Continuity properties for commutators of multilinear type operators on product of certain Hardy spaces

Similar to the property of a linear Calderdn-Zygmund operator, a linear fractional type operator Is associated with a BMO function b fails to satisfy the continuity from the Hardy space Hp into Lp for p ≤ 1. Thus, an alternative result was given by Y. Ding, S. Lu and P. Zhang, they proved that [b,Iα] is continuous from an atomic Hardy space Hp b into Lp, where Hp b is a subspace of the Hardy space Hp for n/（n ＋ 1） 〈 p ≤ 1. In this paper, we study the commutators of multilinear fractional type operators on product of certain Hardy spaces. The endpoint （Hp1 b1 ×... × HP2, Lp） boundedness for multilinear fractional type operators is obtained. We also give the boundedness for the commutators of multilinear Calderdn-Zygmund operators and multilinear fractional type operators on product of certain Hardy spaces when b ∈ （Lipβ）m（Rn）.     

Uniqueness of weak solutions for fractional Navier-Stokes equations

We prove that if u is a weak solution of the d dimensional fractional Navier-Stokes equations for some initial data u₀and if u belongs to path space p=Lq(0,T;Bp,∞r)or p=L1(0,T;B∞,∞r), then u is unique in the class of weak solutions when α>1. The main tools are Bony decomposition and Fourier localization technique. The results generalize and improve many recent known results.     

Local regularity properties for 1D mixed nonlinear Schrödinger equations on half-line

The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrödinger equation u_t={\mathrm{i}\alpha u}_{xx}+\beta u^{2-}{u}_x+\gamma {\left|u\right|}^2{u}_x+\mathrm{i}{\left|u\right|}^{2}u on the half-line with inhomogeneous boundary condition. We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces. Moreover, we show that the nonlinear part of the solution on the half-line is smoother than the initial data.     

Fourier transform of anisotropic mixed-norm Hardy spaces

Let a:=(a1,…,an)∈[1,∞)n,p:=(p1,…,pn)∈(0,1]n,Hpa(Rⁿ)be the anisotropic mixed-norm Hardy space associated with adefined via the radial maximal function,and let f belong to the Hardy space Hpa(Rⁿ).In this article,we show that the Fourier transform fcoincides with a continuous function g on?n in the sense of tempered distributions and,moreover,this continuous function g,multiplied by a step function associated with a,can be pointwisely controlled by a constant multiple of the Hardy space norm of f.These proofs are achieved via the known atomic characterization of Hpa(Rⁿ)and the establishment of two uniform estimates on anisotropic mixed-norm atoms.As applications,we also conclude a higher order convergence of the continuous function gat the origin.Finally,a variant of the Hardy-Littlewood inequality in the anisotropic mixed-norm Hardy space setting is also obtained.All these results are a natural generalization of the well-known corresponding conclusions of the classical Hardy spaces Hp(Rⁿ)with p∈0,1],and are even new for isotropic mixed-norm Hardy spaces on∈n.     

Low dimensional modules over quantum complete intersections in two variables

We classify all the indecomposable modules of dimension ≤ 5 over the quantum exterior algebra k(x, y)/(x^2, y^2, xy + qyx) in two variables, and all the indecomposable modules of dimension ≤3 over the quantum complete intersection k(x,y)/(x^m,y^n,xy + qyx) in two variables, where m or n ≥3, by giving explicitly their diagram presentations.     

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schr?dinger operators

Let φ be a growth function, and let A:=-(?-ia)?(?-ia)+V be a magnetic Schr?dinger operator on L2(?n),n≥2, where α:=(α1,α2,?,αn)∈Lloc2(?n,?n) and 0≤V∈Lloc1(?n). We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space HA,φ(?n), defined by the Lusin area function associated with {e-t2A}t>0, in terms of the Lusin area function associated with {e-tA}t>0, the radial maximal functions and the nontangential maximal functions associated with {e-t2A}t>0 and {e-tA}t>0, respectively. The boundedness of the Riesz transforms LkA-1/2,k∈{1,2,?,n}, from HA,φ(?n) to Lφ(?n) is also presented, where L_k is the closure of ??xk-iαk in L2(?n). These results are new even when φ(x,t):=ω(x)tp for all x∈?nand t ∈(0,+∞) with p ∈(0, 1] and ω∈A∞(?n) (the class of Muckenhoupt weights on ?n).     

Applications of multiresolution analysis in Besov-Q type spaces and Triebel-Lizorkin-Q type spaces

In this survey, we give a neat summary of the applications of the multi-resolution analysis to the studies of Besov-Q type spaces {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and Triebel-Lizorkin-Q type spaces {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n). We will state briefly the recent progress on the wavelet characterizations, the boundedness of Calderón-Zygmund operators, the boundary value problem of {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n). We also present the recent developments on the well-posedness of fluid equations with small data in {\dot{B}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n) and {\dot{F}}_{p,q}^{{\gamma }_1,{\gamma }_2}({\text{ℝ}}^n).     

Dynamical behaviors of non-autonomous fractional FitzHugh-Nagumo system driven by additive noise in unbounded domains

The regularity of random attractors is considered for the non-autonomous fractional stochastic FitzHugh-Nagumo system.We prove that the system has a pullback random attractor that is compact in Hs(Rⁿ)×L²(Rⁿ)and attracts all tempered random sets of L²(Rⁿ)×L²(Rⁿ)in the topology of Hs(Rⁿ)×L²(Rⁿ)with s∈(0,1).By the idea of positive and negative truncations,spectral decomposition in bounded domains,and tail estimates,we achieved the desired results.     

Sum of squares methods for minimizing polynomial forms over spheres and hypersurfaces

This paper studies the problem of minimizing a homogeneous polynomial (form) f(x) over the unit sphere Sn-1={x∈?n:‖x‖2=|1}. The problem is NP-hard when f(x) has degree 3 or higher. Denote by f_min (resp. f_max) the minimum (resp. maximum) value of f(x) on Sn-1. First, when f(x) is an even form of degree 2d, we study the standard sum of squares (SOS) relaxation for finding a lower bound of the minimum f_min:max? γ s.t. f(x)-γ·‖x‖22d? is SOS.Let f_sos be the above optimal value. Then we show that for all n≥2d,1≤fmax?-fsosfmax?-fmin?≤C(d)(n2d).Here, the constant C(d) is independent of n. Second, when f(x) is a multi-form and Sn-1 becomes a multi-unit sphere, we generalize the above SOS relaxation and prove a similar bound. Third, when f(x) is sparse, we prove an improved bound depending on its sparsity pattern; when f(x) is odd, we formulate the problem equivalently as minimizing a certain even form, and prove a similar bound. Last, for minimizing f(x) over a hypersurface H(g)={x∈?n:g(x)=1} defined by a positive definite form g(x), we generalize the above SOS relaxation and prove a similar bound.     

On multivariable Zassenhaus formula

We give a recursive algorithm to compute the multivariable Zassenhaus formula e^X1+X2+…+Xn=e^X1eX2…e^Xn∏∞k=2e^Wk and derive an effective recursion formula of Wk.     

Exponential sums involving automorphic forms for GL(3) over arithmetic progressions

Let f be a Hecke-Maass cusp form for SL(3;\mathrm{\mathbb{Z}}) with Fourier coefficients A_f(m; n); and let \varphi (x) be a C^{\infty } -function supported on [1; 2] with derivatives bounded by {\varphi }^{(j)}(x)\ll j 1. We prove an asymptotic formula for the nonlinear exponential sum {\mathrm{\Sigma }}_{n\equiv l\mathrm{mod}q}{A}_f(m,n)\phi (n/X)e{\left(3\left(kn\right)\right)}^{1/3}/q, where e(z)={\mathrm{e}}^{\mathrm{2}\pi iz} and k\in {\mathrm{\mathbb{Z}}}^{+}.     

Intrinsic square function characterizations of Hardy spaces associated with ball quasi-Banach function spaces

Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and H_X({ℝ}^n) the associated Hardy-type space. In this article, we first establish the finite atomic characterization of H_X({ℝ}^n). As an application, we prove that the dual space of H_X({ℝ}^n) is the Campanato space associated with X. For any given \alpha \in (\mathrm{0},\mathrm{1}] and s\in {\mathbb{Z}}_{+}, using the atomic and the Littlewood–Paley function characterizations of H_X({ℝ}^n),we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function S_{\alpha ,s},the intrinsic g-function g_{\alpha ,s},and the intrinsic g_{\lambda }^{\ast }-function g_{\lambda ,\alpha ,s}^{\ast }, where λ coincides with the best known range.     

Boundary behavior of harmonic functions on metric measure spaces with non-negative Ricci curvature

Let (X, d, μ) be a metric measure space with non-negative Ricci curvature. This paper is concerned with the boundary behavior of harmonic function on the (open) upper half-space X\times {\text{ℝ}}_{+}. We derive that a function f of bounded mean oscillation (BMO) is the trace of harmonic function u(x,t) on X\times {\text{ℝ}}_{+},u(x,0)=f\left(x\right), whenever u satisfies the following Carleson measure condition where \nabla =({\nabla }_x,{\partial }_t) denotes the total gradient and B(x_B,r_B) denotes the (open) ball centered at x_B with radius r_B. Conversely, the above condition characterizes all the harmonic functions whose traces are in BMO space.     

Factorization of simple modules for certain restricted two-parameter quantum groups

We study the representations of the restricted two-parameter quantum groups of types B and G. For these restricted two-parameter quantum groups, we give some explicit conditions which guarantee that a simple module can be factored as the tensor product of a one-dimensional module with a module that is naturally a module for the quotient by central group-like elements. That is, given θ a primitive lth root of unity, the factorization of simple ?θy,θz,( )-modules is possible, if and only if (2(y - z), l) = 1 for =??2n+1; (3(y - z), l) = 1 for g= G₂.     

Derivative estimates of averaging opera tors and extension

We study the derivative operator of the generalized spherical mean S^γt. By considering a more general multiplier m^Ωγ,b=Vn-2/2+γ(|ξ|)|ξ|^bΩ(ξ') and finding the smallest γ such that m^Ωγ,b is an Hp multiplier, we obtain the optimal range of exponents (γ,β,p)to ensure the H^p(R^n) boundedness of a^βS^γ1f(x). As an application, we obtain the derivative estimates for the solution for the Cauchy problem of the wave equation on H^p(R^n) spaces.     

Strong law of large numbers for supercritical superprocesses under second moment condition

Consider a supercritical superprocess X = {X_t, t≥0} on a locally compact separable metric space (E,m). Suppose that the spatial motion of X is a Hunt process satisfying certain conditions and that the branching mechanism is of the form ψ(x,λ)=-a(x)λ+b(x)λ2+∫(0,+∞)(e-λy-1+λy)n(x,dy),?x∈E,λ>0, where a∈Bb(E),b∈Bb+(E), and n is a kernel from E to (0,+∞) satisfying sup?x∈E∫0+∞y2n(x,dy)<+∞. Put Ttf(x)=Pδx?f,Xt?. Suppose that the semigroup {T_t; t≥0}is compact. Let λ₀ be the eigenvalue of the (possibly non-symmetric) generator L of {T_t}that has the largest real part among all the eigenvalues of L, which is known to be real-valued. Let ?0 and ?^0 be the eigenfunctions of L and L^(the dual of L) associated with λ₀, respectively. Assume λ₀>0. Under some conditions on the spatial motion and the ?0-transform of the semigroup {T_t}, we prove that for a large class of suitable functions f, lim?t→+∞e-λ0t?f,Xt?=W∞∫E?^0(y)f(y)m(dy),?Pμ-a.s., for any finite initial measure μ on E with compact support, where W_∞ is the martingale limit defined by W∞:=lim?t→+∞e-λ0t??0,Xt?. Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f.     

Oscillation and variation for Riesz transform in setting of Bessel operators on H1 and BMO

Let \lambda >\mathrm{0} and let the Bessel operator \mathrm{\Delta }\lambda ：=-{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}}{{\mathrm{d}x}^{\mathrm{2}}}}-{\displaystyle \frac{\mathrm{2}\lambda }{x}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}x}} defined on {{\displaystyle ℝ}}_{+}:=(\mathrm{0},\infty ). We show that the oscillation and \rho-variation operators of the Riesz transform {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} associated with {\mathrm{\Delta }}_{\lambda } are bounded on BMO({{\displaystyle ℝ}}_{+},{{\displaystyle dm}}_{\lambda }), where \rho >\mathrm{2} and {{\displaystyle \mathrm{d}m}}_{\lambda }=x^{\mathrm{2}\lambda }\mathrm{d}x. Moreover, we construct a {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-atom as a counterexample to show that the oscillation and \rho-variation operators of {{\displaystyle R}}_{{\mathrm{\Delta }}_{\lambda }} are not bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }). Finally, we prove that the oscillation and the {{\displaystyle (\mathrm{1},\infty )}}_{{\mathrm{\Delta }}_{\lambda }}-variation operators for the smooth truncations associated with Bessel operators {{\displaystyle \tilde{R}}}_{{\mathrm{\Delta }}_{\lambda }} are bounded from H^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }) to L^{\mathrm{1}}({ℝ}^{+},{{\displaystyle \mathrm{d}m}}_{\lambda }).     

Intrinsic contractivity properties of Feynman-Kac semigroups for symmetric jump processes with infinite range jumps

Let (X_t)_t≥0 be a symmetric strong Markov process generated by non-local regular Dirichlet form (D, D(D)) as follows: D(f,g)=∫?d∫?d(f(x)-f(y))(g(x)-g(y))J(x,y)dxdy,?f,g∈D(D), where J(x, y) is a strictly positive and symmetric measurable function on ?d×?d. We study the intrinsic hypercontractivity, intrinsic supercontractivity, and intrinsic ultracontractivity for the Feynman-Kac semigroup TtV(f)(x)=Ex(exp?(-∫0tV(Xs)ds)f(Xt)),?x∈?d,f∈L2(?d;dx). In particular, we prove that for J(x,y)≈|x-y|-d-al{|x-y|≤1}+e-|x-y|l{|x-y|>1} with α ∈(0, 2) and V(x)=|x|λ with λ>0, (TtV)t≥0 is intrinsically ultracontractive if and only if λ>1; and that for symmetric α-stable process (X_t)_t≥0 with α ∈(0, 2) and V(x)=log?λ(1+|x|) with some λ>0, (TtV)t≥0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ>1, and (TtV)t≥0 is intrinsically hypercontractive if and only if λ≥1. Besides, we also investigate intrinsic contractivity properties of (TtV)t≥0 for the case that lim inf?|x|→+∞V(x)<+∞     

Regularity for anisotropic solutions to some nonlinear elliptic system

This paper deals with anisotropic solutions u∈W1,(pi)(Ω,?N) to the nonlinear elliptic system −Σi=1nDi(aiα(χ,Du(χ)))=−Σi=1nDiFiα(χ), α=1,2,...,N, We present a monotonicity inequality for the matrix a=(aiα)∈?N×n,whichguarantees global pointwise bounds for anisotropic solutionsu.