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2.
In this paper we investigate harmonic Hardy-Orlicz and Bergman-Orlicz b
φ,α
( B) spaces, using an identity of Hardy-Stein type. We also extend the notion of the Lusin property by introducing ( φ, α)-Lusin property with respect to a Stoltz domain. The main result in the paper is as follows: Let be a nonnegative increasing convex function twice differentiable on (0, ∞), and u a harmonic function on the unit ball B in . Then the following statements are equivalent:
| (a) |
.
|
| (b) |
.
|
| (c) |
u has (φ, α)-Lusin property with respect to a Stoltz domain with half-angle β, for any .
|
| (d) |
u has (φ, α)-Lusin property with respect to a Stoltz domain with half-angle β, for some .
|
相似文献
3.
If denotes the error term in the classical Rankin-Selberg problem, then it is proved that
where Δ 1( x) = ∫
x
0 Δ( u) du. The latter bound is, up to ‘ɛ’, best possible.
Received: 8 February 2007 相似文献
4.
We analyse degenerate, second-order, elliptic operators H in divergence form on L
2( R
n
× R
m
). We assume the coefficients are real symmetric and a
1
H
δ
≥ H ≥ a
2
H
δ
for some a
1, a
2 > 0 where
Here x
1 ∈ R
n
, x
2 ∈ R
m
and are positive measurable functions such that behaves like as x → 0 and as with and . Our principal results state that the submarkovian semigroup is conservative and its kernel K
t
satisfies bounds
where | B( x; r)| denotes the volume of the ball B( x; r) centred at x with radius r measured with respect to the Riemannian distance associated with H. The proofs depend on detailed subelliptic estimations on H, a precise characterization of the Riemannian distance and the corresponding volumes and wave equation techniques which exploit
the finite speed of propagation. We discuss further implications of these bounds and give explicit examples that show the
kernel is not necessarily strictly positive, nor continuous. 相似文献
5.
Abstract
Let ξ
i
∈ (0, 1) with 0 <
ξ 1 < ξ 2 <
··· < ξ
m−2 < 1,
a
i
, b
i
∈ [0,∞) with
and
. We consider the
m-point boundary-value
problem
where f( x, y) ≥ − M, and M is a positive constant. We show the
existence and multiplicity of positive solutions by applying the
fixed point theorem in cones.
*Supported by the NSFC (10271095).
GG-110-10736-1003, NWNU-KJCXGC-212 and the Foundation of Major
Project of Science and Technology of Chinese Education
Ministry 相似文献
6.
Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0. 相似文献
7.
Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the
type
we prove the following prime analogue of a conjecture of Erd?s & Pomerance (1985) concerning the exponent function fa( p) (defined to be the minimal exponent e for which ae ≡ 1 modulo p):
where
The main result is obtained by computing all the higher moments corresponding to ω( fa( p)), and by comparing them, via the Fréchet-Shohat theorem, with estimates due to Halberstam concerning the moments of ω( p − 1).
Received: 25 October 2004; revised: 12 February 2005 相似文献
8.
Abstract This paper develops the model theory of ordered structures that satisfy Keisler’s regularity scheme and its strengthening REF
(the reflection scheme) which is an analogue of the reflection principle of Zermelo-Fraenkel set theory. Here is a language with a distinguished linear order <, and REF
consists of formulas of the form where φ is an -formula, φ
<x
is the -formula obtained by restricting all the quantifiers of φ to the initial segment determined by x, and x is a variable that does not appear in φ. Our results include:
Theorem
The following five conditions are equivalent for a complete first order theory T in a countable language
with a distinguished linear order:
| (1) |
Some model of T has an elementary end extension with a first new element.
|
| (2) |
T ⊢ REF
.
|
| (3) |
T has an ω
1-like model that continuously embeds ω
1.
|
| (4) |
For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ.
|
| (5) |
For some regular uncountable cardinal κ, T has a κ-like model
that has an elementary extension in which the supremum of M exists.
|
Moreover, if κ is a regular cardinal satisfying κ = κ
<κ
, then each of the above conditions is equivalent to:
| (6) |
T has a κ
+ -like model that continuously embeds a stationary subset of κ.
|
相似文献
9.
Let τ( n) be the Ramanujan τ-function, x ≥ 10 be an integer parameter. We prove that
We also show that
where ω( n) is the number of distinct prime divisors of n and p denotes prime numbers. These estimates improve several results from [6, 9].
Received: 23 November 2006 相似文献
10.
This paper deals with the concept of exponentiability for a special class of multivalued maps. To be more precise, we discuss
the exponentiability of a multivalued map F: X⇉ X expressible in the form F( x) = { Ax: A ∈ Ξ}, with Ξ denoting a collection of linear continuous operators defined on a Banach space X. Among other results, we prove that, under suitable assumptions on Ξ, the Painlevé–Kuratowski limit
exists for all x ∈ X, and it admits the representation [exp F]( x) = {e
A
x: A ∈ clco( Ξ)}. The operation of exponentiation has therefore a convexification effect on Ξ. By exploiting the above-mentioned representation formula, we derive general properties for the semigroup { S
F
( t)}
t⩾0 defined by
By way of application, we obtain a formula of exponential type for the reachable set associated to the differential inclusion
相似文献
11.
The following regularity of weak solutions of a class of elliptic equations of the form are investigated. 相似文献
12.
In this paper, the existence of unbounded solutions for the following nonlinear asymmetric oscillator
is discussed, where α, β are positive constants satisfying
for some ω ∈ R+ / Q, h( t) ∈ L∞ [0, 2π ] is 2π-periodic, x±=max {± x, 0 }.
Received: 23 September 2004 相似文献
13.
The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special
class of functionals: with respect to the convergence u n → u in measure, v n ⇀ v in L p(Ω;ℝ d)
in W −1,p(Ω), and χ n ⇀ χ in L p(Ω), where χ n ∈ Z:= {χ ∈ L ∞(Ω): 0 ≤ χ(x) ≤ 1 for a.e. x}. Here
is a constant-rank partial differential operator. The main result is that if the characteristic cone of
has the full dimension, then the l.s.c. is equivalent to the fact that the F ± are both
-quasiconvex and for a.e. x ∈ Ω and for all u ∈ ℝ d. As a corollary, we obtain several results for the functional with respect to the same convergence. We show that this functional is l.s.c. iff Bibliography: 14 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 100–119. 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.
We characterize the composition operators mapping Blochs boundedly into the weighted Bergman spaces of logarithmic weight.
For 0 < p < ∞, 1 < α < ∞, let Ap, log α denote the space of holomorphic functions F in the unit disc D for which
and let Ap, log ασ denote the class of holomorphic self maps f of D for which
Then for the Bloch pullback operator Cf, the following are equivalent:
| (1) |
Cf maps Bloch space boundedly into A2p, log α |
| (2) |
|
| (3) |
.
|
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion
Fund) (KRF-2007-313-C00026). 相似文献
16.
This paper is concerned with the large time behavior of traveling wave solutions to the Cauchy problem of generalized Benjamin–Bona–Mahony–Burgers
equations
with prescribed initial data Here v( > 0), β are constants, u
± are two given constants satisfying u
+ ≠ u
− and the nonlinear function f( u) ∈ C
2( R) is assumed to be either convex or concave. An algebraic time decay rate to traveling waves of the solutions of the Cauchy
problem of generalized Benjamin-Bona-Mahony-Burgers equation is obtained by employing the weighted energy method developed
by Kawashima and Matsumura in [6] to discuss the asymptotic behavior of traveling wave solutions to the Burgers equation.
revised: May 23 and August 8, 2007 相似文献
17.
Let
be a simply connected domain in
, such that
is connected. If g is holomorphic in Ω and every derivative of g extends continuously on
, then we write g ∈ A∞ (Ω). For g ∈ A∞ (Ω) and
we denote
. We prove the existence of a function f ∈ A∞(Ω), such that the following hold:
| i) |
There exists a strictly increasing sequence μn ∈ {0, 1, 2, …}, n = 1, 2, …, such that, for every pair of compact sets Γ, Δ ⊂
and every l ∈ {0, 1, 2, …} we have
|
| ii) |
For every compact set
with
and Kc connected and every function
continuous on K and holomorphic in K0, there exists a subsequence
of
, such that, for every compact set
we have
|
相似文献
18.
We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving
the p( x)-Laplacian of the form
where Ω is a bounded smooth domain in , and p( x) > 1 for with and φ ≢ 0 on ∂Ω. Using the sub-supersolution method and the variational method, under appropriate assumptions on f, we prove that, there exists λ * > 0 such that the problem has at least two positive solutions if λ = λ *, has at least one positive solution if λ = λ *, and has no positive solution if λ = λ *. To prove the result we establish a special strong comparison principle for the Neumann problems.
The research was supported by the National Natural Science Foundation of China 10371052,10671084). 相似文献
19.
Iterated Brownian motion Zt serves as a physical model for diffusions in a crack. If τ D( Z) is the first exit time of this processes from a domain D⊂ℝ n, started at z∈ D, then Pz[τ D( Z)> t] is the distribution of the lifetime of the process in D. In this paper we determine the large time asymptotics of
which gives exponential integrability of
for parabola-shaped domains of the form Pα={( x, Y)∈ℝ×ℝ n−1: x>0, | Y|< Axα}, for 0<α<1, A>0. We also obtain similar results for twisted domains in ℝ 2 as defined in DeBlassie and Smits: Brownian motion in twisted domains, Preprint, 2004. In particular, for a planar iterated
Brownian motion in a parabola
we find that for z∈℘
Mathematics Subject Classifications (2000) 60J65, 60K99.
Erkan Nane: Supported in part by NSF Grant # 9700585-DMS. 相似文献
20.
The main result is that for sets , the following are equivalent:
| (1) |
The shuffle sum σ(S) is computable.
|
| (2) |
The set S is a limit infimum set, i.e., there is a total computable function g(x, t) such that enumerates S.
|
| (3) |
The set S is a limitwise monotonic set relative to 0′, i.e., there is a total 0′-computable function satisfying such that enumerates S.
|
Other results discuss the relationship between these sets and the sets.
The author’s research was partially supported by a VIGRE grant fellowship. The author thanks Denis Hirschfeldt and Steffen
Lempp for an insightful conversation about L IMI NF sets; Christopher Alfeld and Robert Owen for numerous comments and suggestions; and his thesis advisor Steffen Lempp for
his guidance. 相似文献
|
