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1.
The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity 总被引:1,自引:0,他引:1
Andreas Fr?hlich 《Mathematische Annalen》2007,339(2):287-316
In this paper we establish a general weighted L
q
-theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L
q
-estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates
imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L
p
-regularity of in the respective weighted L
q
-spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L
p
-regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L
p
-regularity. Preprint (1999)] with the fact that for L
q
-spaces -boundedness is implied by weighted estimates. 相似文献
2.
Ali Ben Amor 《Mathematische Zeitschrift》2007,255(3):627-647
Let
be a regular Dirichlet form on L
2(X,m), μ a positive Radon measure charging no sets of zero capacity and Φ an N-function. We prove that the Sobolev-Orlicz inequality(SOI)
for every
is equivalent to a capacitary-type inequality. Further we show that if
is continuously embedded into L
2(X,μ), the latter one implies some integrability condition, which is nothing else but the classical uniform integrability condition
if μ is finite. We also prove that a SOI for
yields a Nash-type inequality and if further μ = m and Φ is admissible, it yields the ultracontractivity of the corresponding semigroup. After, in the spirit of SOIs, we derive
criteria for
to be compactly embedded into L
2(μ), provided μ is finite. As an illustration of the theory, we shall relate the compactness of the latter embedding to the
discreteness of the spectrum of the time changed Dirichlet form and shall derive lower bounds for its eigenvalues in term
of Φ.
This work has been supported by the Deutsche Forschungsgemeinschaft. 相似文献
3.
In this paper we investigate vector-valued parabolic initial boundary value problems , subject to general boundary conditions in domains G in with compact C
2m
-boundary. The top-order coefficients of are assumed to be continuous. We characterize optimal L
p
-L
q
-regularity for the solution of such problems in terms of the data. We also prove that the normal ellipticity condition on
and the Lopatinskii–Shapiro condition on are necessary for these L
p
-L
q
-estimates. As a byproduct of the techniques being introduced we obtain new trace and extension results for Sobolev spaces
of mixed order and a characterization of Triebel-Lizorkin spaces by boundary data.
相似文献
4.
J. Sunklodas 《Lithuanian Mathematical Journal》2009,49(2):216-221
In the paper, we present upper bounds of L
p
norms of order (
X)-1/2 for all 1 ≤ p ≤ ∞ in the central limit theorem for a standardized random variable (X−
X)/ √
X, where a random variable X is distributed by the Poisson distribution with parameter λ > 0 or by the standard gamma distribution Γ(α, 0, 1) with parameter
α > 0.
The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-70/09. 相似文献
5.
Robert Whitley 《Integral Equations and Operator Theory》2008,61(3):433-448
A result due to Nyman establishes the equivalence of the Riemann hypothesis with the density of a set of functions in L
2[0, 1]. Here a large class of analytic functions is considered, which includes the Riemann zeta function and the Dirichlet
L-functions as well as functions not given by a Dirichlet series. For each such function there is an associated integral operator T on L
2[0, 1] such that has no zeros in Re(s) > 1/2 iff the operator T has dense range iff a specified set of functions is dense in L
2[0, 1].
相似文献
6.
F. Bastianelli 《manuscripta mathematica》2009,130(1):113-120
Let C be a very general curve of genus g and let C
(2) be its second symmetric product. This paper concerns the problem of describing the convex cone of all numerically effective -divisors classes in the Néron–Severi space . In a recent work, Julius Ross improved the bounds on in the case of genus five. By using his techniques and by studying the gonality of the curves lying on C
(2), we give new bounds on the nef cone of C
(2) when C is a very general curve of genus 5 ≤ g ≤ 8.
This work has been partially supported by (1) PRIN 2007 “Spazi di moduli e teorie di Lie”; (2) Indam (GNSAGA); (3) FAR 2008 (PV) “Varietá algebriche, calcolo algebrico, grafi orientati e topologici”. 相似文献
7.
The study of harmonic functions on a locally compact group G has recently been transferred to a “non-commutative” setting in two different directions: Chu and Lau replaced the algebra
L
∞(G) by the group von Neumann algebra VN(G) and the convolution action of a probability measure μ on L
∞(G) by the canonical action of a positive definite function σ on VN(G); on the other hand, Jaworski and the first author replaced L
∞(G) by to which the convolution action by μ can be extended in a natural way. We establish a link between both approaches. The action
of σ on VN(G) can be extended to . We study the corresponding space of “σ-harmonic operators”, i.e., fixed points in under the action of σ. We show, under mild conditions on either σ or G, that is in fact a von Neumann subalgebra of . Our investigation of relies, in particular, on a notion of support for an arbitrary operator in that extends Eymard’s definition for elements of VN(G). Finally, we present an approach to via ideals in , where denotes the trace class operators on L
2(G), but equipped with a product different from composition, as it was pioneered for harmonic functions by Willis.
M. Neufang was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach.
V. Runde was supported by NSERC and the Mathematisches Forschungsinstitut Oberwolfach. 相似文献
8.
J. J. Grobler 《Integral Equations and Operator Theory》2007,57(1):83-99
For a probability space
we denote the marginal measures of
, defined on Σ and Λ respectively, by
and
. If ρ is a function norm defined on
marginal function norms ρ1 and ρ2 are defined on
and
. We find conditions which guarantee Lρ 1 + Lρ 2 to be embedded in Lρ as a closed subspace. The problem is encountered in Statistics when estimating a bivariate distribution with known marginals.
We find a condition which, applied to the binormal distribution in L2, improves some known conditions. 相似文献
9.
J. -P. Navarrete 《Geometriae Dedicata》2006,122(1):1-13
Let G be a discrete subgroup of PU(2,1); G acts on
preserving the unit ball , equipped with the Bergman metric. Let be the limit set of G in the sense of Chen–Greenberg, and let be the limit set of the G-action on in the sense of Kulkarni. We prove that L(G) = Λ(G) ∩ S
3 and Λ(G) is the union of all complex projective lines in which are tangent to S
3 at a point in L(G). 相似文献
10.
Christian Wegner 《manuscripta mathematica》2009,128(4):469-481
Let X be a finite aspherical CW-complex whose fundamental group π
1(X) possesses a subnormal series with a non-trivial elementary amenable group G
0. We investigate the L
2-invariants of the universal covering of such a CW-complex X. The main result is the proof of the vanishing of the L
2-torsion under the condition that π
1(X) has semi-integral determinant. We further show that the Novikov–Shubin invariants are positive. 相似文献
11.
Zhongwei Shen 《Mathematische Annalen》2006,336(3):697-725
We study the homogeneous elliptic systems of order $2\ellWe study the homogeneous elliptic systems of order with real constant coefficients on Lipschitz domains in, . For any fixed p > 2, we show that a reverse H?lder condition with exponent p is necessary and sufficient for the solvability of the Dirichlet problem with boundary data in L
p
. We also obtain a simple sufficient condition. As a consequence, we establish the solvability of the L
p
Dirichlet problem for and . The range of p is known to be sharp if and . For the polyharmonic equation, the sharp range of p is also found in the case n = 6, 7 if , and if .Research supported in part by the NSF. 相似文献
12.
Given
, a compact abelian group G and a function
, we identify the maximal (i.e. optimal) domain of the convolution
operator
(as an operator from Lp(G) to itself). This is the
largest Banach function space (with order continuous norm) into which Lp(G)
is embedded and to which
has a continuous extension, still with values
in Lp(G). Of course, the optimal domain depends on p and g. Whereas
is compact, this is not always so for the extension of
to its optimal domain.
Several characterizations of precisely when this is the case are presented. 相似文献
13.
Marian Nowak 《Positivity》2009,13(1):193-199
We study compactness properties of linear operators from an Orlicz space LΦ provided with a natural mixed topology to a Banach space (X, || · ||X). We derive that every Bochner representable operator is -compact. In particular, it is shown that every Bochner representable operator is (τ(L∞, L1), || · ||X)-compact.
相似文献
14.
In this paper, we establish lower bounds for n-term approximations in the metric of L
2(I
2
) of characteristic functions of plane convex subsets of the square I
2
with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than
. 相似文献
15.
J. Casado-Díaz T. Chacón Rebollo V. Girault M. Gómez Mármol F. Murat 《Numerische Mathematik》2007,105(3):337-374
In this paper we consider, in dimension d≥ 2, the standard finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L
∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an
hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs
to L
1(Ω), we prove that the unique solution of the discrete problem converges in (for every q with ) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon
measure. In the case where the dimension is d = 2 or d = 3 and where the coefficients are smooth, we give an error estimate in when the right-hand side belongs to L
r
(Ω) for some r > 1. 相似文献
16.
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
17.
Michael Christ 《Journal of Geometric Analysis》1991,1(3):193-230
For a large class of subharmonicφ, the equation
is studied in
. Pointwise upper bounds are derived for the distribution kernels of the canonical solution operator and of the orthogonal
projection onto the space of entire functions inH. Existence theorems inL
p norms are derived as a corollary. A class of counterexamples, related to the failure of
to be analytic-hypoelliptic on certain CR manifolds, is discussed.
Communicated by Steven Krantz 相似文献
18.
Recently the first author presented exact formulas for the number of 2
n
-periodic binary sequences with given 1-error linear complexity, and an exact formula for the expected 1-error linear complexity
and upper and lower bounds for the expected k-error linear complexity, k ≥ 2, of a random 2
n
-periodic binary sequence. A crucial role for the analysis played the Chan–Games algorithm. We use a more sophisticated generalization
of the Chan–Games algorithm by Ding et al. to obtain exact formulas for the counting function and the expected value for the
1-error linear complexity for p
n
-periodic sequences over prime. Additionally we discuss the calculation of lower and upper bounds on the k-error linear complexity of p
n
-periodic sequences over .
相似文献
19.
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
20.
A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems
We consider the 2m-th order elliptic boundary value problem Lu = f (x, u) on a bounded smooth domain with Dirichlet boundary conditions on ∂Ω. The operator L is a uniformly elliptic operator of order 2m given by . For the nonlinearity we assume that , where are positive functions and q > 1 if N ≤ 2m, if N > 2m. We prove a priori bounds, i.e, we show that for every solution u, where C > 0 is a constant. The solutions are allowed to be sign-changing. The proof is done by a blow-up argument which relies on
the following new Liouville-type theorem on a half-space: if u is a classical, bounded, non-negative solution of ( − Δ)
m
u = u
q
in with Dirichlet boundary conditions on and q > 1 if N ≤ 2m, if N > 2m then .
相似文献