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).     

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.     

Matchings extend to Hamiltonian cycles in hypercubes with faulty edges

We consider the problem of existence of a Hamiltonian cycle containing a matching and avoiding some edges in an n-cube {{\displaystyle Q}}_n, and obtain the following results. Let n\ge \mathrm{3},\hspace{0.17em}M\subset E({{\displaystyle Q}}_n), and F\subset E({{\displaystyle Q}}_n)\backslash M with \mathrm{1}\le \left|F\right|\le \mathrm{2}n-\mathrm{4}-\left|M\right|. If M is a matching and every vertex is incident with at least two edges in the graph {{\displaystyle Q}}_n-F, then all edges of M lie on a Hamiltonian cycle in {{\displaystyle Q}}_n-F. Moreover, if \left|M\right|=\mathrm{1} or \left|M\right|=\mathrm{2}, then the upper bound of number of faulty edges tolerated is sharp. Our results generalize the well-known result for \left|M\right|=\mathrm{1}.     

Hardy space estimates for bi-parameter Littlewood-Paley square functions

Suppose that g(f) are bi-parameter Littlewood-Paley square functions which were introduced by H. Martikainen. It is known that the L^{\mathrm{2}}({ℝ}^n\times {ℝ}^m) boundedness and the H^{\mathrm{1}}({ℝ}^n\times {ℝ}^m)-L^{\mathrm{1}}({ℝ}^n\times {ℝ}^m) boundedness of g(f) have been proved by H. Martikainen and by Z. Li and Q. Xue, respectively. In this paper, we apply the vector-valued theory, the atomic decomposition of product Hardy spaces, and Journe's covering lemma to show that g(f) are bounded from H^p({ℝ}^n\times {ℝ}^m) to L^p({ℝ}^n\times {ℝ}^m) with p smaller than 1.     

Averages of shifted convolution sums for arithmetic functions

Let f be a full-level cusp form for GLm(Z) with Fourier coefficients Af(cm-2,…, c1, n): Let λ(n) be either the von Mangoldt function Λ(n) or the k-th divisor function τk(n): We consider averages of shifted convolution sums of the type Σ|h|≤H |ΣX相似文献   

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.     

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.     

Lower deviations for supercritical branching processes with immigration

For a supercritical branching processes with immigration \{{{\displaystyle Z}}_n\}; it is known that under suitable conditions on the offspring and immigration distributions, Z_n/mⁿ converges almost surely to a finite and strictly positive limit, where m is the offspring mean. We are interested in the limiting properties of P({{\displaystyle Z}}_n={{\displaystyle k}}_n) with {{\displaystyle k}}_n=o(m^n) as n\to \infty. We give asymptotic behavior of such lower deviation probabilities in both Schröder and Böttcher cases, unifying and extending the previous results for Galton-Watson processes in literature.     

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.     

Oscillatory hyper Hilbert transforms along variable curves

For n = 2 or 3 and x\in {ℝ}^n, we study the oscillatory hyper Hilbert transform{{\displaystyle T}}_{\alpha ,\beta }f(x)={{\displaystyle \int }}_{ℝ}f(x-\Gamma (t,x)){\mathrm{e}}^{{-i\left|t\right|}^{-\beta }}{\left|t\right|}^{-\mathrm{1}-\alpha }\mathrm{d}talong an appropriate variable curve \mathrm{\Gamma }(t,x) in {ℝ}^n (namely, \mathrm{\Gamma }(t,x) is a curve in {ℝ}^n for each fixed x), where \alpha >\beta >\mathrm{0}. We obtain some L^p boundedness theorems of {{\displaystyle T}}_{\alpha ,\beta }, under some suitable conditions on \alphaand \beta. These results are extensions of some earlier theorems. However, {{\displaystyle T}}_{\alpha ,\beta }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.     

Dimension of divergence sets for dispersive equation

Consider the generalized dispersive equation defined by{iδtu+Ф(√-△)u=0,(x,t)∈R^n×R,u(x,0)=f(x),F∈F(R^n),(*)whereФ(√-△)is a pseudo-differential operator with symbolФ(|ζ|).In the present paper,assuming thatФsatisfies suitable growth conditions and the initial data in H^s(R^n),we bound the Hausdorff dimension of the sets on which the pointwise convergence of solutions to the dispersive equations(*)fails.These upper bounds of Hausdorff dimension shall be obtained via the Kolmogorov-Seliverstov-Plessner method.     

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.     

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.     

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)<+∞     

Superminimal surfaces in hyperquadric Q2

We study a superminimal surface M immersed into a hyperquadric Q₂ in several cases classified by two global defined functions {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y, which were introduced by X. X. Jiao and J. Wang to study a minimal immersion f : M\to {{\displaystyle Q}}_{\mathrm{2}}. In case both {{\displaystyle \tau }}_X and {{\displaystyle \tau }}_Y are not identically zero, it is proved that f is superminimal if and only if f is totally real or i\circ f:M\to {ℂ\mathrm{P}}^{\mathrm{3}} is also minimal, where i:{{\displaystyle Q}}_{\mathrm{2}}\to {ℂ\mathrm{P}}^{\mathrm{3}} is the standard inclusion map. In the rest case that {{\displaystyle \tau }}_X\equiv \mathrm{0} or {{\displaystyle \tau }}_Y\equiv \mathrm{0}, the minimal immersion f is automatically superminimal. As a consequence, all the superminimal two-spheres in Q₂ are completely described.     

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.     

Pure projective modules and FP-injective modules over Morita rings

Let {\mathrm{\Lambda }}_{(0,0)}=\left(\begin{array}{cc}A& {}_A{N}_B\\ {}_B{N}_A& B\end{array}\right) be a Morita ring, where the bimodule homomorphisms \varphiand \psi are zero. We study the finite presentedness, locally coherence, pure projectivity, pure injectivity, and FP-injectivity of modules over {\mathrm{\Lambda }}_{(\mathrm{0},\mathrm{0})}. Some applications are then given.     

Waring-Goldbach problems for unlike powers with almost equal variables

We prove that almost all positive even integers n can be written as n=p22+p33+p44+p55 with |pkk-N4|≤N321325+? for 2≤k≤5. Moreover, it is proved that each sufficiently large odd integer N can be represented as N=p1+p22+p33+p44+p55 with |pkk-N5|≤N321325+?for 1≤k≤5.     

Fixed points of smoothing transformation in random environment

At each time n\in \mathbb{N},\mathrm{l}\mathrm{e}\mathrm{t}{\overline{Y}}^{(n)}(\xi )=(y_1^{(n)}(\xi ),y_2^{(n)}(\xi ),\cdots ) be a random sequence of non-negative numbers that are ultimately zero in a random environment\xi \text{=}（{\xi }_n{）}_{n\in \mathbb{N}}. The existence and uniqueness of the nonnegative fixed points of the associated smoothing transformation in random environment are considered. These fixed points are solutions to the distributional equation for a.e.\xi ,Z(\xi )\stackrel{\text{d}}{=}{\sum }_{i\in \mathbb{N}+}{y}_i^{(\mathrm{0})}(\xi )Z_i^{(\mathrm{1})}(\xi ),where ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝ are random variables in random environment which satisfy that for any environment\xi; under P_{\xi }; ｛Z_i^{(\mathrm{1})}:i\in {\mathbb{N}}_{+}｝are independent of each other and {\bar{Y}}^{(\mathrm{0})}(\xi ), and have the same conditional distribution P_{\xi }(Z_i^{(1)}(\xi )\in \cdot )=P_{T\xi }(Z(T\xi )\in \cdot ) where T is the shift operator. This extends the classical results of J. D. Biggins [J. Appl. Probab., 1977, 14: 25-37] to the random environment case. As an application, the martingale convergence of the branching random walk in random environment is given as well.     

On B6- and B7-groups

A finite group G is said to be a Bn-group if any n-element subset A = {a₁, a₂,..., a_n} of G satisfies \left|A^2\right|=\left|\left\{a_i{a}_j|1\le i,j\le n\right\}\right|\le n(n+1)/2. In this paper, the characterizations of the B₆- and B₇-groups are given.