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1.
Michael Bildhauer 《manuscripta mathematica》2003,110(3):325-342
Suppose that f: ℝ
nN
→ℝ is a strictly convex energy density of linear growth, f(Z)=g(|Z|2) if N>1. If f satisfies an ellipticity condition of the form
then, following [Bi3], there exists a unique (up to a constant) solution of the variational problem
provided that the given boundary data u
0
W
1
1
(ω;ℝ
N
) are additionally assumed to be of class L
∞(ω;ℝ
N
). Moreover, if μ<3, then the boundedness of u
0 yields local C
1,α-regularity (and uniqueness up to a constant) of generalized minimizers of the problem
In our paper we show that the restriction u
0L
∞(ω;ℝ
N
) is superfluous in the two dimensional case n=2, hence we may prescribe boundary values from the energy class W
1
1
(ω;ℝ
N
) and still obtain the above results.
Received: 12 February 2002 / Revised version: 7 October 2002 Published online: 14 February 2003
Mathematics Subject Classification (2000): 49N60, 49N15, 49M29, 35J 相似文献
2.
Liguang Liu 《Frontiers of Mathematics in China》2007,2(4):599-611
Let ℐ(ℝn) be the Schwartz class on ℝn and ℐ∞(ℝn) be the collection of functions ϕ ∊ ℐ(ℝn) with additional property that
for all multiindices γ. Let (ℐ(ℝn))′ and (ℐ∞(ℝn))′ be their dual spaces, respectively. In this paper, it is proved that atomic Hardy spaces defined via (ℐ(ℝn))′ and (ℐ∞(ℝn))′ coincide with each other in some sense. As an application, we show that under the condition that the Littlewood-Paley
function of f belongs to L
p(ℝn) for some p ∊ (0,1], the condition f ∊ (ℐ∞(ℝn))′ is equivalent to that f ∊ (ℐ(ℝn))′ and f vanishes weakly at infinity. We further discuss some new classes of distributions defined via ℐ(ℝn) and ℐ∞(ℝn), also including their corresponding Hardy spaces.
相似文献
3.
Jorge J. Betancor Juan C. Fariña Teresa Martinez Lourdes Rodríguez-Mesa 《Arkiv f?r Matematik》2008,46(2):219-250
In this paper we investigate Riesz transforms R
μ
(k) of order k≥1 related to the Bessel operator Δμ
f(x)=-f”(x)-((2μ+1)/x)f’(x) and extend the results of Muckenhoupt and Stein for the conjugate Hankel transform (a Riesz transform of order one). We
obtain that for every k≥1, R
μ
(k) is a principal value operator of strong type (p,p), p∈(1,∞), and weak type (1,1) with respect to the measure dλ(x)=x
2μ+1
dx in (0,∞). We also characterize the class of weights ω on (0,∞) for which R
μ
(k) maps L
p
(ω) into itself and L
1(ω) into L
1,∞(ω) boundedly. This class of weights is wider than the Muckenhoupt class of weights for the doubling measure dλ. These weighted results extend the ones obtained by Andersen and Kerman. 相似文献
4.
Emmanuel Rio 《Probability Theory and Related Fields》2000,118(3):342-348
The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L
1(ℝ/ℤ), then the averages
converge to the mean value of ƒ over [0, 1] for almost every x in [0, 1]. In this paper we prove that, for ƒ in L
1(ℝ/ℤ), the averages A
n
a
ƒ(x) converge a.e. to the integral of ƒ over [0, 1] for almost every a > 1. Furthermore we obtain convergence rates in this strong law of large numbers.
Received: 1 March 1999 / Revised version: 20 October 1999 / Published online: 12 October 2000 相似文献
Lois fortes des grands nombres presque s?res pour les sommes de Riesz–Raikov English title: Almost sure versions of the Riesz–Raikov strong law of large numbers
Received: 1 March 1999 / Revised version: 20 October 1999 / Published online: 12 October 2000 相似文献
5.
We consider an operator K˚ϕ = Lϕ−: <CDU(x), Dϕ> in a Hilbert space H, where L is an Ornstein–Uhlenbeck operator, U∈W
1,4(H, μ) and μ is the invariant measure associated with L. We show that K˚ is essentially self-adjoint in the space L
2(H, ν) where ν is the “Gibbs” measure ν(dx) = Z
−:1
e
−:2U(x)
dx. An application to Stochastic quantization is given.
Received: 13 August 1998 / Revised version: 20 September 1999 / Published online: 8 August 2000 相似文献
6.
Robert S. Strichartz 《Journal of Geometric Analysis》1991,1(3):269-289
Let μ be a measure on ℝn that satisfies the estimate μ(B
r(x))≤cr
α for allx ∈ ℝn and allr ≤ 1 (B
r(x) denotes the ball of radius r centered atx. Let ϕ
j,k
(ɛ)
(x)=2
nj2ϕ(ɛ)(2
j
x-k) be a wavelet basis forj ∈ ℤ, κ ∈ ℤn, and ∈ ∈E, a finite set, and letP
j
(T)=Σɛ,k
<T,ϕ
j,k
(ɛ)
>ϕ
j,k
(ɛ)
denote the associated projection operators at levelj (T is a suitable measure or distribution). Iff ∈Ls
p(dμ) for 1 ≤p ≤ ∞, we show thatP
j(f dμ) ∈ Lp(dx) and ||P
j
(fdμ)||L
p(dx)≤c2
j((n-α)/p′))||f||L
p(dμ) for allj ≥ 0. We also obtain estimates for the limsup and liminf of ||P
j
(fdμ)||L
p(dx) under more restrictive hypotheses.
Communicated by Guido Weiss 相似文献
7.
Rostom Getsadze 《Journal d'Analyse Mathématique》2007,102(1):209-223
Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E
0 ⊂ I
2 is any Lebesgue measurable set such that μ2
E
0 > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L1(I
2) and a set E
0
′
, ⊂ E
0, μ2
E
0
′
> 0, such that
and the sequence of double Cesàro means of Fourier series of f with respect to the system {ϕn(x)ϕm(y): n,m = 1, 2,...} is unbounded in the sense of Pringsheim (by rectangles) on E
0
′
. This statement gives critical integrability conditions for the Cesàro summability a.e. of Fourier series in the class of
all complete orthonormal systems of the type {ϕ n(x)ϕm(y): n,m = 1, 2,...}. 相似文献
8.
SAUGATA BANDYOPADHYAY 《Proceedings Mathematical Sciences》2011,121(3):339-348
Let Ω ⊂ ℝ
n
be a smooth, bounded domain. We study the existence and regularity of diffeomorphisms of Ω satisfying the volume form equation
f*(g)=f, \textin W, \phi^\ast(g)=f, \quad \text{in }\Omega, 相似文献
9.
Ferenc Móricz 《Acta Mathematica Hungarica》2012,134(3):356-368
We consider complex-valued functions f∈L
1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform [^(f)]s\hat{f}_{s} and the cosine Fourier transform [^(f)]c\hat{f}_{c} belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere. 相似文献
10.
Qihui Zhang 《分析论及其应用》2006,22(3):271-282
The behavior on the space L∞((R)n) 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 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). It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L∞((R)n), TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO((R)n). 相似文献
11.
In this paper, the boundedness of Toeplitz operator T
b(f) related to strongly singular Calderón-Zygmund operators and Lipschitz function b ε
(ℝn) is discussed from L
p(ℝn) to L
q(ℝn),
, and from L
p(ℝn) to Triebel-Lizorkin space
. We also obtain the boundedness of generalized Toeplitz operator Θ
α0
b
from L
p(ℝn) to L
q(ℝn),
. All the above results include the corresponding boundedness of commutators. Moreover, the boundedness of Toeplitz operator
T
b(f) related to strongly singular Calderón-Zygmund operators and BMO function b is discussed on L
p(ℝn), 1 < p < ∞. 相似文献
12.
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
13.
Let Ω ⊂ ℝ
d
be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae
of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal
Voronoi tessellation of Ω. In this connection we come across a class of operators of the form Ln[f](x): = ?i=1n fi(x)(f(yi) + á?f(yi), x-yi?)L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle), where y1,..., yn\boldsymbol{y}_1,\dots, \boldsymbol{y}_n are distinct points in Ω and {ϕ
1, ..., ϕ
n
} is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators L
n
with a smallest constant in an L
p
error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable
and strictly convex function f. It allows us to describe a best operator L
n
for approximating f by L
n
[f] with respect to the L
p
norm. 相似文献
14.
Summary. We address the following problem from the intersection of dynamical systems and stochastic analysis: Two SDE dx
t
= ∑
j
=0
m
f
j
(x
t
)∘dW
t
j
and dx
t
=∑
j
=0
m
g
j
(x
t
)∘dW
t
j
in ℝ
d
with smooth coefficients satisfying f
j
(0)=g
j
(0)=0 are said to be smoothly equivalent if there is a smooth random diffeomorphism (coordinate transformation) h(ω) with h(ω,0)=0 and Dh(ω,0)=id which conjugates the corresponding local flows,
15.
We prove that if a symmetric submarkovian semigroup (T
t
)
t>0
satisfies an estimate of the form
16.
Let T:x↦2x (mod 1) be the doubling map of the circle ?=ℝ/ℤ. We construct a trigonometric polynomial f:?→ℝ with the following property: ∫f
dμ≥0 for every T-invariant probability measure μ, so that f is cohomologous to a non-negative Lipschitz function, yet f is not cohomologous to any non-negative C
1 function.
Oblatum 28-VI-2001 & 4-X-2001?Published online: 18 January 2002 相似文献
17.
Let ? be the genealogical tree of a supercritical multitype Galton–Watson process, and let Λ be the limit set of ?, i.e., the set of all infinite self-avoiding paths (called ends) through ? that begin at a vertex of the first generation. The limit set Λ is endowed with the metric d(ζ, ξ) = 2
−n
where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence Φ(ζ) of
types of the vertices of ζ. Let Ω be the space of all such sequences. For any ergodic, shift-invariant probability measure
μ on Ω, define Ωμ to be the set of all μ-generic sequences, i.e., the set of all sequences ω such that each finite sequence v occurs in ω with limiting frequency μ(Ω(v)), where Ω(v) is the set of all ω′?Ω that begin with the word v. Then the Hausdorff dimension of Λ∩Φ−1 (Ωμ) in the metric d is
18.
Song Li 《中国科学A辑(英文版)》2003,46(3):364-375
The purpose of this paper is to investigate the refinement equations of the form
19.
In this paper, sufficient conditions are obtained, so that the second order neutral delay differential equation
20.
E. L. Alexandrov 《Mathematical Notes》2000,67(6):679-685
Explicit formulas are derived for the spectral function of double multiplication operator containing a multiplicative evolution
inL
2(X, μ)-space and a convolution-type operator inL
2(ℝ
n
)-spaces. Symmetric convolution and multiplication operators are considered inL
2(X, μ) andL
2(ℝ
n
)-spaces.
Translated fromMatematicheskie Zametki, Vol. 67, No. 6, pp. 803–810, June, 2000. 相似文献
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