDERN-ZYGMUND OPERATORS ON BESOV SPACES AND ITS APPLICATION

In this article, the author introduces a class of non-convolution Calder′on-Zygmund operators whose kernels are certain sums involving the products of Meyer wavelets and their convolutions. The boundedness on Besov spaces Bp^0 ,q（1 ≤p,q ≤∞） is also obtained. Moreover, as an application, the author gives a brief proof of the known result that Hrmander condition can ensure the boundedness of convolution-type Calder′on-Zygmund operators on Besov spaces B^p0 ,q（1 ≤p,q ≤∞）. However, the proof is quite different from the previous one.

Continuity of a Class of Calderón-Zygmund Operators on Certain Besov Spaces

In this paper, we introduce a class of non-convolution-type Calderón-Zygmund operators, whose kernels are certain sums involving the products of the Daubechies wavelets and their convolutions. And we obtain the continuity on the Besov spaces B 0,q p (1 ≤ p, q ≤∞), which is mainly dependent on the properties of the Daubechies wavelets and Lemari's T1 theorem for Besov spaces.

Gevrey Regularity and Time Decay of Fractional Porous Medium Equation in Critical Besov Spaces

In this paper, we show the existence and regularity of mild solutions depending on the small initial data in Besov spaces to the fractional porous medium equation. When 1 < α ≤ 2, we prove global well-posedness for initial data with 1 ≤ p < ∞, 1 ≤ q ≤ ∞, and analyticity of solutions with 1 < p < ∞, 1 ≤ q ≤ ∞. In particular, we also proved that when α = 1, both u and belong to . We solve this equation through the contraction mapping method based on Littlewood-Paley theory and Fourier multiplier. Furthermore, we can get time decay estimates of global solutions in Besov spaces, which is as t → ∞.

A DISCRETE CHARACTERIZATION OF BESOV SPACES

The characterization of isotropic Besov spaces for in terms of progressive differences of a function on dyadic points is obtained. Moreover, for withan analogous characterization of anisotropic Besov spaces is presented.

WEIGHTED HOLOMORPHIC BESOV SPACES AND THEIR BOUNDARY VALUES

We study weighted holomorphic Besov spaces and their boundary values. Under certain restrictions on the weighted function and parameters, we establish the equivalent norms for holomorphic functions in terms of their boundary functions. Some results about embedding and interpolation are also included.

T1 THEOREM FOR BESOV SPACES ON NONHOMOGENEOUS SPACES

Suppose μ is a Radon measure on R^d, which may be non doubling. The only condition assumed on μ is a growth condition, namely, there is a constant Co 〉 0 such that for all x ∈ supp（μ） and r 〉 0, μ（B（x, r）） ≤ Cor^n, where 0 〈 n ≤ d. We prove T1 theorem for non doubling measures with weak kernel conditions. Our approach yields new results for kernels satisfying weakened regularity conditions, while recovering previously known Tolsa＇s results. We also prove T1 theorem for Besov spaces on nonhomogeneous spaces with weak kernel conditions given in [7] .

GENERALIZED BESOV SPACES AND TRIEBEL-LIZORKIN SPACES

In this paper the classical Besov spaces B^sp.q and Triebel-Lizorkin spaces F^sp.q for s∈R are generalized in an isotropy way with the smoothness weights { ｜2j｜^α→ln }7=0. These generalized Besov spaces and Triebel-Lizorkin spaces, denoted by B^α→p.q and F^α→p.q for α^→ E Nk and k ∈N, respectively, keep many interesting properties, such as embedding theorems （with scales property for all smoothness weights）, lifting properties for all parameters 5, and duality for index 0 〈 p 〈∞ By constructing an example, it is shown that there are infinitely many generalized Besov spaces and generalized Triebel-Lizorkin spaces lying between B^sp.q and ∪t〉s B^tp.q, and between F^sp.q and ∪t〉s F^tp.q, respectively.

Characterizations of Weighted Besov Spaces with Variable Exponents

In this paper,we first give characterizations of weighted Besov spaces with variable exponents via Peetre’s maximal functions.Then we obtain decomposition characterizations of these spaces by atom,molecule and wavelet.As an application,we obtain the boundedness of the pseudo-differential operators on these spaces.

Singular Integrals and Weighted Triebel-Lizorkin and Besov Spaces of Arbitrary Number of Parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon＇s identity. This is inspired by the work of discrete Littlewood- Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe＇s covering lemma in multiparameter setting.

Some Applications of Besov Spaces on Fractals

Let F be a compact d-set in R^n with 0 〈 d ≤ n, which includes various kinds of fractals. The author establishes an embedding theorem for the Besov spaces Bpq^s（F） of Triebel and the Sobolev spaces W^1,P（F,d,μ） of Hajtasz when s 〉 1, 1 〈 p 〈∞ and 0 〈 q ≤ ∞. The author also gives some applications of the estimates of the entropy numbers in the estimates of the eigenvalues of some fractal pseudodifferential operators in the spaces Bpq^0（F） and Fpq^0（F）.

The Fejer-Riesz Inequality for the Besov Spaces

A Fejer Riesz inequality for the weighted Besov spaces Bp,q is given. Some characteriza- tions of functions in Bp.q in terms of their Taylor coefficients are obtained.

Regularity Condition of Solutions to the Quasi-geostrophic Equations in Besov Spaces with Negative Indices

With a Hǒlder type inequality in Besov spaces, we show that every strong solution θ（t, x） on （0, T ） of the dissipative quasi-geostrophic equations can be continued beyond T provided that ⊥θ（t, x） ∈L 2γ/γ-2δ （（0, T ）; B^δ-γ/2 ∞∞（R^2）） for 0 〈 δ 〈 γ/2 .

BESOV SPACES AND SELF-SIMILAR SOLUTIONS FOR NONLINEAR EVOLUTION EQUATIONS

In this paper, we establish the existence of global self-similar solutions for the heat and convection-diffusion equations. This we do in some homogeneous Besov spaces using the theory of Besov spaces and the Strichartz estimates. Further, the structure of the self-similar solutions has also been established by using an equivalent norm for Besov spaces.

Pointwise multipliers for Besov spaces of dominating mixed smoothness-Ⅱ

We continue our investigations on pointwise multipliers for Besov spaces of dominating mixed smoothness. This time we study the algebra property of the classes S_(p,q)~rB(R^d) with respect to pointwise multiplication. In addition, if p≤q, we are able to describe the space of all pointwise multipliers for S_(p,q)~rB(R^d).

Existence and Gevrey regularity for a two-species chemotaxis system in homogeneous Besov spaces

We study the Cauchy problem of a two-species chemotactic model. Using the Fourier frequency localization and the Bony paraproduct decomposition, we establish a unique local solution and blow-up criterion of the solution, when the initial data（u0, v0, w0） belongs to homogeneous Besov spaces B^˙p,1^-2＋3/p（R^3） ×B^˙r,1^-2＋3/r（R^3） ×B^˙q,1^3/q（R^3） for p, q and r satisfying some technical assumptions. Furthermore, we prove that if the initial data is sufficiently small, then the solution is global. Meanwhile, based on the so-called Gevrey estimates, we particularly prove that the solution is analytic in the spatial variable. In addition, we analyze the long time behavior of the solution and obtain some decay estimates for higher derivatives in Besov and Lebesgue spaces.

The Boundedness of Composition Operators on Triebel–Lizorkin and Besov Spaces with Different Homogeneities

In this paper, we introduce new Triebel–Lizorkin and Besov Spaces associated with the different homogeneities of two singular integral operators. We then establish the boundedness of composition of two Calder′on–Zygmund singular integral operators with different homogeneities on these Triebel–Lizorkin and Besov spaces.

Global Well-posedness for the Non-viscous MHD Equations with Magnetic Diffusion in Critical Besov Spaces

In this paper,we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion.We first establish the local well-posedness(existence,uniqueness and continuous dependence)with initial data(u_(0),b_(0))in critical Besov spaces B_(p,1)^(d/p+1)×B_(p,1)^(d/p)with 1≤p≤∞,and give a lifespan T of the solution which depends on the norm of the Littlewood–Paley decomposition(profile)of the initial data.Then,we prove the global existence in critical Besov spaces.In particular,the results of global existence also hold in Sobolev space C([0,∞);H~s(S~2))×(C([0,∞);H^(s-1)(S~2))∩L~2([0,∞);H~s(S~2)))with s>2,when the initial data satisfies∫_(S~2)b_(0)dx=0 and||u_(0)||B_(()∞,1~((S~2)))~1+||b_(0)||B_(()∞,1^(S~2))~0≤ε.It’s worth noting that our results imply some large and low regularity initial data for the global existence.

Littlewood-Paley Functions and Triebel-Lizorkin Spaces,Besov Spaces

We establish Littlewood-Paley charaterizations of Triebel-Lizorkin spaces and Besov spaces in Euclidean spaces using several square functions defined via the spherical average,the ball average,the Bochner-Riesz means and some other well-known operators.We provide a simple proof so that we are able to extend and improve many results published in recent papers.

Sharp Estimates in Bergman and Besov Spaces on Bounded Symmetric Domains

Sharp estimates of the point-evaluation functional in weighted Bergman spaces L_~p(Ω, dv_) and for the point-evaluation derivalive functional in Besov spaces B^p(Ω) are obtained for bounded symmetric domains Ω in C^n.

A Class of Oscillatory Singular Integrals with Hardy Kernels on Triebel-Lizorkin Spaces and Besov Spaces

In this paper,the boundedness is obtained on the Triebel-Lizorkin spaces and the Besov spaces for a class of oscillatory singular integrals with Hardy kernels.