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.     

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.     

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.     

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.     

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.     

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 }).     

Convergence of complex martingale for a branching random walk in an independent and identically distributed environment

We consider anR^d-valued discrete time branching random walk in an independent and identically distributed environment indexed by time n∈N.Let W_n(z)(z∈C^d)be the natural complex martingale of the process.We show necessary and sufficient conditions for the L^α-convergence of W_n(z)forα>1,as well as its uniform convergence region.     

Weighted product Hardy space associated with operators

Assuming that the operators L₁, L₂ are self-adjoint and {\mathrm{e}}^{-{{\displaystyle tL}}_i}(i=\mathrm{1},\mathrm{2}) satisfy the generalized Davies-Gaffney estimates, we shall prove that the weighted Hardy space {{\displaystyle H}}_{{{\displaystyle L}}_{\mathrm{1}},{{\displaystyle L}}_{\mathrm{2}},\omega }^{\mathrm{1}}({ℝ}^{{{\displaystyle n}}_{\mathrm{1}}}\times {ℝ}^{{{\displaystyle n}}_{\mathrm{2}}}) associated to operators L₁, L₂ on product domain, which is defined in terms of area function, has an atomic decomposition for some weight \omega.     

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   

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.     

Dynamical behaviors for generalized pendulum type equations with p-Laplacian

We consider a pendulum type equation with p-Laplacian {({\varphi }_p(x^{\prime }))}^{\prime }+{G_x}^{\prime }(t,x)=p(t), where {{\displaystyle \varphi }}_p(u)={\left|u\right|}^{p-\mathrm{2}}u,p>\mathrm{1},G(t,x) and p(t) are 1-periodic about every variable. The solutions of this equation present two interesting behaviors. On the one hand, by applying Moser's twist theorem, we find infinitely many invariant tori whenever {\displaystyle {\int }_{\mathrm{0}}^{\mathrm{1}}p(t)}\mathrm{d}t=\mathrm{0}, which yields the bounded-ness of all solutions and the existence of quasi-periodic solutions starting at t = 0 on the invariant tori. On the other hand, if p(t) = 0 and {G_x}^{\prime }(t,x) has some specific forms, we find a full symbolic dynamical system made by solutions which oscillate between any two different trivial solutions of the equation. Such chaotic solutions stay close to the trivial solutions in some fixed intervals, according to any prescribed coin-tossing sequence.     

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).     

An isometrical CPn-theorem

Let M^n(n\ge \mathrm{3}) be a complete Riemannian manifold with {{\displaystyle \sec ?}}_M\ge \mathrm{1}, and let {{\displaystyle M}}_i^{{{\displaystyle n}}_i}(i=\mathrm{1},\mathrm{2}) be two complete totally geodesic submanifolds in M. We prove that if n₁ + n₂ = n − 2 and if the distance \left|{{\displaystyle M}}_{\mathrm{1}}{{\displaystyle M}}_{\mathrm{2}}\right|\ge \pi /\mathrm{2}, then M_i is isometric to {\mathbb{S}}^{{{\displaystyle n}}_i}/{{\displaystyle \mathrm{?}}}_h,{\mathrm{?}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}}, or {\mathrm{?}\mathrm{P}}^{{{\displaystyle n}}_i/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}} with the canonical metric when n_i>0, and thus, M is isometric to {\mathbb{S}}^n/{{\displaystyle \mathrm{?}}}_h,{\mathrm{?}\mathrm{P}}^{n/\mathrm{2}}, or {\mathrm{?}\mathrm{P}}^{n/\mathrm{2}}/{{\displaystyle \mathrm{?}}}_{\mathrm{2}} except possibly when n = 3 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}{\mathbb{S}}^{\mathrm{1}}/{{\displaystyle \mathrm{?}}}_h with h\ge \mathrm{2} or n = 4 and M₁ (or M₂) \stackrel{\mathrm{i}\mathrm{s}\mathrm{o}}{{\displaystyle \cong }}\mathrm{?}{\mathrm{P}}^{\mathrm{2}}.     

Forward and symmetric Wick-Itô integrals with respect to fractional Brownian motion

Let B^H=\{{{\displaystyle B}}_t^H,t\ge \mathrm{0}\} be a fractional Brownian motion with Hurst index H\in (\mathrm{0},\mathrm{1}). Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to B^H as follows: {\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{-}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_s^H)\mathrm{d}s,{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond {\mathrm{d}}^{°}{{\displaystyle B}}_s^H=\underset{\epsilon \downarrow \mathrm{0}}{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}\epsilon }}{\displaystyle {\int }_{\mathrm{0}}^t{{\displaystyle u}}_s}\diamond ({{\displaystyle B}}_{s+\epsilon }^H-{{\displaystyle B}}_{(s-\epsilon )\vee \mathrm{0}}^H)\mathrm{d}s,in probability, where ◊ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of B^H for all H\in (\mathrm{0},\mathrm{1}).     

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}}}^{+}.     

Existence and multiplicity results for nonlinear Schrödinger-Poisson equation with general potential

This paper is concerned with the Schrödinger-Poisson equation-\mathrm{\Delta }u+V(x)u+\phi (x)u=f(x,u),x\in {ℝ}^{\mathrm{3}},-\mathrm{\Delta }\phi =u^{\mathrm{2}},\underset{\left|x\right|\to +\infty }{\lim }\phi (x)=\mathrm{0.}Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of solutions are obtained via variational methods.     

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.     

Ground state solutions for a non-autonomous nonlinear Schrödinger-KdV system

We study the Schrödinger-KdV system\{\begin{array}{ll}-\mathrm{\Delta }u+{\lambda }_{\mathrm{1}}(x)u=u^{\mathrm{3}}+\beta uv,& u\in H^{\mathrm{1}}({ℝ}^N),\\ -\mathrm{\Delta }v+{\lambda }_{\mathrm{2}}(x)v={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}^{\mathrm{2}}+{\displaystyle \frac{\beta }{\mathrm{2}}}{u}^{\mathrm{2}},& v\in H^{\mathrm{1}}({ℝ}^N),\end{array}where N=\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{}{\lambda }_i(x)\in C({ℝ}^N,ℝ),{\lim }_{\left|x\right|\to \infty }{\lambda }_i(x)={\lambda }_i(\infty ), and {\lambda }_i(x)\le {\lambda }_i(\infty ),i= 1,2,a.e. x\in {ℝ}^N.We obtain the existence of nontrivial ground state solutions for the above system by variational methods and the Nehari manifold.     

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.