共查询到20条相似文献,搜索用时 937 毫秒
1.
We study the rough bilinear fractional integral
$
\tilde B_{\Omega ,\alpha } (f,g)(x) = \int_{\mathbb{R}^n } {f(x + y)g(x - y)\frac{{\Omega (x,y')}}
{{\left| y \right|^{n - \alpha } }}dy} ,
$
\tilde B_{\Omega ,\alpha } (f,g)(x) = \int_{\mathbb{R}^n } {f(x + y)g(x - y)\frac{{\Omega (x,y')}}
{{\left| y \right|^{n - \alpha } }}dy} ,
相似文献
2.
In this paper,the parameterized Marcinkiewicz integrals with variable kernels defined by μΩ^ρ(f)(x)=(∫0^∞│∫│1-y│≤t Ω(x,x-y)/│x-y│^n-p f(y)dy│^2dt/t1+2p)^1/2 are investigated.It is proved that if Ω∈ L∞(R^n) × L^r(S^n-1)(r〉(n-n1p'/n) is an odd function in the second variable y,then the operator μΩ^ρ is bounded from L^p(R^n) to L^p(R^n) for 1 〈 p ≤ max{(n+1)/2,2}.It is also proved that,if Ω satisfies the L^1-Dini condition,then μΩ^ρ is of type(p,p) for 1 〈 p ≤ 2,of the weak type(1,1) and bounded from H1 to L1. 相似文献
3.
Guan-zhen Zhou 《高校应用数学学报(英文版)》2010,25(1):43-47
The inequality plays an important role in Fourier analysis and approximation theory. It has recently been generalized by Telyakovskii and Leindler. This paper further generalizes and improves their results by introducing a new class of sequences called γ-piecewise bounded variation sequence (γ-PBVS). 相似文献
4.
Local and Global Existence of Solutions to Initial Value Problems of Nonlinear Kaup-Kupershmidt Equations 总被引:6,自引:0,他引:6
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(4):881-892
This paper is devoted to studying the initial value problems of the nonlinear Kaup Kupershmidt equations δu/δt + α1 uδ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x,t)∈ E R^2, and δu/δt + α2 δu/δx δ^2u/δx^2 + βδ^3u/δx^3 + γδ^5u/δx^5 = 0, (x, t) ∈R^2. Several important Strichartz type estimates for the fundamental solution of the corresponding linear problem are established. Then we apply such estimates to prove the local and global existence of solutions for the initial value problems of the nonlinear Kaup- Kupershmidt equations. The results show that a local solution exists if the initial function u0(x) ∈ H^s(R), and s ≥ 5/4 for the first equation and s≥301/108 for the second equation. 相似文献
5.
Wei HUANG Ye JIANG Li Xin ZHANG 《数学学报(英文版)》2005,21(5):1057-1070
Let {X,Xn;n ≥ 1} be a strictly stationary sequence of ρ-mixing random variables with mean zeros and finite variances. Set Sn =∑k=1^n Xk, Mn=maxk≤n|Sk|,n≥1.Suppose limn→∞ESn^2/n=:σ^2〉0 and ∑n^∞=1 ρ^2/d(2^n)〈∞,where d=2 if 1≤r〈2 and d〉r if r≥2.We prove that if E|X|^r 〈∞,for 1≤p〈2 and r〉p,then limε→0ε^2(r-p)/2-p ∑∞n=1 n^r/p-2 P{Mn≥εn^1/p}=2p/r-p ∑∞k=1(-1)^k/(2k+1)^2(r-p)/(2-p)E|Z|^2(r-p)/2-p,where Z has a normal distribution with mean 0 and variance σ^2. 相似文献
6.
ShuXing Chen 《中国科学A辑(英文版)》2009,52(9):1829-1843
In this paper we discuss the fundamental solution of the Keldysh type operator $
L_\alpha u \triangleq \frac{{\partial ^2 u}}
{{\partial x^2 }} + y\frac{{\partial ^2 u}}
{{\partial y^2 }} + \alpha \frac{{\partial u}}
{{\partial y}}
$
L_\alpha u \triangleq \frac{{\partial ^2 u}}
{{\partial x^2 }} + y\frac{{\partial ^2 u}}
{{\partial y^2 }} + \alpha \frac{{\partial u}}
{{\partial y}}
, which is a basic mixed type operator different from the Tricomi operator. The fundamental solution of the Keldysh type operator
with $
\alpha > - \frac{1}
{2}
$
\alpha > - \frac{1}
{2}
is obtained. It is shown that the fundamental solution for such an operator generally has stronger singularity than that
for the Tricomi operator. Particularly, the fundamental solution of the Keldysh type operator with $
\alpha < \frac{1}
{2}
$
\alpha < \frac{1}
{2}
has to be defined by using the finite part of divergent integrals in the theory of distributions. 相似文献
7.
Xing-youZhang YongHuang 《应用数学学报(英文版)》2005,21(1):93-100
In this paper we study the homogenization of degenerate quasilinear parabolic equations: where a(t, y, a, λ) is periodic in (t, y). 相似文献
8.
Let Ω ? 0 be an open bounded domain in R N (N ≥ 3) and $2^* (s) = \tfrac{{2(N - s)}} {{N - 2}}$ , 0 < s < 2. We consider the following elliptic system of two equations in H 0 1 (Ω) × H 0 1 (Ω): $$- \Delta u - t\frac{u} {{\left| x \right|^2 }} = \frac{{2\alpha }} {{\alpha + \beta }}\frac{{\left| u \right|^{\alpha - 2} u\left| v \right|^\beta }} {{\left| x \right|^s }} + \lambda u, - \Delta v - t\frac{v} {{\left| x \right|^2 }} = \frac{{2\beta }} {{\alpha + \beta }}\frac{{\left| u \right|^\alpha \left| v \right|^{\beta - 2} v}} {{\left| x \right|^s }} + \mu v,$$ where λ, µ > 0 and α, β > 1 satisfy α + β = 2*(s). Using the Moser iteration, we prove the asymptotic behavior of solutions at the origin. In addition, by exploiting the Mountain-Pass theorem, we establish the existence of solutions. 相似文献
9.
Local and Global Existence of Solutions to
Initial Value Problems of Modified Nonlinear
Kawahara Equations 总被引:3,自引:0,他引:3
Shuang Ping TAO Shang Bin CUI 《数学学报(英文版)》2005,21(5):1035-1044
This paper is devoted to studying the initial value problem of the modified nonlinear Kawahara equation the first partial dervative of u to t ,the second the third +α the second partial dervative of u to x ,the second the third +β the third partial dervative of u to x ,the second the thire +γ the fifth partial dervative of u to x = 0,(x,t)∈R^2.We first establish several Strichartz type estimates for the fundamental solution of the corresponding linear problem. Then we apply such estimates to prove local and global existence of solutions for the initial value problem of the modified nonlinear Karahara equation. The results show that a local solution exists if the initial function uo(x) ∈ H^s(R) with s ≥ 1/4, and a global solution exists if s ≥ 2. 相似文献
10.
In the “lost notebook”, Ramanujan recorded infinite product expansions for
|