LOCAL ESTIMATE ABOUT SCHRODINGER MAXIMAL OPERATOR ON H-TYPE GROUPS

Let ? be full Laplacian on H-type group G. Then for every compact set D ■ G,a local estimate of the Schr¨odinger maximal operator holds, that is, ∫_(D)sup |e~(it?)f(x)|~2dx ■||f ||_(H~s)~2, s 1/2.We also show that the above inequality fails when s 1/4.     

Composition of fractional Orlicz maximal operators and A 1-weights on spaces of homogeneous type

For a Young function θ with 0 ≤α 〈 1, let Mα,θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type （X, d, μ） by Mα,θf（x） = supx∈（B）α ｜｜f｜｜θ,B, where ｜｜f｜｜θ,B is the mean Luxemburg norm of f on a ball B. When α= 0 we simply denote it by Me. In this paper we prove that if Ф and ψare two Young functions, there exists a third Young function θ such that the composition Mα,ψ o MФ is pointwise equivalent to Mα,θ. As a consequence we prove that for some Young functions θ, if Mα,θf 〈∞a.e. and δ ∈（0,1） then （Mα,θf）δ is an A1-weight.     

关于特定同余式方程解数的余项（英文）

Let f(x) be an irreducible polynomial of degree m ≥ 2 with integer coefficients,and let r(n) denote the number of solutions x of the congruence f(x) ≡ 0(mod n) satisfying0 ≤ x n. Define ?(x) =Σ 1≤n≤x r(n)-αx, where α is the residue of the Dedekind zeta function ζ(s, K) at its simple pole s = 1. In this paper it is shown that ∫_1~X?~2(x)dx? ε{X~(3-6/m+3+ε)if m ≥ 3,X~(2+ε) if m = 2,for any non-Abelian polynomial f(x) and any ε 0. This result constitutes an improvement upon that of Lü for the error terms on average.     

<Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-estimates of Littlewood-Paley and Lusin functions for Jacobi analysis

For α≥β≥ -1/2 let Δ(x) = (2shx)2α+1(2chx)2β+1 denote the weight function on R+ and L1(Δ) the space of integrable functions on R+ with respect to Δ(x)dx, equipped with a convolution structure. For a suitable φ∈ L1(Δ), we put φt(x) = t-1Δ(x)-1Δ(x/t)φ(x/t) for t > 0 and define the radial maximal operator Mφ as usual manner. We introduce a real Hardy space H1(Δ) as the set of all locally integrable functions f on R+ whose radial maximal function Mφ(f) belongs to L1(Δ). In this paper we obtain a relation between...     

Extracting outer function part from Hardy space function

Any analytic signal fa(e~(it)) can be written as a product of its minimum-phase signal part(the outer function part) and its all-phase signal part(the inner function part). Due to the importance of such decomposition, Kumarasan and Rao(1999), implementing the idea of the Szeg?o limit theorem(see below),proposed an algorithm to obtain approximations of the minimum-phase signal of a polynomial analytic signal fa(e~(it)) = e~(iN0t)M∑k=0a_k~(eikt),(0.1)where a_0≠ 0, a_M≠ 0. Their method involves minimizing the energy E(f_a, h_1, h_2,..., h_H) =1/(2π)∫_0~(2π)|1+H∑k=1h_k~(eikt)|~2|fa(e~(it))|~2dt(0.2) with the undetermined complex numbers hk's by the least mean square error method. In the limiting procedure H →∞, one obtains approximate solutions of the minimum-phase signal. What is achieved in the present paper is two-fold. On one hand, we rigorously prove that, if fa(e~(it)) is a polynomial analytic signal as given in(0.1),then for any integer H≥M, and with |fa(e~(it))|~2 in the integrand part of(0.2) being replaced with 1/|fa(e~(it))|~2,the exact solution of the minimum-phase signal of fa(e~(it)) can be extracted out. On the other hand, we show that the Fourier system e~(ikt) used in the above process may be replaced with the Takenaka-Malmquist(TM) system, r_k(e~(it)) :=((1-|α_k|~2e~(it))/(1-α_ke~(it))~(1/2)∏_(j=1)~(k-1)(e~(it)-α_j/(1-α_je~(it))~(1/2), k = 1, 2,..., r_0(e~(it)) = 1, i.e., the least mean square error method based on the TM system can also be used to extract out approximate solutions of minimum-phase signals for any functions f_a in the Hardy space. The advantage of the TM system method is that the parameters α_1,..., α_n,...determining the system can be adaptively selected in order to increase computational efficiency. In particular,adopting the n-best rational(Blaschke form) approximation selection for the n-tuple {α_1,..., α_n}, n≥N, where N is the degree of the given rational analytic signal, the minimum-phase part of a rational analytic signal can be accurately and efficiently extracted out.     

A subclass of Ockham algebras

Here we introduce a subclass of the class of Ockham algebras ( L ; f ) for which L satisfies the property that for every x ∈ L , there exists n ≥ 0 such that fn ( x ) and fn+1 ( x ) are complementary. We characterize the structure of the lattice of congruences on such an algebra ( L ; f ). We show that the lattice of compact congruences on L is a dual Stone lattice, and in particular, that the lattice Con L of congruences on L is boolean if and only if L is finite boolean. We also show that L is congruence coherent if and only if it is boolean. Finally, we give a sufficient and necessary condition to have the subdirectly irreducible chains.     

ON GENERALIZED FEYNMAN-KAC TRANSFORMATION FOR MARKOV PROCESSES ASSOCIATED WITH SEMI-DIRICHLET FORMS

Suppose that X is a right process which is associated with a semi-Dirichlet form(ε,D(ε)) on L~2(E;m).Let J be the jumping measure of(ε,D(ε)) satisfying J(E×E-d) ∞.Let u ∈ D(ε)_b:= D(ε)∩ L~∞(E;m),we have the following Fukushima's decomposition u(X_t)-u(X_0) =M_t~u+N_t~u.Define P_t~uf(x)=E_x[e~(N_t~u)f(X_t)].Let Q~u(f,g) =ε(f,g)+ε(u,fg)for f,g∈ D(ε)_b.In the first part,under some assumptions we show that(Q~u,D(ε)_b) is lower semi-bounded if and only if there exists a constant α_0≥0 such that ‖P_t~u‖2≤e~(α_0~t) for every t0.If one of these assertions holds,then(P_t~u)t≥0 is strongly continuous on L~2(E;m).If X is equipped with a differential structure,then under some other assumptions,these conclusions remain valid without assuming J(E×E-d)∞.Some examples are also given in this part.Let A_t be a local continuous additive functional with zero quadratic variation.In the second part,we get the representation of A_t and give two sufficient conditions for P_t~A f(x) = E_x[e~(A_t) f(X_t)]to be strongly continuous.     

一类振荡积分算子在Wiener 共合空间上的有界性

假设a,b0并且K_(a,b)(x)=(e~(i|x|~(-b)))/(|x|~(n+a))定义强奇异卷积算子T如下:Tf(x)=(K_(a,b)*f)(x),本文主要考虑了如上定义的算子T在Wiener共合空间W(FL~p,L~q)(R~n)上的有界性.另一方面,设α,β0并且γ(t)=|t|~k或γ(t)=sgn(t)|t|~k.利用振荡积分估计,本文还研究了算子T_(α,β)f(x,y)=p.v∫_(-1)~1f(x-t,y-γ(t))(e~(2πi|t|~(-β)))/(t|t|~α)dt及其推广形式∧_(α,β)f(x,y,z)=∫_(Q~2)f(x-t,y-s,z-t~ks~j)e~(-2πit)~(-β_1_s-β_2)t~(-α_1-1)s~(-α_2-1)dtds在Wiener共合空间W(FL~p,L~q)上的映射性质.本文的结论足以表明,Wiener共合空间是Lebesgue空间的一个很好的替代.     

THE ASYMPTOTIC PROPERTIES OF WEIGHTED MARKOV OPERATORS

IInttodllct1OllLet(X,E,u)be a a－inite me。ure svace．L‘＝{f:人*(x)I。(d。)＜co}．D＝{j。L‘:f＞0川f【＝1};any function f E D is called a density,and。j(A)＝jA f(x)。(dx)(A E Z)isthe Prob——hi,measurt。Sociattd WZth/．/。*‘,…0＝人/(t)…dt);j“()＝ti(厂 j(t》;and f－(x)＝max(0,一/(x))．By the support of a function g E L‘we simply mean the set ofall。such that g(x)一 0,that is,suppo＝《x:g(x)f 0}．Notice that suppg Is unique up to azero measure s戌．A s虹 M C LI Is said to be wea…     

Weighted mean convergence of Hakopian interpolation on the disk

In this paper, we study weighted mean integral convergence of Hakopian interpolation on the unit disk D. We show that the inner product between Hakopian interpolation polynomial Hn(f;x,y) and a smooth function g(x,y) on D converges to that of f(x,y) and g(x,y) on D when n →∞, provided f(x,y) belongs to C(D) and all first partial derivatives of g(x,y) belong to the space LipαM(0 ＜α≤ 1). We further show that provided all second partial derivatives of g(x,y) also belong to the space LipαM and f(x,y) belongs to C1 (D), the inner product between the partial derivative of Hakopian interpolation polynomial (6)/(6)xHn(f;x,y) and g(x,y) on D converges to that between (6)/(6)xf(x,y) and g(x,y) on D when n →∞.     

On the bilinear square Fourier multiplier operators and related multilinear square functions

Let n 1 and Tm be the bilinear square Fourier multiplier operator associated with a symbol m,which is defined by Tm(f1, f2)(x) =(∫_0~∞︱∫_((Rn)2)e~(2πix·(ξ1+ξ2))m(tξ1, tξ2)?f1(ξ1)?f2(ξ2)dξ1dξ2︱~2(dt)/t) ~(1/2).Let s be an integer with s ∈ [n + 1, 2n] and p0 be a number satisfying 2n/s p0 2. Suppose that νω=∏_i~2=1ω_i~(p/pi) and each ω_i is a nonnegative function on Rn. In this paper, we show that under some condition on m, Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~p(ν_ω) if p0 p1, p2 ∞ with 1/p = 1/p1 + 1/p2. Moreover,if p0 2n/s and p1 = p0 or p2 = p0, then Tm is bounded from L~(p1)(ω_1) × L~(p2)(ω_2) to L~(p,∞)(ν_ω). The weighted end-point L log L type estimate and strong estimate for the commutators of Tm are also given. These were done by considering the boundedness of some related multilinear square functions associated with mild regularity kernels and essentially improving some basic lemmas which have been used before.     

Endp oint Estimates for Commutators of Fractional Integrals Asso ciated to Op erators with Heat Kernel Bounds

Let L be the infinitesimal generator of an analytic semigroup on L~2(R~n)with pointwise upper bounds on heat kernel,and denote by L~(-α/2)the fractional integrals of L.For a BMO function b(x),we show a weak type Llog L estimate of the commutators [b,L~(-α/2)](f)(x) = b(x)L~(-α/2)(f)(x)-L~(-α/2)(bf)(x).We give applications to large classes of differential operators such as the Schr¨odinger operators and second-order elliptic operators of divergence form.     

BOUNDEDNESS OF GENERALIZED MARCINKIEWICZ INTEGRAL ON THE H~p-SOBOLEV SPACES

§ 1　IntroductionIn this paper,we consider the following general Marcinkiewicz integral:μΩ,αf (x) =∫∞0 ∫|x- y|≤ tΩ (x -y)| x -y| n- 1 f(y) dy2 dtt3+2α1 2 (1 .1 )for all f∈ S(Rn) ,α≥ 0 ,andΩ is a distribution kernel.Whenα=0 ,it is the classicalMacinkiewicz integral operator many mathematicians have studied.There are a lot ofreferences about the topic that you can find.Here we take a simple list of some results ofthis topic.In 1 958,Stein[8] firstproved that ifΩ∈ Lipα(Sn- 1…     

Steady Ricci solitons with horizontally ε-pinched Ricci curvature

In this paper,we prove that any κ-noncollapsed gradient steady Ricci soliton with nonnegative curvature operator and horizontally κ-pinched Ricci curvature must be rotationally symmetric.As an application,we show that any κ-noncollapsed gradient steady Ricci soliton(Mn,g,f) with nonnegative curvature operator must be rotationally symmetric if it admits a unique equilibrium point and its scalar curvature R(x) satisfies lim_(ρ(x)→∞) R(x)f(x)=C_0 sup_(x∈M) R(x) with C_0n-2/2.     

Existence and multiplicity of normalized solutions for a class of fractional Choquard equations

In this paper, we study the existence and multiplicity of solutions with a prescribed L2-norm for a class of nonlinear fractional Choquard equations in RN:(-△)su-λu =(κα*|u|p)|u|p-2u,where N≥3,s∈(0,1),α∈(0,N),p∈(max{1 +(α+2s)/N,2},(N+α)/(N-2s)) and κα(x)=|x|α-N. To get such solutions,we look for critical points of the energy functional I(u) =1/2∫RN|(-△)s/2u|2-1/(2p)∫RN(κα*|u|p)|u|p on the constraints S(c)={u∈Hs(RN):‖u‖L2(RN)2=c},c >0.For the value p∈(max{1+(α+2s)/N,2},(N+α)/(N-2s)) considered, the functional I is unbounded from below on S(c). By using the constrained minimization method on a suitable submanifold of S(c), we prove that for any c>0, I has a critical point on S(c) with the least energy among all critical points of I restricted on S(c). After that,we describe a limiting behavior of the constrained critical point as c vanishes and tends to infinity. Moreover,by using a minimax procedure, we prove that for any c>0, there are infinitely many radial critical points of I restricted on S(c).     

NEW PROOFS OF THE DECAY ESTIMATE WITH SHARP RATE OF THE GLOBAL WEAK SOLUTION OF THE n-DIMENSIONAL INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Consider the Cauchy problem for the n-dimensional incompressible NavierStokes equations:??tu-α△u+(u·?)u+?p = f(x, t), with the initial condition u(x, 0) = u0(x) and with the incompressible conditions ? · u = 0, ? · f = 0 and ? · u0= 0. The spatial dimension n ≥ 2.Suppose that the initial function u0∈ L1(Rn) ∩ L2(Rn) and the external force f ∈ L1(Rn× R+) ∩ L1(R+, L2(Rn)). It is well known that there holds the decay estimate with sharp rate:(1 + t)1+n/2∫Rn|u(x, t)|2 dx ≤ C, for all time t 0, where the dimension n ≥ 2, C 0 is a positive constant, independent of u and(x, t).The main purpose of this paper is to provide two independent proofs of the decay estimate with sharp rate, both are complete, systematic, simplified proofs, under a weaker condition on the external force. The ideas and methods introduced in this paper may have strong influence on the decay estimates with sharp rates of the global weak solutions or the global smooth solutions of similar equations, such as the n-dimensional magnetohydrodynamics equations, where the dimension n ≥ 2.     

EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/|x|s|u|q-2u,(x,t)∈Ω×(0,∞),u(x,0)=f(x),x∈Ω,u(x,t)=0,(x,t)∈Ω×(0,∞),(P)}where-△pu ≡-div(|▽u|p-2▽u),1相似文献   

关于有界单叶函数的第三项系数

1.在单位圆|z|<1中的单叶正则函数,满足f(0)=0,f′(0)=1的全体,成一函数族s,我们熟知有准确估计及-1≤|C_3|-|c_2|≤Q≈1.05,这里C_n表示f(z)展开成幂级数时z~n项的系数S中的函数f(z)满足|f(z)|相似文献   

The Gradient Estimate of a Neumann Eigenfunction on a Compact Manifold with Boundary

Let e_λ(x) be a Neumann eigenfunction with respect to the positive Laplacian A on a compact Riemannian manifold M with boundary such that △e_λ=λ~2e_λ in the interior of M and the normal derivative of e\ vanishes on the boundary of M.Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M.The authors show the following gradient estimate for χλf as λ≥1:‖▽χλ f‖∞≤C(λ‖χλ f‖∞+λ~(-1)‖△χλf‖∞),where C is a positive constant depending only on M,As a corollary,the authors obtain the gradient estimate of e_λ:For every λ≥1,it holds that ‖▽e_λ‖∞≤Cλ‖e_λ‖∞.     

Vector cascade algorithms with infinitely supported masks in weighted L 2-spaces

In this paper, we shall study the solutions of functional equations of the form Φ =∑α∈Zsa(α)Φ(M·-α), where Φ = (φ1, . . . , φr)T is an r×1 column vector of functions on the s-dimensional Euclidean space, a:=(a(α))α∈Zs is an exponentially decaying sequence of r×r complex matrices called refinement mask and M is an s×s integer matrix such that limn→∞M-n=0. We are interested in the question, for a mask a with exponential decay, if there exists a solution Φ to the functional equation with each function φj,j=1, . . . ,r, belonging to L2(Rs) and having exponential decay in some sense? Our approach will be to consider the convergence of vector cascade algorithms in weighted L2 spaces. The vector cascade operator Qa,M associated with mask a and matrix M is defined by Qa,Mf:=∑α∈Zsa(α)f (M·-α),f= (f1, . . . , fr)T∈(L2,μ(Rs))r.The iterative scheme (Qan,Mf)n=1,2,... is called a vector cascade algorithm or a vector subdivision scheme. The purpose of this paper is to provide some conditions for the vector cascade algorithm to converge in (L2,μ(Rs))r , the weighted L2 space. Inspired by some ideas in [Jia,R.Q.,Li,S.: Refinable functions with exponential decay: An approach via cascade algorithms. J. Fourier Anal. Appl., 17, 1008-1034 (2011)], we prove that if the vector cascade algorithm associated with a and M converges in (L2(Rs))r, then its limit function belongs to (L2,μ(Rs))r for some μ0.