共查询到20条相似文献,搜索用时 46 毫秒
1.
V. A. Yudin 《Proceedings of the Steklov Institute of Mathematics》2011,273(1):188-189
It is established that H. Bohr’s inequality \(\sum\nolimits_{k = 0}^\infty {\left| {{{f^{\left( k \right)} \left( 0 \right)} \mathord{\left/ {\vphantom {{f^{\left( k \right)} \left( 0 \right)} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {2^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} k!} \right)}}} \right| \leqslant \sqrt 2 \left\| f \right\|_\infty }\) is sharp on the class H ∞. 相似文献
2.
S. B. Gashkov 《Mathematical Notes》2016,100(5-6):666-676
For the linear positive Korovkin operator \(f\left( x \right) \to {t_n}\left( {f;x} \right) = \frac{1}{\pi }\int_{ - \pi }^\pi {f\left( {x + t} \right)E\left( t \right)dt} \), where E(x) is the Egervary–Szász polynomial and the corresponding interpolation mean \({t_{n,N}}\left( {f;x} \right) = \frac{1}{N}\sum\limits_{k = - N}^{N - 1} {{E_n}\left( {x - \frac{{\pi k}}{N}} \right)f\left( {\frac{{\pi k}}{N}} \right)} \), the Jackson-type inequalities \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \left( {1 + \pi } \right){\omega _f}\left( {\frac{1}{n}} \right),\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant 2{\omega _f}\left( {\frac{\pi }{{n + 1}}} \right)\), where ωf (x) denotes the modulus of continuity, are proved for N > n/2. For ωf (x) ≤ Mx, the inequality \(\left\| {{t_{n,N}}\left( {f;x} \right) - f\left( x \right)} \right\| \leqslant \frac{{\pi M}}{{n + 1}}\). is established. As a consequence, an elementary derivation of an asymptotically sharp estimate of the Kolmogorov width of a compact set of functions satisfying the Lipschitz condition is obtained. 相似文献
3.
Natalia Budarina 《Lithuanian Mathematical Journal》2011,51(4):461-471
In this paper, we show that if the volume sum \( \sum\nolimits_{h = 1}^\infty {{h^{n - 1}}{\Psi^t}(h)} \) converges for a function ψ (not necessarily monotonic), then the set of points \( \left( {x,{w_1}, \ldots, {w_{t - 1}}} \right) \in {\mathbb R} \times {{\mathbb Q}_{{p_1}}} \times \ldots \times {{\mathbb Q}_{{p_{t - 1}}}} \) that simultaneously satisfy the inequalities \( \left| {P(x)} \right| < \Psi (H) {\text{and}} {\left| {P\left( {{w_i}} \right)} \right|_{{p_i}}} < \Phi (H), 1 \leqslant i \leqslant t - 1 \), for infinitely many integer polynomials P has measure zero. 相似文献
4.
B. F. Ivanov 《Vestnik St. Petersburg University: Mathematics》2017,50(3):265-273
Let m ≥ 2, the numbers p 1,…, p m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ...\frac{1}{{{p_m}}} < 1\), and γ1 ∈ L p1(?1), …, γ m ∈ \({L^{{p_m}}}\)(?1). We prove that, if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both concepts have been introduced by the author for functions of spaces L p (?1), p ∈ (1, +∞]), we have the inequality \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\int\limits_a^b {\prod\limits_{k = 1}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]} d\tau } } \right| \leqslant C{\prod\limits_{k = 1}^m {\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|} _{L_{{a_k}}^{{p_k}}}}\left( {{\mathbb{R}^1}} \right)\), where the constant C > 0 is independent of functions \(\Delta {\gamma _k} \in L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right)\) and \(L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right) \subset {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)\), 1 ≤ k ≤ m are some specially constructed normed spaces. In addition, we give a boundedness condition for the integral of the product of functions over a subset of ?1. 相似文献
5.
In this paper, we consider the two-dimensional Hausdorff operators on the power weighted Hardy space H_(|x|α)~1(R~2) ( -1 ≤α≤0), defined by H_(Φ,A)f(x)=∫R~2Φ(u)f(A(u)x)du,where Φ∈L_loc~1(R~2),A(u) = (α_(ij)(u))_(i,j=1)~2 is a 2×2 matrix, and each α_(i,j) is a measurablefunction.We obtain that HΦ,A is bounded from H_(|x|~α)~1(R~2) ( -1≤α≤0) to itself, if∫R2|Φ(u)‖det A~(-1)(u)|‖A(u)‖~(-α)ln(1+‖A~(-1)(u)‖~2/|det A~(-1)(u)|)du∞.This result improves some known theorems, and in some sense it is sharp. 相似文献
6.
In this paper, we consider the logarithmically improved regularity criterion for the supercritical quasi-geostrophic equation in Besov space \(\dot B_{\infty ,\infty }^{ - r}\left( {{\mathbb{R}^2}} \right)\). The result shows that if θ is a weak solutions satisfies then θ is regular at t = T. In view of the embedding \({L^{\frac{2}{r}}} \subset M_{\frac{2}{r}}^p \subset \dot B_{\infty ,\infty }^{ - r}\) with \(2 \leqslant p < \frac{2}{r}\) and 0 ≤ r < 1, we see that our result extends the results due to [20] and [31].
相似文献
$$\int_0^T {\frac{{\left\| {\nabla \theta ( \cdot ,s)} \right\|_{\dot B_{\infty ,\infty }^{ - r} }^{\tfrac{\alpha }{{\alpha - r}}} }}{{1 + \ln \left( {e + \left\| {\nabla ^ \bot \theta ( \cdot ,s)} \right\|_{L^{\tfrac{2}{r}} } } \right)!}}ds < \infty for some 0 < r < \alpha and 0 < \alpha < 1,}$$
7.
8.
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
9.
In this paper, we study the global scattering result of the solution for the generalized Davey–Stewartson system(i?_tu + Δu = |u|~2u + uv_(x1),(t, x) ∈ R × R~3,-Δv =(|u|~2)_(x1).)The main difficulties are the failure of the interaction Morawetz estimate and the asymmetrical structure of nonlinearity(in particular, the nonlinearity is non-local). To compensate, we utilize the strategy derived from concentration-compactness idea, which was first introduced by Kenig and Merle [Invent.Math., 166, 645–675(2006)]. 相似文献
10.
Timo S. Hänninen 《Israel Journal of Mathematics》2017,219(1):71-114
We study the operator-valued positive dyadic operator where the coefficients {λ Q : C → D} Q∈D are positive operators from a Banach lattice C to a Banach lattice D. We assume that the Banach lattices C and D* each have the Hardy–Littlewood property. An example of a Banach lattice with the Hardy–Littlewood property is a Lebesgue space.
$${T_\lambda }\left( {f\sigma } \right): = \sum\limits_{Q \in D} {{\lambda _Q}} \int_Q {fd\sigma 1Q}, $$
In the two-weight case, we prove that the L C p (σ) → L D q (ω) boundedness of the operator T λ( · σ) is characterized by the direct and the dual L ∞ testing conditions: , .
Here L C p (σ) and L D q (ω) denote the Lebesgue–Bochner spaces associated with exponents 1 < p ≤ q < ∞, and locally finite Borel measures σ and ω.$$\left\| {{1_Q}{T_\lambda }} \right\|{\left( {{1_Q}f\sigma } \right)||_{L_D^q\left( \omega \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\sigma } \right)}}\sigma {\left( Q \right)^{1/p}}$$
$${\left\| {{1_Q}{T_\lambda }*\left( {{1_{Qg\omega }}} \right)} \right\|_{L_{C*}^{p'}\left( \sigma \right)}} \lesssim {\left\| g \right\|_{L_{D*}^\infty \left( {Q,\omega } \right)}}\omega {\left( Q \right)^{1/q'}}$$
In the unweighted case, we show that the L C p (μ) → L D p (μ) boundedness of the operator T λ( · μ) is equivalent to the end-point direct L ∞ testing condition: .
This condition is manifestly independent of the exponent p. By specializing this to particular cases, we recover some earlier results in a unified way. 相似文献
$${\left\| {{1_Q}{T_\lambda }\left( {{1_Q}f\mu } \right)} \right\|_{L_D^1\left( \mu \right)}} \lesssim {\left\| f \right\|_{L_C^\infty \left( {Q,\mu } \right)}}\left( {Q,\mu } \right)\mu \left( Q \right)$$
11.
Tadej Kotnik 《Advances in Computational Mathematics》2008,29(1):55-70
The paper describes a systematic computational study of the prime counting function π(x) and three of its analytic approximations: the logarithmic integral \({\text{li}}{\left( x \right)}: = {\int_0^x {\frac{{dt}}{{\log \,t}}} }\), \({\text{li}}{\left( x \right)} - \frac{1}{2}{\text{li}}{\left( {{\sqrt x }} \right)}\), and \(R{\left( x \right)}: = {\sum\nolimits_{k = 1}^\infty {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} \mathord{\left/ {\vphantom {{\mu {\left( k \right)}{\text{li}}{\left( {x^{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}} } \right)}} k}} \right. \kern-\nulldelimiterspace} k} }\), where μ is the Möbius function. The results show that π(x)x) for 2≤x≤1014, and also seem to support several conjectures on the maximal and average errors of the three approximations, most importantly \({\left| {\pi {\left( x \right)} - {\text{li}}{\left( x \right)}} \right|} < x^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}\) and \( - \frac{2}{5}x^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} < {\int_2^x {{\left( {\pi {\left( u \right)} - {\text{li}}{\left( u \right)}} \right)}du < 0} }\) for all x>2. The paper concludes with a short discussion of prospects for further computational progress. 相似文献
12.
This paper is a continuation of [3]. Suppose f∈Hp(T), 0σ r σ f,σ=1/p?1. When p=1, it is just the partial Fourier sums Skf. In this paper we establish the sharp estimations on the degree of approximation: $$\left\{ { - \frac{1}{{logR}}\int\limits_1^R {\left\| {\sigma _r^\delta f - f} \right\|_{H^p (T)}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqq C{\mathbf{ }}{}_p\omega \left( {f,{\mathbf{ }}( - \frac{1}{{logR}})^{1/p} } \right)_{H^p (T)} ,0< p< 1,$$ and \(\frac{1}{{\log L}}\sum\limits_{k - 1}^L {\frac{{\left\| {S_k f - f} \right\|_H 1_{(T)} }}{k} \leqq Cp\omega (f; - \frac{1}{{\log L}})_H 1_{(T)} } \) Where $$\omega (f,{\mathbf{ }}h)_{H^p (T)} \begin{array}{*{20}c} { = Sup} \\ {0 \leqq \left| u \right| \leqq h} \\ \end{array} \left\| {f( \cdot + u) - f( \cdot )} \right\|_{H^p (T).} $$ . 相似文献
13.
Hermann König 《Israel Journal of Mathematics》2014,203(1):23-57
Let S j : (Ω, P) → S 1 ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S 1. We determine the best constants a p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p 0 }~ 0.4756. For p ≥ 1 the best constants a p have been known for some time. The result implies for a norm 1 sequence x ∈ ? n , ‖x‖2 = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse. 相似文献
14.
In this paper, we consider the ground-states of the following M-coupled system: where \(p_{ij} + q_{ij} = 2*: = \frac{{2N}}{{N - 2}}(N \geqslant 3)\). We prove the existence of ground-states to the M-coupled system. At the same time, we not only give out the characterization of the ground-states, but also study the number of the ground-states, containing the positive ground-states and the semi-trivial ground-states, which may be the first result studying the number of not only positive ground-states but also semi-trivial ground-states.
相似文献
$$\left\{ {\begin{array}{*{20}{c}}{ - \Delta {u_i} = \sum\limits_{j = 1}^M {{k_{ij}}\frac{{2{q_{ij}}}}{{2*}}{{\left| {{u_j}} \right|}^{{p_{ij}}}}{{\left| {{u_i}} \right|}^{{q_{ij}} - {2_{{u_i}}}}},x \in {\mathbb{R}^N},} } \\{{u_i} \in {D^{1,2}}\left( {{\mathbb{R}^N}} \right),i = 1,2, \ldots ,M,}\end{array}} \right.$$
15.
A remark on the existence of entire large and bounded solutions to a (<Emphasis Type="Italic">k</Emphasis><Subscript>1</Subscript>, <Emphasis Type="Italic">k</Emphasis><Subscript>2</Subscript>)-Hessian system with gradient term 下载免费PDF全文
Dragos Patru Covei 《数学学报(英文版)》2017,33(6):761-774
In this paper, we study the existence of positive entire large and bounded radial positive solutions for the following nonlinear system Here \({S_{{k_i}}}\left( {\lambda \left( {{D^2}{u_i}} \right)} \right)\) is the k i -Hessian operator, a 1, p 1, f 1, a 2, p 2 and f 2 are continuous functions.
相似文献
$$\left\{ {\begin{array}{*{20}c}{S_{k_1 } \left( {\lambda \left( {D^2 u_1 } \right)} \right) + a_1 \left( {\left| x \right|} \right)\left| {\nabla u_1 } \right|^{k_1 } = p_1 \left( {\left| x \right|} \right)f_1 \left( {u_2 } \right)} & {for x \in \mathbb{R}^N ,} \\{S_{k_2 } \left( {\lambda \left( {D^2 u_2 } \right)} \right) + a_2 \left( {\left| x \right|} \right)\left| {\nabla u_2 } \right|^{k_2 } = p_2 \left( {\left| x \right|} \right)f_2 \left( {u_1 } \right)} & {for x \in \mathbb{R}^N .} \\\end{array} } \right.$$
16.
Hengcai Tang 《Lithuanian Mathematical Journal》2011,51(1):75-81
In this paper, we improve the previous results of the authors [G. Lü and H. Tang, On some results of Hua in short intervals, Lith. Math. J., 50(1):54–70, 2010] by proving that each sufficiently large integer N satisfying some congruence conditions can be written aswhere U = N 1/2?η+ε with \( \eta = \frac{1}{{2k\left( {{K^2} + 1} \right)}} \) and K = 2k ?1, k ? 2.
相似文献
$ \left\{ {\begin{array}{*{20}{c}} {N = p_1^2 + p_2^2 + p_3^2 + p_4^2 + {p^k},} \hfill \\ {\left| {{p_j} - \sqrt {{\frac{N}{5}}} } \right| \leqslant U,\quad \left| {p - {{\left( {\frac{N}{5}} \right)}^{\frac{1}{k}}}} \right| \leqslant U\,{N^{ - \frac{1}{2} + \frac{1}{k}}},\quad j = 1,\,2,\,\,3,\,4,} \hfill \\ \end{array} } \right. $
17.
Following an idea of Lin, we prove that if A and B are two positive operators such that 0 mI ≤ A ≤m'I≤ M'I ≤ B ≤ MI, then Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2Φ~2(A≠B) and Φ~2(A+B/2)≤K~2(h)/(1+(logM'/m'/g))~2(Φ(A)≠Φ(B))~2 where K(h)=(h+1)~2/4 and h = M/m and Φ is a positive unital linear map. 相似文献
18.
D. T. Luyen 《Mathematical Notes》2017,101(5-6):815-823
In this paper, we study the existence of multiple solutions for the boundary-value problem where Ω is a bounded domain with smooth boundary in R N (N ≥ 2) and Δ γ is the subelliptic operator of the type We use the three critical point theorem.
相似文献
$${\Delta _\gamma }u + f\left( {x,u} \right) = 0in\Omega ,u = 0on\partial \Omega ,$$
$${\Delta _\gamma }u = \sum\limits_{j = 1}^N {{\partial _{{x_j}}}\left( {\gamma _j^2{\partial _{{x_j}}}u} \right)} ,{\partial _{{x_j}}}u = \frac{{\partial u}}{{\partial {x_j}}},\gamma = \left( {{\gamma _1},{\gamma _2}, \ldots ,{\gamma _N}} \right).$$
19.
Mei Yue JIANG 《数学学报(英文版)》2005,21(5):1219-1228
In this paper, we give a Landesman-Lazer type theorem for periodic solutions of the asymmetric 1-dimensional p-Laplacian equation -(|x'|^p-2x')'=λ|x|^p-2x++μ|x|^p-2x-+f(t,x)with periodic boundary value. 相似文献
20.
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.
- Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
- Let P + and P ? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
- We establish the sharp versions of the estimates above in the nonperiodic case.