共查询到20条相似文献,搜索用时 15 毫秒
1.
Let φ be a growth function, and let A : = - ( ? - i a ) ? ( ? - i a ) + V be a magnetic Schr?dinger operator on L 2 ( ? n ) , n ≥ 2 , where α : = ( α 1 , α 2 , ? , α n ) ∈ L l o c 2 ( ? n , ? n ) and 0 ≤ V ∈ L l o c 1 ( ? n ) . We establish the equivalent characterizations of the Musielak-Orlicz-Hardy space H A , φ ( ? n ) , defined by the Lusin area function associated with { e - t 2 A } t > 0 , in terms of the Lusin area function associated with { e - t A } t > 0 , the radial maximal functions and the nontangential maximal functions associated with { e - t 2 A } t > 0 and { e - t A } t > 0 , respectively. The boundedness of the Riesz transforms L k A - 1 / 2 , k ∈ { 1 , 2 , ? , n } , from H A , φ ( ? n ) to L φ ( ? n ) is also presented, where Lk is the closure of ? ? x k - i α k in L 2 ( ? n ) . These results are new even when φ ( x , t ) : = ω ( x ) t p for all x ∈ ? n and t ∈(0,+∞) with p ∈(0, 1] and ω ∈ A ∞ ( ? n ) (the class of Muckenhoupt weights on ? n ). 相似文献
2.
Consider a supercritical superprocess X = {Xt, 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 ) λ 2 + ∫ ( 0 , + ∞ ) ( e - λ y - 1 + λ y ) n ( x , d y ) , ? x ∈ E , λ > 0 , where a ∈ B b ( E ) , b ∈ B b + ( E ) , and n is a kernel from E to (0,+∞) satisfying sup ? x ∈ E ∫ 0 + ∞ y 2 n ( x , d y ) < + ∞ . Put T t f ( x ) = P δ x ? f , X t ? . Suppose that the semigroup {Tt; t≥0}is compact. Let λ0 be the eigenvalue of the (possibly non-symmetric) generator L of {Tt}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 {Tt}, we prove that for a large class of suitable functions f, lim ? t → + ∞ e - λ 0 t ? f , X t ? = W ∞ ∫ E ? ^ 0 ( y ) f ( y ) m ( d y ) , ? P μ - 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 - λ 0 t ? ? 0 , X t ? . Moreover, the exceptional set in the above limit does not depend on the initial measure μ and the function f. 相似文献
3.
Let (Xt)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 ) d x d y , ? 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 T t V ( f ) ( x ) = E x ( exp ? ( - ∫ 0 t V ( X s ) d s ) f ( X t ) ) , ? x ∈ ? d , f ∈ L 2 ( ? d ; d x ) . In particular, we prove that for J ( x , y ) ≈ | x - y | - d - a l { | x - y | ≤ 1 } + e - | x - y | l { | x - y | > 1 } with α ∈(0, 2) and V ( x ) = | x | λ with λ>0, ( T t V ) t ≥ 0 is intrinsically ultracontractive if and only if λ>1; and that for symmetric α-stable process (Xt)t≥0 with α ∈(0, 2) and V ( x ) = log ? λ ( 1 + | x | ) with some λ>0, ( T t V ) t ≥ 0 is intrinsically ultracontractive (or intrinsically supercontractive) if and only if λ>1, and ( T t V ) t ≥ 0 is intrinsically hypercontractive if and only if λ ≥ 1 . Besides, we also investigate intrinsic contractivity properties of ( T t V ) t ≥ 0 for the case that lim inf ? | x | → + ∞ V ( x ) < + ∞ 相似文献
4.
We prove some transcendence results for the sums of some multivariate serms of the form ∑j1,j2,...,jm=0 ^∞Cj1j2...jm(r1^j1r2^j2...rm^jm) for n = 1, 2, where Cj1j2...jm are some rational functions of j1 + j2 + ... + jm. 相似文献
5.
Shangquan BU 《Frontiers of Mathematics in China》2015,10(2):239
Using known operator-valued Fourier multiplier results on vectorvalued H?lder continuous function spaces, we completely characterize the wellposedness of the degenerate differential equations ( M u ) ' ( t ) = A u ( t ) + f ( t ) for t ∈ R in H?lder continuous function spaces C a ( 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 ) . 相似文献
6.
We consider a branching random walk on N with a random environment in time (denoted by ξ). Let Zn be the counting measure of particles of generation n, and let Zn(t) be its Laplace transform. We show the convergence of the free energy n-llog Zn(t), large deviation principles, and central limit theorems for the sequence of measures {Zn}, and a necessary and sufficient condition for the existence of moments of the limit of the martingale Zn(t)/E[Zn(t)ξ]. 相似文献
7.
Jiawang NIE 《Frontiers of Mathematics in China》2012,7(2):321-346
This paper studies the problem of minimizing a homogeneous polynomial (form) f (x ) over the unit sphere S n - 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 S n - 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 ‖ 2 2 d ? i s S O S . Let f sos be the above optimal value. Then we show that for all n ≥2d ,1 ≤ f max ? - f s o s f max ? - f min ? ≤ C ( d ) ( n 2 d ) . Here, the constant C (d ) is independent of n . Second, when f (x ) is a multi-form and S n - 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. 相似文献
8.
At each time be a random sequence of non-negative numbers that are ultimately zero in a random environment . 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 ,where are random variables in random environment which satisfy that for any environment ; under ; are independent of each other and , and have the same conditional distribution 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. 相似文献
9.
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 = L q ( 0 , T ; B p , ∞ r ) o r p = L 1 ( 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. 相似文献
10.
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. 相似文献
11.
We give the explicit formulas of the minimizers of the anisotropic Rudin-Osher-Fatemi models E 1 φ ( u ) = ∫ Ω φ o ( D u ) d x + λ ∫ Ω | u − f | d x , u ∈ B V ( Ω ) , E 2 φ ( u ) = ∫ Ω φ o ( D u ) d x + λ ∫ Ω ( u − f ) 2 d x , u ∈ B V ( Ω ) , where Ω ⊂ ? 2 is a domain, φ o is an anisotropic norm on ? 2 , and f is a solution of the anisotropic 1-Laplacian equations. 相似文献
12.
This paper deals with anisotropic solutions u ∈ W 1 , ( p i ) ( Ω , ? N ) to the nonlinear elliptic system − Σ i = 1 n D i ( a i α ( χ , D u ( χ ) ) ) = − Σ i = 1 n D i F i α ( χ ) , α = 1 , 2 , ... , N , We present a monotonicity inequality for the matrix a = ( a i α ) ∈ ? N × n , whichguarantees global pointwise bounds for anisotropic solutionsu . 相似文献
13.
Let be a fractional Brownian motion with Hurst index . Inspired by pathwise integrals and Wick product, in this paper, we consider the forward and symmetric Wick-Itô integrals with respect to BH as follows: in probability, where ◊ denotes the Wick product. We show that the two integrals coincide with divergence-type integral of BH for all . 相似文献
14.
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 | a 3 − λ a 2 2 | ≤ 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. 相似文献
15.
Let be the solution to a linear stochastic heat equation driven by a Gaussian noise, which is a Brownian motion in time and a fractional Brownian motion in space with Hurst parameter : For any given , we show a decomposition of the stochastic process as the sum of a fractional Brownian motion with Hurst parameter H/2 (resp., H) and a stochastic process with C∞-continuous trajectories. Some applications of those decompositions are discussed. 相似文献
16.
17.
Let c>1 and : We study the solubility of the Diophantine inequality in Piatetski-Shapiro primes p1,p2, .., ps of the form for some , and improve the previous results in the cases s = 2, 3, 4. 相似文献
18.
We consider a pendulum type equation with p-Laplacian , where 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 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 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. 相似文献
19.
Liqun HU 《Frontiers of Mathematics in China》2015,10(5):1101
For the normalized Fourier coefficients of Maass cusp forms λ(n) and the normalized Fourier coefficients of holomorphic cusp forms a(n), we give the bound of ∑ m 1 2 + m 2 2 + m 3 2 ≤ x λ ( m 1 2 + m 2 2 + m 3 2 ) Λ ( m 1 2 + m 2 2 + m 3 2 ) and ∑ m 1 2 + m 2 2 + m 3 2 ≤ x a ( m 1 2 + m 2 2 + m 3 2 ) Λ ( m 1 2 + m 2 2 + m 3 2 ) . 相似文献
20.
Let X be a ball quasi-Banach function space satisfying some mild additional assumptions and the associated Hardy-type space. In this article, we first establish the finite atomic characterization of . As an application, we prove that the dual space of is the Campanato space associated with X. For any given and , using the atomic and the Littlewood–Paley function characterizations of ,we also establish its s-order intrinsic square function characterizations, respectively, in terms of the intrinsic Lusin-area function ,the intrinsic g-function ,and the intrinsic -function , where λ coincides with the best known range. 相似文献