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1.
Let
be a nondecreasing sequence of positive numbers and let l
1,α be the space of real sequences
for which
. We associate every sequence ξ from l
1,α with a sequence
, where ϕ(·) is a permutation of the natural series such that
, j ∈ ℕ. If p is a bounded seminorm on l
1,α and
, then
Using this equality, we obtain several known statements.
__________
Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 1002–1006, July, 2005. 相似文献
2.
Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters 下载免费PDF全文
Jiao Chen 《数学学报(英文版)》2017,33(8):1083-1106
The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces. 相似文献
3.
Singular Integrals and Commutators in Generalized Morrey Spaces 总被引:1,自引:0,他引:1
Lubomiea Softova 《数学学报(英文版)》2006,22(3):757-766
4.
Xiu Gui LIU 《数学学报(英文版)》2007,23(6):1025-1032
Abstract Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by (b0hn-h1bn-1)∈ ExtA^3,(p^n+p)q(Zp,Zp) in the Adams spectral sequence. At the same time, he proved that i.(hlhn) ∈ExtA^2,(p^n+P)q(H^*M, Zp) is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element ξn∈π(p^n+p)q-2M. In this paper, with Lin's results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj′j^-γsi^-i′ξn in the stable homotopy groups of spheres. The new one is of degree p^nq + sp^2q + spq + (s - 2)q + s - 6 and is represented up to a nonzero scalar by hlhnγ-s in the E2^s+2,*-term of the Adams spectral sequence, where p ≥ 7, q = 2(p - 1), n ≥ 4 and 3 ≤ s 〈 p. 相似文献
5.
Hu Ke 《数学年刊B辑(英文版)》1980,1(34):421-427
Let \[f(z) = z + \sum\limits_{n = 1}^\infty {{a_n}{z^n} \in S} {\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \log \frac{{f(z) - f(\xi )}}{{z - \xi }} - \frac{{z\xi }}{{f(z)f(\xi )}} = \sum\limits_{m,n = 1}^\infty {{d_{m,n}}{z^m}{\xi ^n},} \], we denote \[{f_v} = f({z_v})\] , \[\begin{array}{l}
{\varphi _\varepsilon }({z_u}{z_v}) = {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\frac{1}{{(1 - {z_u}{{\bar z}_v})}},\g_m^\varepsilon (z) = - {F_m}(\frac{1}{{f(z)}}) + \frac{1}{{{z^m}}} + \varepsilon {{\bar z}^m},
\end{array}\], where \({F_m}(t)\) is a Faber polynomial of degree m.
Theorem 1. If \[f(z) \in S{\kern 1pt} {\kern 1pt} {\kern 1pt} and{\kern 1pt} {\kern 1pt} {\kern 1pt} \sum\limits_{u,v = 1}^N {{A_{u,v}}{x_u}{{\bar x}_v} \ge 0} \] and then \[\begin{array}{l}
\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}} \right|^\varepsilon }\exp \{ \alpha {F_l}({z_u},{z_v})\} \ \le \sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} \varphi _\varepsilon ^\alpha ({z_u}{z_v})l = 1,2,3,
\end{array}\], where \[\begin{array}{l}
{F_1}({z_u},{z_v}) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} g_n^\varepsilon ({z_u})\bar g_n^\varepsilon ({z_v}),\{F_2}({z_u},{z_v}) = \frac{1}{{1 + {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}),\{F_3}({z_u},{z_v}) = \frac{1}{{1 - {\varepsilon _n}R{d_{n,n}}}}Rg_n^\varepsilon ({z_u})Rg_n^\varepsilon ({z_v}).
\end{array}\] The \[F({z_u},{z_v}) = \frac{1}{2}{g_1}({z_u}){{\bar g}_2}({z_v})\] is due to Kungsun.
Theorem 2. If \(f(z) \in S\) ,then \[P(z) + \left| {\sum\limits_{u,v = 1}^N {{A_{u,v}}{\lambda _u}{{\bar \lambda }_v}} {{\left| {\frac{{{f_u} - {f_v}}}{{{z_u} - {z_v}}}\frac{{{z_u}{z_v}}}{{{f_u}{f_v}}}} \right|}^\varepsilon }} \right| \le \sum\limits_{u,v = 1}^N {{\lambda _u}{{\bar \lambda }_v}} \frac{1}{{1 - {z_u}{{\bar z}_v}}}\], where \[\begin{array}{l}
P(z) = \frac{1}{2}\sum\limits_{n = 1}^\infty {\frac{1}{n}} {G_n}(z),\{G_n}(z) = {\left| {\left| {\sum\limits_{n = 1}^N {{\beta _u}({F_n}(\frac{1}{{f({z_u})}}) - \frac{1}{{z_u^n}})} } \right| - \left| {\sum\limits_{n = 1}^N {{\beta _u}z_u^n} } \right|} \right|^2},
\end{array}\], \(P(z) \equiv 0\) is due to Xia Daoxing. 相似文献
6.
S. Norvidas 《Lithuanian Mathematical Journal》2009,49(2):185-189
For a compact set K in ℝ
n
, let B
2
K
be the set of all functions f ∈ L
2(ℝ2) bandlimited to K, i.e., such that the Fourier transform f̂ of f is supported by K. We investigate the question of approximation of f ∈ B
2
K
by finite exponential sums
in the space , as τ → ∞. 相似文献
7.
8.
S. V. Kislyakov 《Journal of Mathematical Sciences》2009,156(5):824-833
Let 1 < r < 2 and let b is a weight on ℝ such that satisfies the Muckenhoupt condition Ar′/2 (r′ is the exponent conjugate to r). If fj are functions whose Fourier transforms are supported on mutually disjoint intervals, then
for 0 < p ≤ r. Bibliography: 9 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 180–198. 相似文献
9.
Jay Taylor 《Israel Journal of Mathematics》2017,217(1):435-475
In this paper we establish the following estimate: where ω ≥ 0, 0 < ε < 1 and Φ(t) = t(1 + log+(t)). This inequality relies upon the following sharp L p estimate: where 1 < p < ∞, ω ≥ 0 and 0 < δ < 1. As a consequence we recover the following estimate essentially contained in [18]: We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.
相似文献
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant \frac{{{c_T}}}{{{\varepsilon ^2}}}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right){M_{L{{\left( {\log L} \right)}^{1 + \varepsilon }}}}} \omega \left( x \right)dx$$
$${\left\| {\left[ {b,T} \right]f} \right\|_{{L^p}\left( \omega \right)}} \leqslant {c_T}{\left( {p'} \right)^2}{p^2}{\left( {\frac{{p - 1}}{\delta }} \right)^{\frac{1}{{p'}}}}{\left\| b \right\|_{BMO}}{\left\| f \right\|_{{L^p}\left( {{M_{L{{\left( {{{\log }_L}} \right)}^{2p - 1 + {\delta ^\omega }}}}}} \right)}}$$
$$\omega \left( {\left\{ {x \in {\mathbb{R}^n}:\left| {\left[ {b,T} \right]f\left( x \right)} \right| > \lambda } \right\}} \right) \leqslant {c_T}{\left[ \omega \right]_{{A_\infty }}}{\left( {1 + {{\log }^ + }{{\left[ \omega \right]}_{{A_\infty }}}} \right)^2}\int_{{\mathbb{R}^n}} {\Phi \left( {{{\left\| b \right\|}_{BMO}}\frac{{\left| {f\left( x \right)} \right|}}{\lambda }} \right)M} \omega \left( x \right)dx.$$
10.
Multilinear Singular Integrals with Rough Kernel 总被引:9,自引:0,他引:9
ShanZhenLU HuoXiongWU PuZHANG 《数学学报(英文版)》2003,19(1):51-62
For a class of multilinear singular integral operators T
A
,
where R
m
(A; x, y) denotes the m-th Taylor series remainder of A at x expanded about y, A has derivatives of order m − 1 in
is homogeneous of degree zero, the authors prove that T
A
is bounded from L
p
(ℝ
n
) to
and from L
1(ℝ
n
) to L
n/(n−β),∞(ℝ
n
) with the bound
And if Ω has vanishing moments of order m − 1 and satisfies some kinds of Dini regularity otherwise, then T
A
is also bounded from L
p
(ℝ
n
) to
with the bound
Supported by the National 973 Project (G1990751) and SEDF of China (20010027002) 相似文献
11.
A. A. Mogul’skiĭ 《Siberian Advances in Mathematics》2010,20(3):191-200
Let X,X(1),X(2),... be independent identically distributed random variables with mean zero and a finite variance. Put S(n) = X(1) + ... + X(n), n = 1, 2,..., and define the Markov stopping time η
y
= inf {n ≥ 1: S(n) ≥ y} of the first crossing a level y ≥ 0 by the random walk S(n), n = 1, 2,.... In the case $
\mathbb{E}
$
\mathbb{E}
|X|3 < ∞, the following relation was obtained in [8]: $
\mathbb{P}\left( {\eta _0 = n} \right) = \frac{1}
{{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right)
$
\mathbb{P}\left( {\eta _0 = n} \right) = \frac{1}
{{n\sqrt n }}\left( {R + \nu _n + o\left( 1 \right)} \right)
as n → ∞, where the constant R and the bounded sequence ν
n
were calculated in an explicit form. Moreover, there were obtained necessary and sufficient conditions for the limit existence
$
H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
$
H\left( y \right): = \mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
for every fixed y ≥ 0, and there was found a representation for H(y). The present paper was motivated by the following reason. In [8], the authors unfortunately did not cite papers [1, 5] where
the above-mentioned relations were obtained under weaker restrictions. Namely, it was proved in [5] the existence of the limit
$
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
$
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _y = n} \right)
for every fixed y ≥ 0 under the condition
$
\mathbb{E}
$
\mathbb{E}
X
2 < ∞ only; In [1], an explicit form of the limit $
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right)
$
\mathop {\lim }\limits_{n \to \infty } n^{{3 \mathord{\left/
{\vphantom {3 2}} \right.
\kern-\nulldelimiterspace} 2}} \mathbb{P}\left( {\eta _0 = n} \right)
was found under the same condition
$
\mathbb{E}
$
\mathbb{E}
X
2 < ∞ in the case when the summand X has an arithmetic distribution. In the present paper, we prove that the main assertion in [5] fails and we correct the original proof. It worth noting that
this corrected version was formulated in [8] as a conjecture. 相似文献
12.
ON A MULTILINEAR OSCILLATORY SINGULAR INTEGRAL OPERATOR (I) 总被引:2,自引:0,他引:2
ONAMULTILINEAROSCILLATORYSINGULARINTEGRALOPERATOR(I)CHENWENGUHUGUOENLUSHANZHENManuscriptreceivedOctober18,1994.RevisedDece... 相似文献
13.
Axel Grünrock 《Central European Journal of Mathematics》2010,8(3):500-536
The Cauchy problem for the higher order equations in the mKdV hierarchy is investigated with data in the spaces $
\hat H_s^r \left( \mathbb{R} \right)
$
\hat H_s^r \left( \mathbb{R} \right)
defined by the norm
$
\left\| {v_0 } \right\|_{\hat H_s^r \left( \mathbb{R} \right)} : = \left\| {\left\langle \xi \right\rangle ^s \widehat{v_0 }} \right\|_{L_\xi ^{r'} } , \left\langle \xi \right\rangle = \left( {1 + \xi ^2 } \right)^{\frac{1}
{2}} , \frac{1}
{r} + \frac{1}
{{r'}} = 1
$
\left\| {v_0 } \right\|_{\hat H_s^r \left( \mathbb{R} \right)} : = \left\| {\left\langle \xi \right\rangle ^s \widehat{v_0 }} \right\|_{L_\xi ^{r'} } , \left\langle \xi \right\rangle = \left( {1 + \xi ^2 } \right)^{\frac{1}
{2}} , \frac{1}
{r} + \frac{1}
{{r'}} = 1
相似文献
14.
We study cyclicity of operators on a separable Banach space which admit a bicyclic vector such that the norms of its images
under the iterates of the operator satisfy certain growth conditions. A simple consequence of our main result is that a bicyclic
unitary operator on a Banach space with separable dual is cyclic. Our results also imply that if is the shift operator acting on the weighted space of sequences , if the weight ω satisfies some regularity conditions and ω(n) = 1 for nonnegative n, then S is cyclic if . On the other hand one can see that S is not cyclic if the series diverges. We show that the question of Herrero whether either S or S* is cyclic on admits a positive answer when the series is convergent. We also prove completeness results for translates in certain Banach spaces of functions on . 相似文献
15.
Bao-huai Sheng 《应用数学学报(英文版)》2005,21(4):529-536
Let S^1-1,q≥2,be the surface of the unit sphere in the Euclidean space R^1,f(x)∈L^p(S^q-1),f(x)≥0,f absohutely unegual to 0,1≤p≤+∞,Then,it is proved in the present paper that there is a spherical harmonics PN(x) of order≤N and a constant C〉0 such that where ω(f,δ)L^p=sup 0〈t≤δ‖St(f)-f‖L^p is a kind of moduli of continuity and ^‖f-1/PN‖L^p≤Cω(f,N^-1)L^p,St(f,μ)=1/|S^q-2|Sin^2λt ∫-μμ’=t f(μ')dμ' is a translation operator. 相似文献
16.
Qi Keng LU Ke WU 《数学学报(英文版)》2007,23(4):577-598
For an integer m ≥ 4, we define a set of 2[m/2] × 2[m/2] matrices γj (m), (j = 0, 1,..., m - 1) which satisfy γj (m)γk (m) +γk (m)γj (m) = 2ηjk (m)I[m/2], where (ηjk (m)) 0≤j,k≤m-1 is a diagonal matrix, the first diagonal element of which is 1 and the others are -1, I[m/2] is a 2[m/1] × 2[m/2] identity matrix with [m/2] being the integer part of m/2. For m = 4 and 5, the representation (m) of the Lorentz Spin group is known. For m≥ 6, we prove that (i) when m = 2n, (n ≥ 3), (m) is the group generated by the set of matrices {T|T=1/√ξ((I+k) 0 + 0 I-K) ( U 0 0 U), (ii) when m = 2n + 1 (n≥ 3), (m) is generated by the set of matrices {T|T=1/√ξ(I -k^- k I)U,U∈ (m-1),ξ=1-m-2 ∑k,j=0 ηkja^k a^j〉0, K=i[m-3 ∑j=0 a^j γj(m-2)+a^(m-2) In],K^-=i[m-3∑j=0 a^j γj(m-2)-a^(m-2) In]} 相似文献
17.
Suppose f∈Hp(Tn), 0 r δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1. 相似文献
18.
De-xiang Ma Wei-gao Ge Xue-gang Chen 《应用数学学报(英文版)》2005,21(4):661-670
In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{( φp(u'))'+q(t)f(t,u,u')=0,0〈t〈1,u(0)=∑i=1^nαiu(ξi),u'(1)=∑i=1^nβiu'(ξi),whereφp(s)=|s|^p-2s,p≥2;ξi∈(0,1)(i=1,2,…,n),0≤αi,βi〈1(i=1,2,…n),0≤∑i=1^nαi,∑i=1^nβi〈1,and q(t) may be singular at t=0,1,f(t,u,u')may be singular at u'=0 相似文献
19.
Adam Osękowski 《Israel Journal of Mathematics》2012,192(1):429-448
The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.
20.
Simon Raulot 《Mathematische Zeitschrift》2009,261(2):321-349
Let M be an n-dimensional connected compact manifold with non-empty boundary equipped with a Riemannian metric g, a spin structure σ and a chirality operator Γ. We define and study some properties of a spin conformal invariant given by:
|