共查询到20条相似文献,搜索用时 15 毫秒
1.
We show that any pointwise multiplier for BMO(ℝn) generates a function p from the class (ℝn) of those functions for which the Hardy-Littlewood maximal operator is bounded on the variable Lp space. In particular, this gives a positive answer to Diening's conjecture saying that there are discontinuous functions
which nevertheless belong to (ℝn). 相似文献
2.
Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent 总被引:1,自引:0,他引:1
We prove the boundedness of the maximal operator Mr in the spaces L^p(·)(Г,p) with variable exponent p(t) and power weight p on an arbitrary Carleson curve under the assumption that p(t) satisfies the log-condition on Г. We prove also weighted Sobolev type L^p(·)(Г, p) → L^q(·)(Г, p)-theorem for potential operators on Carleson curves. 相似文献
3.
Péter Simon 《Monatshefte für Mathematik》2000,131(4):321-334
The one- and two-parameter Walsh system will be considered in the Paley as well as in the Kaczmarz rearrangement. We show
that in the two-dimensional case the restricted maximal operator of the Walsh–Kaczmarz (C, 1)-means is bounded from the diagonal Hardy space H
p
to L
p
for every . To this end we consider the maximal operator T of a sequence of summations and show that the p-quasi-locality of T implies the same statement for its two-dimensional version T
α. Moreover, we prove that the assumption is essential. Applying known results on interpolation we get the boundedness of T
α as mapping from some Hardy–Lorentz spaces to Lorentz spaces. Furthermore, by standard arguments it will be shown that the
usual two-parameter maximal operators of the (C, 1)-means are bounded from L
p
spaces to L
p
if . As a consequence, the a.e. convergence of the (C, 1)-means will be obtained for functions such that their hybrid maximal function is integrable. Of course, our theorems from
the two-dimensional case can be extended to higher dimension in a simple way.
(Received 20 April 2000; in revised form 25 September 2000) 相似文献
4.
Andrei K. Lerner 《Archiv der Mathematik》2005,85(6):538-543
We show that the Hardy-Littlewood maximal operator and a class of Calderón-Zygmund singular integrals satisfy the strong type
modular inequality in variable Lp spaces if and only if the variable exponent p(x) ∼ const.
Received: 15 September 2004 相似文献
5.
We study a class of kernels associated to functions of a distinguished Laplacian on the solvable group AN occurring in the Iwasawa decomposition G = ANK of a noncompact semisimple Lie group G. We determine the maximal ideal space of a commutative subalgebra of L1, which contains the algebra generated by the heat kernel, and we prove that the spectrum of the Laplacian is the same on all Lp spaces, 1 ≤ p < ∞. When G is complex, we derive a formula that enables us to compute the Lp norm of these kernels in terms of a weighted Lp norm of the corresponding kernels for the Euclidean Laplacian on the tangent space. We also prove that, when G is either rank one or complex, certain Hardy-Littlewood maximal operators, which are naturally associated with these kernels, are weak type (1, 1). 相似文献
6.
We prove a time hierarchy theorem for inverting functions computable in a slightly nonuniform polynomial time. In particular,
we prove that if there is a strongly one-way function, then for any k and for any polynomial p, there is a function f computable in linear time with one bit of advice such that there is a polynomial-time probabilistic adversary that inverts
f with probability ≥ 1/p(n) on infinitely many lengths of input, while all probabilistic O(n
k
)-time adversaries with logarithmic advice invert f with probability less than 1/p(n) on almost all lengths of input. We also prove a similar theorem in the worst-case setting, i.e., if P ≠ NP, then for every l > k ≥ 1
Bibliography: 21 titles.
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 54–76. 相似文献
7.
Tiba 《Applied Mathematics and Optimization》2008,47(1):45-58
Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W
0
l,p
(Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary
value problems of arbitrary order, in arbitrary dimension and with general cost functionals. 相似文献
8.
Turdebek N. Bekjan 《Journal of Mathematical Analysis and Applications》2006,322(1):87-96
We define the Hardy-Littlewood maximal function of τ-measurable operators and obtain weak (1,1)-type and (p,p)-type inequalities for the Hardy-Littlewood maximal function. 相似文献
9.
Tiba 《Applied Mathematics and Optimization》2003,47(1):45-58
Abstract. We prove that for bounded open sets Ω with continuous boundary, Sobolev spaces of type W
0
l,p
(Ω ) are characterized by the zero extension outside of Ω . Combining this with a compactness result for domains of class C, we obtain a general existence theorem for shape optimization problems governed by nonlinear nonhomogenous Dirichlet boundary
value problems of arbitrary order, in arbitrary dimension and with general cost functionals. 相似文献
10.
Integration and approximation in arbitrary dimensions 总被引:13,自引:0,他引:13
We study multivariate integration and approximation for various classes of functions of d variables with arbitrary d. We consider algorithms that use function evaluations as the information about the function. We are mainly interested in
verifying when integration and approximation are tractable and strongly tractable. Tractability means that the minimal number
of function evaluations needed to reduce the initial error by a factor of ɛ is bounded by C(d)ɛ−p
for some exponent p independent of d and some function C(d). Strong tractability means that C(d) can be made independent of d. The ‐exponents of tractability and strong tractability are defined as the smallest powers of ɛ{-1} in these bounds.
We prove that integration is strongly tractable for some weighted Korobov and Sobolev spaces as well as for the Hilbert space
whose reproducing kernel corresponds to the covariance function of the isotropic Wiener measure. We obtain bounds on the ‐exponents,
and for some cases we find their exact values. For some weighted Korobov and Sobolev spaces, the strong ‐exponent is the same
as the ‐exponent for d=1, whereas for the third space it is 2.
For approximation we also consider algorithms that use general evaluations given by arbitrary continuous linear functionals
as the information about the function. Our main result is that the ‐exponents are the same for general and function evaluations.
This holds under the assumption that the orthonormal eigenfunctions of the covariance operator have uniformly bounded L∞ norms. This assumption holds for spaces with shift-invariant kernels. Examples of such spaces include weighted Korobov spaces.
For a space with non‐shift‐invariant kernel, we construct the corresponding space with shift-invariant kernel and show that
integration and approximation for the non-shift-invariant kernel are no harder than the corresponding problems with the shift-invariant
kernel. If we apply this construction to a weighted Sobolev space, whose kernel is non-shift-invariant, then we obtain the
corresponding Korobov space. This enables us to derive the results for weighted Sobolev spaces.
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
11.
Yu. A. Dubinskii 《Proceedings of the Steklov Institute of Mathematics》2010,269(1):106-126
We prove a Hardy-type inequality that provides a lower bound for the integral ∫0∞|f(r)|
p
r
p−1
dr, p > 1. In the scale of classical Hardy inequalities, this integral corresponds to the value of the exponential parameter for
which neither direct nor inverse Hardy inequalities hold. However, the problem of estimating this integral and its multidimensional
generalization from below arises in some practical questions. These are, for example, the question of solvability of elliptic
equations in the scale of Sobolev spaces in the whole Euclidean space ℝ
n
, some questions in the theory of Sobolev spaces, hydrodynamic problems, etc. These questions are studied in the present paper. 相似文献
12.
In this paper we introduce a generalized Sobolev space by defining a semi-inner product formulated in terms of a vector distributional
operator P consisting of finitely or countably many distributional operators P
n
, which are defined on the dual space of the Schwartz space. The types of operators we consider include not only differential
operators, but also more general distributional operators such as pseudo-differential operators. We deduce that a certain
appropriate full-space Green function G with respect to L := P
*T
P now becomes a conditionally positive function. In order to support this claim we ensure that the distributional adjoint operator
P
* of P is well-defined in the distributional sense. Under sufficient conditions, the native space (reproducing-kernel Hilbert space)
associated with the Green function G can be embedded into or even be equivalent to a generalized Sobolev space. As an application, we take linear combinations
of translates of the Green function with possibly added polynomial terms and construct a multivariate minimum-norm interpolant
s
f,X
to data values sampled from an unknown generalized Sobolev function f at data sites located in some set
X ì \mathbbRd{X \subset \mathbb{R}^d}. We provide several examples, such as Matérn kernels or Gaussian kernels, that illustrate how many reproducing-kernel Hilbert
spaces of well-known reproducing kernels are equivalent to a generalized Sobolev space. These examples further illustrate
how we can rescale the Sobolev spaces by the vector distributional operator P. Introducing the notion of scale as part of the definition of a generalized Sobolev space may help us to choose the “best”
kernel function for kernel-based approximation methods. 相似文献
13.
If f∈L2[0, 1] and g*∈L2[0, 1] is the best non-decreasing approximation to f, then it's shown that ‖f−g*‖2=‖f−θ(f)‖2, where θ(f) denotes the Hardy-Littlewood maximal function of f. 相似文献
14.
We consider the problem −Δu=|u|
p−1u+λu in Ω with
on δΩ, where Ω is a bounded domain inR
N
,p=(N+2)/(N−2) is the critical Sobolev exponent,n the outward pointing normal and λ a constant. Our main result is that if Ω is a ball inR
N
, then for every λ∈R the problem admits infinitely many solutions. Next we prove that for every bounded domain Ω inR
3, symmetric with respect to a plane, there exists a constant μ>0 such that for every λ<μ this problem has at least one non-trivial
solution.
This work was supported by the Paris VI-Leiden exchange program
Supported by the Netherlands organisation for scientific research NWO, under number 611-306-016. 相似文献
15.
A. Fiorenza J. M. Rakotoson 《Calculus of Variations and Partial Differential Equations》2006,25(2):187-203
We study some generalized small Lebesgue spaces and their associated Sobolev spaces. In particular, we prove that small Lebesgue-Sobolev
spaces W1,(p(Ω) are compactly embedded in
, p < n. As an application, we study variational problems involving critical exponents under multiple constraints.
Mathematics Subject Classification (2000) 46E30, 46E35, 46B70, 26D07, 35J60. 相似文献
16.
Julián Fernández Bonder Pablo Groisman Julio D. Rossi 《Annali di Matematica Pura ed Applicata》2007,186(2):341-358
The best Sobolev trace constant is given by the first eigenvalue of a Steklov-like problem. We deal with minimizers of the
Rayleigh quotient ‖u‖2
H
1
(Ω)
2/‖u‖2
L
2
(∂Ω) for functions that vanish in a subset A⊂ Ω, which we call the hole. We look for holes that minimize the best Sobolev trace constant among subsets of Ω with prescribed
volume. First, we find a formula for the first variation of the first eigenvalue with respect to the hole. As a consequence
of this formula, we prove that when Ω is a ball the symmetric hole (a centered ball) is critical when we consider deformations
that preserves volume but is not optimal. Finally, we prove that by the Finite Element Method we can approximate the optimal
configuration and, by means of the shape derivative, we design an algorithm to compute the discrete optimal holes.
Mathematics Subject Classification (2000) 35P15, 49K20, 49M25, 49Q10 相似文献
17.
B. Bojarski 《Ukrainian Mathematical Journal》2007,59(3):379-395
We present two fundamental facts from the jet theory for Sobolev spaces W
m, p
. One of these facts is that the formal differentiation of the k-jets theory is compatible with the pointwise definition of Sobolev (m − 1)-jet spaces on regular subsets of the Euclidean spaces ℝn. The second result describes the Sobolev imbedding operator of Sobolev jet spaces increasing the order of integrability of
Sobolev functions up to the critical Sobolev exponent.
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 3, pp. 345–358, March, 2007. 相似文献
18.
We find necessary and sufficient conditions on a Banach spaceX in order for the vector-valued extensions of several operators associated to the Ornstein-Uhlenbeck semigroup to be of weak
type (1, 1) or strong type (p, p) in the range 1<p<∞. In this setting, we consider the Riesz transforms and the Littlewood-Paleyg-functions. We also deal with vector-valued extensions of some maximal operators like the maximal operators of the Ornstein-Uhlenbeck
and the corresponding Poisson semigroups and the maximal function with respect to the gaussian measure.
In all cases, we show that the condition onX is the same as that required for the corresponding harmonic operator: UMD, Lusin cotype 2 and Hardy-Littlewood property.
In doing so, we also find some new equivalences even for the harmonic case.
The first and third authors were partially supported by CONICET (Argentina) and Convenio Universidad Autónoma de Madrid-Universidad
Nacional del Litoral. The second author was partially supported by the European Commission via the TMR network “Harmonic Analysis”. 相似文献
19.
Loukas GRAFAKOS 《中国科学A辑(英文版)》2008,51(12):2253-2284
Let X be an RD-space, i.e., a space of homogeneous type in the sense of Coifman and Weiss, which has the reverse doubling property. Assume that X has a dimension n. For α∈ (0, ∞) denote by Hαp(X ), Hdp(X ), and H?,p(X ) the corresponding Hardy spaces on X defined by the nontangential maximal function, the dyadic maximal function and the grand maximal function, respectively. Using a new inhomogeneous Calder′on reproducing formula, it is shown that all these Hardy spaces coincide with Lp(X ) when p ∈ (1, ∞] a... 相似文献
20.
Simon [J. Approxim. Theory,
127, 39–60 (2004)] proved that the maximal operator σα,κ,* of the (C, α)-means of the Walsh–Kaczmarz–Fourier series is bounded from the martingale Hardy space H
p
to the space L
p
for p > 1 / (1 + α), 0 < α ≤ 1. Recently, Gát and Goginava have proved that this boundedness result does not hold if p ≤ 1 / (1 + α). However, in the endpoint case p = 1 / (1 + α ), the maximal operator σα,κ,* is bounded from the martingale Hardy space H
1/(1+α) to the space weak- L
1/(1+α). The main aim of this paper is to prove a stronger result, namely, that, for any 0 < p ≤ 1 / (1 + α), there exists a martingale f ∈ H
p
such that the maximal operator σα,κ,*
f does not belong to the space L
p
. 相似文献