Anisotropic Hardy-Lorentz spaces and their applications

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on Rn. We introduce the anisotropic Hardy-Lorentz space H~(p,q)_A(R~n) associated with A via the non-tangential grand maximal function and then establish its various real-variable characterizations in terms of the atomic and the molecular decompositions, the radial and the non-tangential maximal functions, and the finite atomic decompositions. All these characterizations except the ∞-atomic characterization are new even for the classical isotropic Hardy-Lorentz spaces on Rn.As applications, we first prove that Hp,q A(Rn) is an intermediate space between H~(p1,q1)_A(Rn) and H~(p2,q2)_A(R~n) with 0 p1 p p2 ∞ and q1, q, q2 ∈(0, ∞], and also between H~(p,q1)_A(Rn) and H~(p,q2)_A(R~n) with p ∈(0, ∞)and 0 q1 q q2 ∞ in the real method of interpolation. We then establish a criterion on the boundedness of sublinear operators from H~(p,q)_A(R~n) into a quasi-Banach space; moreover, we obtain the boundedness of δ-type Calder′on-Zygmund operators from H~(p,∞)_A(R~n) to the weak Lebesgue space L~(p,∞)(R~n)(or to H~p_A(R~n)) in the ln λcritical case, from H~(p,q)_A(R~n) to L~(p,q)(R~n)(or to H~(p,q)_A(R~n)) with δ∈(0,(lnλ)/(ln b)], p ∈(1/(1+,δ),1] and q ∈(0, ∞], as well as the boundedness of some Calderon-Zygmund operators from H~(p,q)_A(R~n) to L~(p,∞)(R~n), where b := | det A|,λ_:= min{|λ| : λ∈σ(A)} and σ(A) denotes the set of all eigenvalues of A.     

Commutators of Singular Integral Operators Related to Magnetic Schrdinger Operators

Let A:=-(▽-ia(向量))·(?-ia(向量))+V be a magnetic Schrdinger operator on L~2(R~n),n≥2,where a(向量)=(a_1,···,a_n)∈L~2_(loc)(R~n,R~n) and 0≤V∈L~1_(loc)(R~n).In this paper,we show that for a function b in Lipschitz space Lip_α(R~n) with α∈(0,1),the commutator[b,V~(1/2)A~(-1/2)] is bounded from L_p(R~n) to L_q(R~n),where p,q∈(1,2] and 1/p-1/q =α/n.     

L~p-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on R~n

Let L=-div(A▽) be a second-order divergent-form elliptic operator,where A is an accretive n×n matrix with bounded and measurable complex coefficients on Herein,we prove that the commutator [b,L~(1/2)]of the Kato square root L~(1/2) and b with ▽b∈L~n(R~n)(n 2),is bounded from the homogenous Sobolev space L_1~p(R~n) to L~p(R~n)(p-(L) pp+(L)).     

On Characterization of Poisson Integrals of Schrödinger Operators with Morrey Traces

Let L be a Schr?dinger operator of the form L =-Δ + V acting on L~2(R~n) where the nonnegative potential V belongs to the reverse H?lder class B_q for some q ≥ n. In this article we will show that a function f ∈ L~(2,λ)(R~n), 0 λ n, is the trace of the solution of L_u =-u_(tt) + L_u =0, u(x, 0) = f(x), where u satisfies a Carleson type condition sup x_B,r_Br_B~(-λ)∫_0~(rB)∫_(B(x_B,r_B))t|u(x,t)|~2dxdt≤C∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces L_L~(2,λ)(R~n) associated to the operator L, i.e.L_L~(2,λ)(R~n)=L~(2,λ)(R~n).Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L~(2,λ)(R~n) for all 0 λ n.     

Carleson measures,BMO spaces and balayages associated to Schrdinger operators

Let L be a Schrdinger operator of the form L =-? + V acting on L~2(R~n), n≥3, where the nonnegative potential V belongs to the reverse Hlder class B_q for some q≥n. Let BMO_L(R~n) denote the BMO space associated to the Schrdinger operator L on R~n. In this article, we show that for every f ∈ BMO_L(R~n) with compact support, then there exist g ∈ L~∞(R~n) and a finite Carleson measure μ such that f(x) = g(x) + S_(μ,P)(x) with ∥g∥∞ + |||μ|||c≤ C∥f∥BMO_L(R~n), where S_(μ,P)=∫(R_+~(n+1))Pt(x,y)dμ(y, t),and Pt(x, y) is the kernel of the Poisson semigroup {e-~(t(L)~(1/2))}t0 on L~2(R~n). Conversely, if μ is a Carleson measure, then S_(μ,P) belongs to the space BMO_L(R~n). This extends the result for the classical John-Nirenberg BMO space by Carleson(1976)(see also Garnett and Jones(1982), Uchiyama(1980) and Wilson(1988)) to the BMO setting associated to Schrdinger operators.     

双线性分数次积分算子交换子在Morrey空间上有界的充分必要条件

In this paper,we obtain that b∈ BMO(R~n) if and only if the commutator[b,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.Also we show that b ∈ Lip_β(R~n) if and only if the commutator[b,I_α]is bounded from the Morrey spaces L~(p_1,λ_1)(R~n)×L~(p_2,λ_2)(R~n) to L~(q,λ)(R~n),for some appropriate indices p,q,λ,μ.     

Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on R~n. Let H_A~(p,q )(R~n) be the anisotropic Hardy-Lorentz spaces associated with A defined via the nontangential grand maximal function. In this article, the authors characterize H_A~(p,q )(R~n) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley g~*_λ-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space L_(p,q)(R~n). All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on R~n. Moreover, the range of λ in the g~*_λ-function characterization of H_A~(p,q )(R~n) coincides with the best known one in the classical Hardy space H~p(R~n) or in the anisotropic Hardy space H_A~p (R~n).     

Real-variable characterizations of anisotropic product Musielak-Orlicz Hardy spaces

Let A :=(A_1, A_2) be a pair of expansive dilations and φ : R~n×R~m×[0, ∞) → [0, ∞) an anisotropic product Musielak-Orlicz function. In this article, we introduce the anisotropic product Musielak-Orlicz Hardy space H~φ_A(R~n× R~m) via the anisotropic Lusin-area function and establish its atomic characterization, the g-function characterization, the g_λ~*-function characterization and the discrete wavelet characterization via first giving out an anisotropic product Peetre inequality of Musielak-Orlicz type. Moreover, we prove that finite atomic decomposition norm on a dense subspace of H~φ_A(R~n× R~m) is equivalent to the standard infinite atomic decomposition norm. As an application, we show that, for a given admissible triplet(φ, q, s), if T is a sublinear operator and maps all(φ, q, s)-atoms into uniformly bounded elements of some quasi-Banach spaces B, then T uniquely extends to a bounded sublinear operator from H~φ_A(R~n× R~m) to B. Another application is that we obtain the boundedness of anisotropic product singular integral operators from H~φ_A(R~n× R~m) to L~φ(R~n× R~m)and from H~φ_A(R~n×R~m) to itself, whose kernels are adapted to the action of A. The results of this article essentially extend the existing results for weighted product Hardy spaces on R~n× R~m and are new even for classical product Orlicz-Hardy spaces.     

GABOR ANALYSIS OF THE SPACES M（p,q,w）（R~d） AND S（p,q,r,w,ω）（R~d）

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤∞. We also investigate the embeddings between these spaces and the dual space of M(p, q, w)(Rd). Later we define the space S(p, q, r, w, ω)(Rd) for 1 < p < ∞, 1 ≤ q ≤∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p, q, r, w, ω)(Rd). At the end of this article, we characterize the multipliers of the spaces M(p, q, w)(Rd) and S(p, q, r, w, ω)(Rd).     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Weighted Compact Commutator of Bilinear Fourier Multiplier Operator

Let T_σ be the bilinear Fourier multiplier operator with associated multiplier σ satisfying the Sobolev regularity that sup κ∈Z∥σ_κ∥W~s(R~(2n)) ∞ for some s ∈ (n, 2n]. In this paper, it is proved that the commutator generated by T_σ and CMO(R~n) functions is a compact operator from L~(p1)(R~n, w_1) × L~(p2)(R~n, w_2) to L~p(R~n, ν_w) for appropriate indices p_1, p_2, p ∈ (1, ∞) with1 p=1/ p_1 +1/ p_2 and weights w_1, w_2 such that w = (w_1, w_2) ∈ A_(p/t)(R~(2n)).     

Global Poincaré Inequalities on the Heisenberg Group and Applications

Let f be in the localized nonisotropic Sobolev space Wloc^1,p （H^n） on the n-dimensional Heisenberg group H^n = C^n ×R, where 1≤ p ≤ Q and Q = 2n ＋ 2 is the homogeneous dimension of H^n. Suppose that the subelliptic gradient is gloablly L^p integrable, i.e., fH^n ｜△H^n f｜^p du is finite. We prove a Poincaré inequality for f on the entire space H^n. Using this inequality we prove that the function f subtracting a certain constant is in the nonisotropic Sobolev space formed by the completion of C0^∞（H^n） under the norm of （∫H^n ｜f｜ Qp/Q-p）^Q-p/Qp ＋ （∫ H^n ｜△H^n f｜^p）^1/p. We will also prove that the best constants and extremals for such Poincaré inequalities on H^n are the same as those for Sobolev inequalities on H^n. Using the results of Jerison and Lee on the sharp constant and extremals for L^2 to L（2Q/Q-2） Sobolev inequality on the Heisenberg group, we thus arrive at the explicit best constant for the aforementioned Poincaré inequality on H^n when p=2. We also derive the lower bound of the best constants for local Poincaré inequalities over metric balls on the Heisenberg group H^n.     

The L~(p,q)-stability of the Shifts of Finitely Many Functions in Mixed Lebesgue Spaces L~(p,q)(R~(d+1))

The stability is an expected property for functions,which is widely considered in the study of approximation theory and wavelet analysis.In this paper,we consider the Lp,q-stability of the shifts of finitely many functions in mixed Lebesgue spaces L~(p,q)(R~(d+1)).We first show that the shiftsφ(·-k)(k∈Z~(d+1))are Lp,q-stable if and only if for anyξ∈R~(d+1),∑_(k∈Z~(d+1))|φ(ξ+2πk)|~20.Then we give a necessary and sufficient condition for the shifts of finitely many functions in mixed Lebesgue spaces L~(p,q)(R~(d+1))to be Lp,q-stable which improves some known results.     

On approximation of smooth functions from null spaces of optimal linear differential operators with constant coefficients

For a real valued function f defined on a finite interval I we consider the problem of approximating f from null spaces of differential operators of the form Ln(ψ) = n ∑ k=0 akψ(k), where the constant coefficients ak ∈ R may be adapted to f . We prove that for each f ∈ C(n)(I), there is a selection of coefficients {a1, ,an} and a corresponding linear combination Sn( f ,t) = n ∑ k=1 bkeλkt of functions ψk(t) = eλkt in the nullity of L which satisfies the following Jackson’s type inequality: f (m) Sn(m )( f ,t) ∞≤ |an|2n|Im|1/1q/ep|λ|λn|n|I||nm1 Ln( f ) p, where |λn| = mka x|λk|, 0 ≤ m ≤ n 1, p,q ≥ 1, and 1p + q1 = 1. For the particular operator Mn(f) = f + 1/(2n) f(2n) the rate of approximation by the eigenvalues of Mn for non-periodic analytic functions on intervals of restricted length is established to be exponential. Applications in algorithms and numerical examples are discussed.     

The Gleason''''s problem for some polyharmonic and hyperbolic harmonic function spaces

LetΩ Rn be a bounded convex domain with C2 boundary. For 0相似文献   

GABOR ANALYSIS OF THE SPACES M(p,q,w)(R~d) AND S(p,q,r,w,ω)(R~d)

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p, q, w)(Rd) to be the subspace of tempered distributions f ∈ S′(Rd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p, q, wdμ) (R2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1 ≤ p, q ≤ ∞. We also investigate the embeddings between these spaces and the...     

Oscillatory Strongly Singular Integral Associated to the Convex Surfaces of Revolution

Here we consider the following strongly singular integral T_(?,γ,α,β)f (x, t) =∫ _(R~n)e~[i|y|~(-β)]?(y/|y|)/|y|~(n+α)f (x - y, t - γ(|y|))dy,where ? ∈ L~p(S~(n-1)), p 1, n 1, α 0 and γ is convex on(0,∞).We prove that there exists A( p, n) 0 such that if β A( p, n)(1 + α), then T_(?,γ,α,β)is bounded from L~2(R~(n+1)) to itself and the constant is independent of γ. Furthermore,when ? ∈ C~∞(S~(n-1)), we will show that T?,γ,α,βis bounded from L~2(R~(n+1)) to itself only if β 2α and the constant is independent of γ.     

Stability of the Equilibrium to the Boltzmann Equation with Large Potential Force

The Boltzmann equation with external potential force exists a unique equilibrium—local Maxwellian. The author constructs the nonlinear stability of the equilibrium when the initial datum is a small perturbation of the local Maxwellian in the whole space R~3. Compared with the previous result [Ukai, S., Yang, T. and Zhao, H.-J.,Global solutions to the Boltzmann equation with external forces, Anal. Appl.(Singap.), 3,2005, 157–193], no smallness condition on the Sobolev norm H~1 of the potential is needed in our arguments. The proof is based on the entropy-energy inequality and the L~2-L~∞ estimates.     

AN EXTENSION OF THE HARDY-LITTLEWOOD-PóLYA INEQUALITY

The Hardy-Littlewood-Pólya (HLP) inequality [1] states that if a∈l~p,b∈l~q and p>1,q>1,1/p + 1/q>1, λ=2-(1/p+1/q),then Σ[(a_rb_s)/︱r-s︱~λ](r≠s)≤C‖a‖_P‖b‖_q.In this article, we prove the HLP inequality in the case where λ= 1, p = q = 2 with a logarithm correction, as conjectured by Ding [2]:Σ[(a_rb_s)/︱r-s︱~λ](r≠s,1≤r,s≤N)≤(2㏑N+1)‖a‖_2‖b‖_2.In addition, we derive an accurate estimate for the best constant for this inequality.