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1.
The commutators of oscillatory singular integral operators with homogeneous kernel
are studied, where Ω is homogeneous of degree zero, has mean value zero on the unit sphere. It is proved that Ω∈L (logL)K+1(Sn-1) is a sufficient condition under which the k-th order commutator is bounded on L2(Rn).
Supported by NSF of China (1997 1010, 1970 1039) 相似文献
2.
GuoEnHU BoLinMA 《数学学报(英文版)》2004,20(5):785-792
The commutators of singular integral operators with homogeneous kernel Ω(x)/|x|^n are studied, where Ω is homogeneous of degree zero, and has mean value zero on the unit sphere. It is proved that Ω∈ L(logL)^(k 1)(S^(n-1)) is a sufficient condition such that the k-th order commutator is bounded on L^2(R^n). 相似文献
3.
Riccardo De Arcangelis 《Annali dell'Universita di Ferrara》1989,35(1):135-145
Summary Letf: (x, z)∈R
n×Rn→f(x, z)∈[0, +∞] be measurable inx and convex inz.
It is proved, by an example, that even iff verifies a condition as|z|
p≤f(x, z)≤Λ(a(x)+|z|q) with 1<p<q,a∈L
loc
s
(R
n),s>1, the functional
that isL
1(Ω)-lower semicontinuous onW
1,1(Ω), does not agree onW
1,1(Ω) with its relaxed functional in the topologyL
1(Ω) given by inf
Riassunto Siaf: (x, z)∈R n×Rn→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| p≤f(x, z)≤Λ(a(x)+|z|q) con 1<p<q,a∈L loc s (R n),s>1, il funzionale , che èL 1(Ω)-semicontinuo inferiormente suW 1,1(Ω), non coincide suW 1,1(Ω) con il suo funzionale rilassato nella topologiaL 1(Ω) definito da inf相似文献
4.
GuoEnHU 《数学学报(英文版)》2003,19(2):397-404
The L^2(R^n) boundedness for the multilinear singular integral operators defined by TAf(x)=∫R^nΩ(x-y)/|x-y|^n 1(A(x)-A(y)-△↓A(y)(x-y))f(y)dy is considered,where Ω is homogeneous of degree zero,integrable on the unit sphere and has vanishing moment of order one,A has derivatives of order one in BMO(R^n) boundedness for the multilinear operator TA is given. 相似文献
5.
Jiao Yulan 《分析论及其应用》2004,20(4):373-382
Lp(Rn) boundedness is considered for the multilinear singular integral operator defined by TAf(x) = ∫Rn Ω(x - y)/|x - y|n 1 (A(x) - A(y) - (△)A(y)(x - y))f(y)dy,where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one. A has derivatives of order one in BMO(Rn). We give a smoothness condition which is fairly weaker than that Ω∈ Lipα(Sn-1) (0 <α≤ 1) and implies the Lp(Rn) (1 < p < oo) boundedness for the operator TA. Some endpoint estimates are also established. 相似文献
6.
Jiang Liya Chen Jiecheng 《高校应用数学学报(英文版)》2006,21(1):69-78
The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i|x|^aΩ(x)|x|^-n is studied,where a∈R,a≠0,1 and Ω∈L^1(S^n-1) is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω(x' )∈Llog+L(S^n-1 ), the Fp^a,q(R^n) boundedness of the above operator is obtained. Meanwhile ,when Ω(x) satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 (R^n). 相似文献
7.
Let Ω be a bounded domain with a smooth C
2 boundary in ℝn (n ≥ 3), 0 ∈
, and υ denote the unit outward normal to ∂Ω. In this paper, we are concerned with the following class of boundary value problems:
where 2* = 2n/(n − 2) is the limiting exponent for the embedding of H
1(Ω) into L
p
(Ω), 2 < p < 2*,
, η ≥ 0 and λ ∈ ℝ1 are parameters, and α(x) ∈ C(∂Ω), α(x) ≥ 0. Through a compactness analysis of the functional corresponding to the problem (*), we obtain the existence of positive
solutions for this problem under various assumptions on the parameters μ, λ and the fact that 0 ∈ Ω or 0 ∈ ∂Ω.
The research was supported by NSFC(10471052, 10571069, 10631030) and the Key Project of Chinese Ministry of Education(107081)
and NCET-07-0350. 相似文献
(*) |
8.
Mihai Mihăilescu 《Czechoslovak Mathematical Journal》2008,58(1):155-172
We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ
N
. Our attention is focused on two cases when , where m(x) = max{p
1(x), p
2(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized
Lebesgue-Sobolev spaces, combined with a ℤ2-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods. 相似文献
9.
Sergio Campanato 《Annali di Matematica Pura ed Applicata》1970,86(1):125-154
Riassunto Si considera un'equazione ellittica non variazionale
aaDau=f a coefficienti continui in un apertogW di ℝn e si dimostrano certe proprietà locali delle soluzioni forti dell'equazione medesima nell'ipotesi che f appartenga allo spazio
di Morrey L
2,λ(Ω) o allo spazio di John-Nirenberg ℰ(Ω). Analoga indagine è svolta supponendoΩ=ℝ
+
n
e supponendo che u verifichi sull'iperpiano xn=0 condizioni di Dirichlet omogenee.
Ricerca svolta nell'ambito dei Contratti di Ricerca del Comitato per la Matematica del C.N.R.
Entrata in Redazione il 17 febbraio 1970. 相似文献
10.
Alessandra Pagano 《Annali dell'Universita di Ferrara》1993,39(1):1-17
We consider a (possibly) vector-valued function u: Ω→R
N, Ω⊂R
n, minimizing the integral
, whereD
iu=∂u/∂x
i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D
1u,…,Dn−1u∈Lq, under suitable assumptions ona
i(x).
Sunto Consideriamo una funzione u: Ω→R N, Ω⊂R n che minimizzi l'integrale , doveD iu=∂u/∂xi, o un funzionale con un comportamento simile; sotto opportune ipotesi sua i(x), dimostriamo la seguente maggiore integrabilità perDu:D 1u,…,Dn−1uεLq.相似文献
11.
In a cylindric domain D = (0, ∞) × Ω where Ω ⊂ ℝ
n+1 is an unbounded domain, the first mixed problem for a high-order parabolic equation
is considered. The boundary values are homogeneous and the initial value is a finite function. In terms of the new geometrical
characteristics of the domain, the upper estimate of L
2-norm ∥u(t)∥ of the solution to the problem is established. In particular, in domains {(x, y) ∈ ℝ
n+1 | x > 0, |y
1| < x
a
}, 0 < a < q/l, under the assumption that the upper and lower symbols of the operator L are separated from zero, this estimate takes the form
. This estimate is determined by minor terms of the equation. The sharpness of the estimate for a wide class of unbounded
domains is proved in the case k = l = m = 1.
__________
Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 4, pp. 113–132, 2006. 相似文献
12.
LiJunjie BianBaojun 《高校应用数学学报(英文版)》2000,15(3):273-280
The following regularity of weak solutions of a class of elliptic equations of the form are investigated. 相似文献
13.
O. V. Matveev 《Mathematical Notes》1997,62(3):339-349
Supposem, n ∈ℕ,m≡n (mod 2),K(x)=|x|
m
form odd,K(x)=|x|
m
In |x| form even (x∈ℝ
n
),P is the set of real polynomials inn variables of total degree ≤m/2, andx
1,...,x
N
∈ℝ
n
. We construct a function of the form
coinciding with a given functionf(x) at the pointsx
1,...,x
N
. Error estimates for the approximation of functionsf∈W
p
k
(Ω) and theirlth-order derivatives in the normsL
q
(Ωε) are obtained for this interpolation method, where Ω is a bounded domain in ℝ
n
, ε>0, and Ωε={x∈Ω:dist(x, ∂∈)>ε}.
Translated fromMatematicheskie Zametki, Vol. 62, No. 3, pp. 404–417, September, 1997.
Translated by N. K. Kulman 相似文献
14.
Let Δ3 be the set of functions three times continuously differentiable on [−1, 1] and such that f″′(x) ≥ 0, x ∈ [−1, 1]. We prove that, for any n ∈ ℕ and r ≥ 5, there exists a function f ∈ C
r
[−1, 1] ⋂ Δ3 [−1, 1] such that ∥f
(r)∥
C[−1, 1] ≤ 1 and, for an arbitrary algebraic polynomial P ∈ Δ3 [−1, 1], there exists x such that
| f(x) - P(x) | 3 C?n \uprhonr(x), \left| {f(x) - P(x)} \right| \geq C\sqrt n {{\uprho}}_n^r(x), 相似文献
15.
This paper deals with the existence of weak solutions in W
01(Ω) to a class of elliptic problems of the form
16.
Flavia Giannetti Tadeusz Iwaniec Jani Onninen Anna Verde 《Journal of Geometric Analysis》2002,12(2):223-254
Let ƒ: Ω → ℝn be a mapping in the Sobolev space W1,n−1(Ω,ℝn), n ≥ 2. We assume that the determinant of the differential matrix Dƒ (x) is nonnegative, while the cofactor matrix D#ƒ satisfies
, where Lp(Ω) is an Orlicz space. We show that, under the natural Divergence Condition on P, see (1.10), the Jacobian lies in L
loc
1
(Ω). Estimates above and below L
loc
1
(Ω) are also studied. These results are stronger than the previously known estimates, having assumed integrability conditions
on the differential matrix. 相似文献
17.
D. R. Heath-Brown 《Proceedings Mathematical Sciences》1994,104(1):13-29
LetF(x) =F[x1,…,xn]∈ℤ[x1,…,xn] be a non-singular form of degree d≥2, and letN(F, X)=#{xεℤ
n
;F(x)=0, |x|⩽X}, where
. It was shown by Fujiwara [4] [Upper bounds for the number of lattice points on hypersurfaces,Number theory and combinatorics, Japan, 1984, (World Scientific Publishing Co., Singapore, 1985)] thatN(F, X)≪X
n−2+2/n
for any fixed formF. It is shown here that the exponent may be reduced ton - 2 + 2/(n + 1), forn ≥ 4, and ton - 3 + 15/(n + 5) forn ≥ 8 andd ≥ 3. It is conjectured that the exponentn - 2 + ε is admissable as soon asn ≥ 3. Thus the conjecture is established forn ≥ 10. The proof uses Deligne’s bounds for exponential sums and for the number of points on hypersurfaces over finite fields.
However a composite modulus is used so that one can apply the ‘q-analogue’ of van der Corput’s AB process.
Dedicated to the memory of Professor K G Ramanathan 相似文献
18.
Harri Ojanen 《Journal of Fourier Analysis and Applications》2000,6(4):427-436
Weighted Lp estimates (1<p<∞) are shown for oscillatory singular integral operators with polynomial phase and a rough kernel of the form
eiP(x,y)Ω(x−y)h(|x−y|)|x−y|−n. We assume that Ω∈L logL(Sn−1) is homogeneous of degree zero and ∫Sn-1Ω=0. The radial factor h has bounded variation. The necessary condition on the weight is similar to the Ap condition but involves rectangles (instead of cubes) arising from a covering of a star-shaped set related to Ω. 相似文献
19.
Approximation to the function |x| plays an important role in approximation theory. This paper studies the approximation to the function xαsgn x, which equals |x| if α = 1. We construct a Newman Type Operator rn(x) and prove max |x|≤1|xαsgn x-rn(x)|~Cn1/4e-π1/2(1/2)αn. 相似文献
20.
姚奎 《高校应用数学学报(英文版)》2001,16(2):161-170
§ 1 PreliminariesWe considerψ( x)∈ L1 ( Rn) satisfying the mean valuezero,i.e.∫Rnψdx=0 ,and definethe square function g( f) on Rnbyg( f) ( x) =( k|ψk* f|2 ) 1 2 ( x)for f∈ S( Rn) ,the Schwartz space,whereψk( x) =ψ2 k( x) . Whenψ has some smooth property,one can obtain the weak type estimate by viewingthe square function g( f) as the vector-valued singularintegrals,which the readercan referto [1 ,2 ] .As for the results aboutthe Lp-estimates,see [3,4 ] .In this paper,we sha… 相似文献
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