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).     

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.     

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.     

Generalized area operators on Lp(?n)

We introduce the generalized area operators by using nonnegative measures defined on upper half-spaces ?+n+1. The characterization of the boundedness and compactness of the generalized area operator from L^p(?n) to L^q(?n) is investigated in terms of s-Carleson measures with 1<p, q<+∞. In the case of p = q = 1, the weak type estimate is also obtained.     

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.     

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)<+∞     

Diophantine inequalities over Piatetski-Shapiro primes

Let c>1 and : We study the solubility of the Diophantine inequality in Piatetski-Shapiro primes p₁,p₂, .., p_s of the form p_j=[m^{\gamma }] for some m\in \mathbb{N}, and improve the previous results in the cases s = 2, 3, 4.     

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.     

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.     

Well-posedness of degenerate differential equations in H?lder continuous function spaces

Using known operator-valued Fourier multiplier results on vectorvalued H?lder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations (Mu)'(t)=Au(t)+f(t) for t∈R in H?lder continuous function spaces Ca(R;X) by the boundedness of the M-resolvent of A, where A and M are closed operators on a Banach space X satisfying D(A)?D(M).     

Slim exceptional set for sums of two squares,two cubes,and two biquadrates of primes

We prove that, with at most O\left(N^{\frac{\mathrm{17}}{\mathrm{192}}+\epsilon }\right) exceptions, all even positive integers up to Nare expressible in the form p_{\mathrm{1}}^{\mathrm{2}}+p_{\mathrm{2}}^{\mathrm{2}}+p_{\mathrm{3}}^{\mathrm{3}}+p_{\mathrm{4}}^{\mathrm{3}}+p_{\mathrm{5}}^{\mathrm{4}}+p_{\mathrm{6}}^{\mathrm{4}},where p_{\mathrm{1}},\mathrm{}{p}_{\mathrm{2}},.\mathrm{}.\mathrm{}.\mathrm{},\mathrm{}{p}_{\mathrm{6}} are prime numbers. This gives large improvement of a recent result O\left(N^{\frac{\mathrm{13}}{\mathrm{16}}+\epsilon }\right) due to M. Zhang and J. J. Li.     

Bicomplex Hermitian Clifford analysis

Complex Hermitian Clifford analysis emerged recently as a refinement of the theory of several complex variables, while at the same time, the theory of bicomplex numbers motivated by the bicomplex version of quantum mechanics is also under full development. This stimulates us to combine the Hermitian Clifford analysis with the theory of bicomplex number so as to set up the theory of bicomplex Hermitian Clifford analysis. In parallel with the Euclidean Clifford analysis, the bicomplex Hermitian Clifford analysis is centered around the bicomplex Hermitian Dirac operator |D:C∞(R4n,W4n)→C∞(R4n,W4n), where W4n is the tensor product of three algebras, i.e., the hyperbolic quaternion B^, the bicomplex number B, and the Clifford algebra Rn. The operator D is a square root of the Laplacian in R4n, introduced by the formula D|=∑j=03Kj?Zj with Kjbeing the basis of B^, and ?Zj denoting the twisted Hermitian Dirac operators in the bicomplex Clifford algebra B?R0,4n whose definition involves a delicate construction of the bicomplexWitt basis. The introduction of the operator D can also overturn the prevailing opinion in the Hermitian Clifford analysis in the complex or quaternionic setting that the complex or quaternionic Hermitiean monogenic functions are described by a system of equations instead of by a single equation like classical monogenic functions which are null solutions of Dirac operator. In contrast to the Hermitian Clifford analysis in quaternionic setting, the Poisson brackets of the twisted real Clifford vectors do not vanish in general in the bicomplex setting. For the operator D, we establish the Cauchy integral formula, which generalizes the Martinelli-Bochner formula in the theory of several complex variables.     

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.     

Exponential sums involving Maass forms

We study the exponential sums involving l：burmr coeffcients ot Maass forms and exponential functions of the form e（anZ）, where 0 ≠ α∈R and 0 〈 β 〈 1. An asymptotic formula is proved for the nonlinear exponential sum ∑x〈n≤2x λg（n）e（αnβ）, when β = 1/2 and ｜α｜ is close to 2√ q C Z＋, where Ag（n） is the normalized n-th Fourier coefficient of a Maass cusp form for SL2 （Z）. The similar natures of the divisor function 7（n） and the representation function r（n） in the circle problem in nonlinear exponential sums of the above type are also studied.     

Values of binary linear forms at prime arguments

value of a given binary linear form at prime arguments. Let λ₁ and λ₂ be positive real numbers such that λ₁/λ₂ is irrational and algebraic. For any (C, c) well-spaced sequence V and δ>0, let E(V, X, δ) denote the number of υ∈V with υ≤X for which the inequality |λ1p1+λ2ρ2−υ|<υ−δ has no solution in primes p₁, p₂. It is shown that for any ε>0,we have E(V, X, δ) «max(X35+2δ+ε,X23+43δ+ε).     

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.     

Fekete and Szegö problem for a subclass of quasi-convex mappings in several complex variables

Let K be the familiar class of normalized convex functions in the unit disk. Keogh and Merkes proved the well-known result that max?f∈K|a3−λa22|≤max?{1/3,|λ−1|},λ∈?, and the estimate is sharp for each λ. We investigate the corresponding problem for a subclass of quasi-convex mappings of type B defined on the unit ball in a complex Banach space or on the unit polydisk in ?n. The proofs of these results use some restrictive assumptions, which in the case of one complex variable are automatically satisfied.     

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}).     

Minimizers of anisotropic Rudin-Osher-Fatemi models

We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models E1φ(u)=∫Ωφo(Du)dx+λ∫Ω|u−f|dx, u∈BV(Ω),E2φ(u)=∫Ωφo(Du)dx+λ∫Ω(u−f)2dx, u∈BV(Ω), where Ω⊂?2 is a domain, φo is an anisotropic norm on ?2, and f is a solution of the anisotropic 1-Laplacian equations.