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1.
Boundedness of generalized higher commutators of Marcinkiewicz integrals   总被引:1,自引:0,他引:1  
Let (b) = (b1,…,bm) be a finite family of locally integrable functions. Then,we introduce generalized higher commutator of Marcinkiwicz integral as follows:μ(b)Ω=(∫∞o|F(b)Ω,t(f)(x)|2et/t)1/2,whereF(b)Ω(f)(x)=1/t∫|x-y|≤tΩ(x-y)/|x-y|n-1m∏j=1(bj(x)-bj(y))f(y)dy.When bj ∈(A)βj, 1≤j≤m, 0<βj<1,m∑j=1βj =β<n, and Ω is homogeneous of degree zero and satisfies the cancelation condition, we prove that μ(b)Ω is bounded from Lp(Rn)to Ls(Rn), where 1 < p < n/β and 1/s = 1/p -β/n. Moreover, if Ω also satisfies some Lq-Dini condition, then μ(b)Ω is bounded from Lp(Rn) to (F)β,∞p(Rn) and on certain Hardy spaces. The article extends some known results.  相似文献   

2.
§1 IntroductionSuppose thatf is analytic in the open unit disc D in the complex plane.We defineMp(r,f) =12π∫2π0 | f(reiθ) | pdθ1 / p,0

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3.
Let (→b)=(b1,…,bm),bi∈Λβi(Rn),1≤i≤m,0<βi<β,0<β<1,[(→b),T]f(x)=∫Rn,(b1(x)-b1(y))…(bm(x)-bm(y)))K(x-y)f(y)dy where K is a Calder(o)n-Zygmund kernel.In this paper,we show that[(→b),T] is bounded from Lp (Rn) to Fβ,∞p(Rn),as well as[(→b,Iα)] from Lp(Rn) to Fβ,∞p(Rn),where 1/q=1/p-α/n.  相似文献   

4.
Let h be a measurable function defined on R+×R+. LetΩ∈ L(log L+)νq(Sn1-1×Sn2-1)(1 ≤νq≤ 2) be homogeneous of degree zero and satisfy certain cancellation conditions. We show that the singular integral T f(x1, x2) = p. v.∫ Rn1+n2Ω(y′1, y′2)h(|y1|, |y2|)|y1|n1|y2|n2f(x1- y1, x2- y2)dy1dy2maps from Sα1, α2p, q˙F(Rn1× Rn2) boundedly to itself for 1 p, q ∞, α1, α2 ∈ R.  相似文献   

5.
In this paper, we study integral operators of the form Tαf(x)=∫Rn|x-A1y|-α1 ··· |x-Amy|-αmf(y)dy,where Ai are certain invertible matrices, αi 0, 1 ≤ i ≤ m, α1 + ··· + αm = n-α, 0 ≤α n. For 1/q = 1/p-α/n , we obtain the Lp (Rn, wp)-Lq(Rn, wq) boundedness for weights w in A(p, q) satisfying that there exists c 0 such that w(Aix) ≤ cw(x), a.e. x ∈ Rn , 1 ≤ i ≤ m.Moreover, we obtain theappropriate weighted BMO and weak type estimates for certain weights satisfying the above inequality. We also give a Coifman type estimate for these operators.  相似文献   

6.
本文研究了Marcinkiewicz积分交换子μΩ,b(f)(x)=(integral from n=0 to ∞|Fb,t(f)(x)|2 dt/t3)1/2, 其中Fb,t(f)(x)=integral from n=|x-y|≤t(Ω(x-y_/|x-y|n-1)b(x)-b(y)f(y)dy及b∈Λβ,证明了算子μΩ,b是Lp(Rn) 到Fβ,∞p(Rn)上的有界算子并且也是Lp(Rn)到Lq(Rn)上的有界算子.  相似文献   

7.
Let→b=(b1,b2,…,bm),bi∈∧βi(Rn),1≤I≤m,βi>0,m∑I=1βi=β,0<β<1,μΩ→b(f)(x)=(∫∞0|F→b,t(f)(x)|2dt/t3)1/2,F→b,t(f)(x)=∫|x-y|≤t Ω(x,x-y)/|x-y|n-1 mΠi=1[bi(x)-bi(y)dy.We consider the boundedness of μΩ,→b on Hardy type space Hp→b(Rn).  相似文献   

8.
Toeplitz operators related to strongly singular Calderón-Zygmund operators   总被引:1,自引:0,他引:1  
In this paper, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and Lipschitz function b∈Λβ0(Rn) is discussed from Lp(Rn) to Lq(Rn), 1/q=1/p-β0/n, and from Lp(Rn) to Triebel-Lizorkin space Fβ0,∞p. We also obtain the boundedness of generalized Toeplitz operatorθbα0 from LP(Rn) to Lq(Rn), 1/q =1/p-α0 β0/n. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator Tb(f) related to strongly singular Calderon-Zygmund operators and BMO function b is discussed on LP(Rn), 1 < p <∞.  相似文献   

9.
Let B be a Banach space, Φ1 , Φ2 be two generalized convex Φ-functions and Ψ 1 , Ψ 2 the Young complementary functions of Φ1 , Φ2 respectively with ∫t t 0 ψ2 (s) s ds ≤ c 0 ψ1 (c 0 t) (t > t 0 ) for some constants c 0 > 0 and t 0 > 0, where ψ1 and ψ2 are the left-continuous derivative functions of Ψ 1 and Ψ 2 , respectively. We claim that: (i) If B is isomorphic to a p-uniformly smooth space (or q-uniformly convex space, respectively), then there exists a constant c > 0 such that for any B-valued martingale f = (f n ) n ≥ 0 , ‖f*‖Φ1 ≤ c‖S (p) (f ) ‖Φ2 (or ‖S (q) (f )‖Φ1 ≤ c‖f*‖Φ2 , respectively), where f and S (p) (f ) are the maximal function and the p-variation function of f respec- tively; (ii) If B is a UMD space, T v f is the martingale transform of f with respect to v = (v n ) n ≥ 0 (v*≤ 1), then ‖(T v f )*‖Φ1 ≤ c ‖f *‖Φ2 .  相似文献   

10.
本文建立了R1中临界版的离散加权型Hardy-Littlewood-Sobolev不等式∑-N≤r,s≤N;r≠0,s≠0;r≠s(1/|r|α*arbs|s|α/|r-s|≤CαλαN‖a‖2‖b‖2),其中α≥0,a=(a-N,...,aN),b=(b-N,...,bN).当α≥1时,我们得到了最佳常数λαN为Nα-1/2,即∑-N≤r,s≤N;r≠0,s≠0;r≠s(1/|r|α*arbs|s|α/|r-s|≤CαNα-1/2‖a‖2‖b‖2).  相似文献   

11.
§ 1  Introduction and resultsL et { X,Xi;i≥ 1} be a sequence of i.i.d.random variables,and set Sn= ni=1 Xi,n≥1.Hsu and Robbins[1 ] introduced the conceptof complete convergence.They together withErdos[2 ] proved n≥ 1 P(|Sn|≥εn) <∞ ,ε>0 (1)if and only if EX=0 and EX2 <∞ .L ater,Spitzer[3] proved n≥ 11n P(|Sn|≥εn) <∞ ,ε>0if and only if EX =0 and E|X|<∞ .More generally,it was shown by Baum and Katz[4 ]that,for 0 0 (…  相似文献   

12.
The Nikolskii type inequality for cardinal splines
is proved, which is exact in the sense of order, where ∈ ℒ m,h , and ℒ m,k is the space of cardinal splines with nodes
Project supported by the National Natural Science Foundation of China (Grant No. 19671012), and Doctoral Programme Foundation of Institution of Higher Education.  相似文献   

13.
§ 1 IntroductionConsiderthenonautonomousdelaylogisticdifferenceequationΔyn =pnyn( 1 - yτ(n) )  ,n =0 ,1 ,2 ,...,( 1 1 )wherepn ∞n =0 isasequenceofpositiverealnumbers ,τ(n) ∞n =0 isanondecreasingsequenceofintegers,τ(n) <nandlimn→∞τ(n) =∞ ,Δyn=yn +1- yn.Motivatedbyplausibleapplications…  相似文献   

14.
Let f(x, y) be a periodic function defined on the region D
with period 2π for each variable. If f(x, y) ∈ C p (D), i.e., f(x, y) has continuous partial derivatives of order p on D, then we denote by ω α,β(ρ) the modulus of continuity of the function
and write
For p = 0, we write simply C(D) and ω(ρ) instead of C 0(D) and ω 0(ρ). Let T(x,y) be a trigonometrical polynomial written in the complex form
We consider R = max(m 2 + n 2)1/2 as the degree of T(x, y), and write T R(x, y) for the trigonometrical polynomial of degree ⩾ R. Our main purpose is to find the trigonometrical polynomial T R(x, y) for a given f(x, y) of a certain class of functions such that
attains the same order of accuracy as the best approximation of f(x, y). Let the Fourier series of f(x, y) ∈ C(D) be
and let
Our results are as follows Theorem 1 Let f(x, y) ∈ C p(D (p = 0, 1) and
Then
holds uniformly on D. If we consider the circular mean of the Riesz sum S R δ (x, y) ≡ S R δ (x, y; f):
then we have the following Theorem 2 If f(x, y) ∈ C p (D) and ω p(ρ) = O(ρ α (0 < α ⩾ 1; p = 0, 1), then
holds uniformly on D, where λ 0 is a positive root of the Bessel function J 0(x) It should be noted that either
or
implies that f(x, y) ≡ const. Now we consider the following trigonometrical polynomial
Then we have Theorem 3 If f(x, y) ∈ C p(D), then uniformly on D,
Theorems 1 and 2 include the results of Chandrasekharan and Minakshisundarm, and Theorem 3 is a generalization of a theorem of Zygmund, which can be extended to the multiple case as follows Theorem 3′ Let f(x 1, ..., x n) ≡ f(P) ∈ C p and let
where
and
being the Fourier coefficients of f(P). Then
holds uniformly. __________ Translated from Acta Scientiarum Naturalium Universitatis Pekinensis, 1956, (4): 411–428 by PENG Lizhong.  相似文献   

15.
Let (z ∈ ℝ). Further let λ denote a large real parameter. We show that for arbitrary real numbersk and α withk>=2.7013 and 0<α≦1,
  相似文献   

16.
Let p be a prime number, n be a positive integer, and ƒ(x) = axk + bx. We put
where e(t) = exp(2πit). This special exponential sum has been widely studied in connection with Waring’s problem. We write n in the form n = Qk + r, where 0 ≤ r ≤ k − 1 and Q ≥ 0. Let α = ord p(k), β = ord p(k − 1), and θ = ord p(b). We define
and J = [ζ]. Moreover, we denote V = min(Q, J). Improving the preceding result, we establish the theorem. Theorem. Let k ≥ 2 and n ≥ 2. If p > 2, then
. An example showing that this result is best possible is given. Bibliography: 15 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 63–75.  相似文献   

17.
Let {X, X n;n≥1} be a strictly stationary sequence of ρ-mixing random variables with mean zero and finite variance. Set . Suppose lim n→∞ and , where d=2, if −1<b<0 and d>2(b+1), if b≥0. It is proved that, for any b>−1,
, where Γ(•) is a Gamma function. Research supported by the National Natural Science Foundation of China (10071072).  相似文献   

18.
Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 < λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.  相似文献   

19.
Some existence and multiplicity of homoelinic orbits for second order Hamiltonian system x-a(t)x f(t,x)=0 are given by means of variational methods, where the function -1/2a(t)|s|^2∫^t0f(t,s)ds is asymptotically quadratic in s at infinity and subquadratic in s at zero, and the function a (t) mainly satisfies the growth condition limt→∞∫^t 1 t a(t)dt= ∞,VI∈R^1.A resonance case as well as a noncompact case is discussed too.  相似文献   

20.
Let{ψμ} be an orthonormal wavelet of L^2(R^d) and the support of a whole of its Fourier transform be Uμsupp{ψμ}=Пi=1^d[Ai, Di]-Пi=1^d(Bi, Ci), Ai≤Bi≤Ci≤Di. Under the weakest condition that each │ψμ│, is continuous for ω ∈ δ(Пi=1^d[Ai, Di]), a characterization of the above support of a whole is given.  相似文献   

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