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1.
Consider a central Gaussian convolution semigroup (μ
t
)
t > 0
on a connected compact semisimple group. Then either the measure μ
t
is singular with respect to Haar measure for all t > 0, or there exists a time t such that μ
t
is absolutely continuous with respect to Haar measure and admits a continuous density.
Received: 6 November 2001 / Revised version: 13 June 2002 / Published online: 30 September 2002
Research partially supported by NSF grant DMS 0102126
Mathematics Subject Classification (2000): 28C10, 28C20, 60B15, 60G30
Keywords or phrases: Gaussian convolution semigroups – Dichotomy – Absolute continuity 相似文献
2.
Valentin Matache 《Complex Analysis and Operator Theory》2008,2(1):169-197
Operators on function spaces acting by composition to the right with a fixed selfmap φ of some set are called composition operators of symbol φ. A weighted composition operator is an operator equal to a composition operator followed by a multiplication operator. We
summarize the basic properties of bounded and compact weighted composition operators on the Hilbert Hardy space on the open
unit disk and use them to study composition operators on Hardy–Smirnov spaces.
Submitted: January 30, 2007. Revised: June 19, 2007. Accepted: July 11, 2007. 相似文献
3.
Andrzej Rozkosz 《Probability Theory and Related Fields》2003,125(3):393-407
We extend the definition of solutions of backward stochastic differential equations to the case where the driving process
is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of
solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic
partial differential equations in Sobolev spaces.
Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002
Research supported by KBN Grant 0253 P03 2000 19.
Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55
Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution 相似文献
4.
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
5.
The limit as N → ∞ of the eigenvalue correlation function is studied in a neighborhood of zero for N × N Hermitian matrices
chosen at random from the Hilbert-Schmidt sphere of an appropriate radius. Dyson’s famous sin π(t1 − t2)/π(t1 − t2)-kernel asymptotic is extended to this class of matrix ensembles. Bibliography: 9 titles.
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 341, 2007, pp. 68–80. 相似文献
6.
We study Lipschitz contraction properties of general Markov kernels seen as operators on spaces of probability measures equipped
with entropy-like ``distances'. Universal quantitative bounds on the associated ergodic constants are deduced from Dobrushin's
ergodic coefficient. Strong contraction properties in Orlicz spaces for relative densities are proved under more restrictive
mixing assumptions. We also describe contraction bounds in the entropy sense around arbitrary probability measures by introducing
a suitable Dirichlet form and the corresponding modified logarithmic Sobolev constants. The interest in these bounds is illustrated
on the example of inhomogeneous Gaussian chains. In particular, the existence of an invariant measure is not required in general.
Received: 31 October 2000 / Revised version: 21 February 2003 /
Published online: 12 May 2003
L. Miclo also thanks the hospitality and support of the Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brasil,
where part of this work was done.
Mathematics Subject Classification (2000): 60J05, 60J22, 37A30, 37A25, 39A11, 39A12, 46E39, 28A33, 47D07
Key words or phrases: Lipschitz contraction – Generalized relative entropy – Markov kernel – Dobrushin's ergodic coefficient – Orlicz norm – Dirichlet
form – Spectral gap – Modified logarithmic Sobolev inequality – Inhomogeneous Gaussian chains – Loose of memory property 相似文献
7.
In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at
the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=〈φ,S
φ〉, through a function ΔS
t
of t, i.e., ΔS[φ]≥ΔS
t
, for all physical states φ with 〈φ,S
φ〉=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric
rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint
extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions
and the function ΔS
t
. We also discuss potential applications in quantum and classical information theory.
相似文献
8.
Gui Qiao XU Yong Sheng SUN Yong Ping LIU 《数学学报(英文版)》2006,22(6):1667-1678
In this paper, we consider the n-widths and average widths of Besov classes in the usual Sobolev spaces. The weak asymptotic results concerning the Kolmogorov n-widths, the linear n-widths, the Gel'fand n-widths, in the Sobolev spaces on T^d, and the infinite-dimensional widths and the average widths in the Sobolev spaces on Ra are obtained, respectively. 相似文献
9.
A. A. Makarov 《Journal of Mathematical Sciences》2009,156(4):617-631
We consider approximate relations in the form of a system of linear algebraic equations that yield B
φ
-splines. We construct Lagrange type splines of the first order and give examples of polynomial, trigonometric, hyperbolic,
and exponential B
φ
-splines. We also construct a system of linear functionals biorthogonal to the B
φ
-splines and resolve an interpolation problem generated by this system. For refined nonuniform grids we establish an embedding
of spaces of B
φ
-splines. The decomposition and reconstruction formulas are obtained. Bibliography: 20 titles.
Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 47–60. 相似文献
10.
Rosalba Di Vincenzo 《Mediterranean Journal of Mathematics》2007,4(1):119-126
Strong solvability in Sobolev spaces is proved for a unilateral boundary value problem for nonlinear parabolic operators.
The operator is assumed to be of Carathéodory type and to satisfy a suitable ellipticity condition; only measurability with
respect to the independent variable X is required.
The main tools of the proof are an estimate for the second derivatives of functions which satisfy the unilateral boundary
conditions and the monotonicity of the operator − u
t
with respect to Δu for the same functions. 相似文献
11.
We present a definition of general Sobolev spaces with respect to arbitrary measures, Wh,p (Ω,μ) for 1 ≤p≤∞. In [RARP] we proved that these spaces are complete under very light conditions. Now we prove that if we consider certain general types of measures, then Cc∞ (R) is dense in thee spaces. As an application to Sobolev orthogonal polynomials, we study the boundedness of the multiplication operator. This gives an estimation of the zeroes of Sobolev orthogonal polynomials. 相似文献
12.
In this paper we investigate some algebra properties of dual Toeplitz operators on the orthogonal complement of the Dirichlet space in the Sobolev space. We completely characterize commuting dual Toeplitz operators with harmonic symbols, and show that a dual Toeplitz operator commutes with a nonconstant analytic dual Toeplitz operator if and only if its symbol is analytic. We also obtain the sufficient and necessary conditions on the harmonic symbols for SφSφψ= Sφψ. 相似文献
13.
Let (ℋ
t
)
t≥0 be the Ornstein-Uhlenbeck semigroup on ℝ
d
with covariance matrix I and drift matrix −λ(I+R), where λ>0 and R is a skew-adjoint matrix and denote by γ
∞ the invariant measure for (ℋ
t
)
t≥0. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L
2(γ
∞). We investigate the weak type 1 estimate of the Riesz transforms for (ℋ
t
)
t≥0. We prove that if the matrix R generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type 1 with respect
to the invariant measure γ
∞. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on
L
p
(γ
∞) if 1<p<∞.
The authors have received support by the Italian MIUR-PRIN 2005 project “Harmonic Analysis” and by the EU IHP 2002-2006 project
“HARP”. 相似文献
14.
B. I. Peleshenko 《Ukrainian Mathematical Journal》2005,57(11):1741-1762
We prove theorems on interpolation of quasilinear operators of weak type (ϕ0, ψ0, ϕ0, ψ1) in Lorentz spaces. The operators under study are analogs of the Calderón operator and the Benett operator for concave and
convex functions ϕ0(t), ψ0(t), ϕ1(t), and ψ1(t).
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 57, No. 11, pp. 1490–1507, November, 2005. 相似文献
15.
A. D. Baranov 《Journal of Mathematical Sciences》2000,101(2):2881-2913
The boundedness conditions for the differentiation operator in Hilbert spaces of entire functions (Branges spaces) and conditions
under which the embedding Kи⊂L2(μ) holds in spaces Kи associated with the Branges spacesH(E) are studied. Measure μ such that the above embedding is isometric are of special interest. It turns out that the condition
E'/E∈H∞(C+) is sufficient for the boundedness of the differentiation operator inH(E). Under certain restrictions on E, this condition is also necessary. However, this fact fails in the general case, which is
demonstrated by the counterexamples constructed in this paper. The convex structure of the set of measures μ such that the
embedding KE
*
/E⊂L2(μ) is isometric (the set of such measures was described by de Brages) is considered. Some classes of measures that are extreme
points in the set of Branges measures are distinguished. Examples of measures that are not extreme points are also given.
Bibliography: 7 titles.
Translated fromProblemy Matematicheskogo Analiza, No. 19, 1999, pp. 27–68. 相似文献
16.
We analyze a class of quasilinear elliptic problems involving a p(·)-Laplace-type operator on a bounded domain
W ì \mathbb RN{\Omega\subset{\mathbb R}^N}, N ≥ 2, and we deal with nonlinear conditions on the boundary. Working on the variable exponent Lebesgue–Sobolev spaces, we
follow the steps described by the “fountain theorem” and we establish the existence of a sequence of weak solutions. 相似文献
17.
N. G. Khoma 《Ukrainian Mathematical Journal》1998,50(12):1917-1923
In three spaces, we find exact classical solutions of the boundary-value periodic problem utt - a2uxx = g(x, t) u(0, t) = u(π, t) = 0, u(x, t + T) = u(x, t), x ∈ ℝ, t ∈ ℝ. We study the periodic boundary-value problem for a quasilinear equation whose left-hand side is the d’Alembert operator
and whose right-hand side is a nonlinear operator.
Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1680–1685, December, 1998. 相似文献
18.
Wu Liming 《Probability Theory and Related Fields》1999,114(4):549-585
Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C
∞
0(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ
d
such that ∇φ∈L
loc
2(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L
1(D, φ2
dx)? (ii) Whether the semigroup determined by ℒ is strong Feller?
Received: 21 October 1997 / Revised version: 3 September 1998 相似文献
19.
Richard A. Alò Andrè de Korvin Charles Roberts 《Annali di Matematica Pura ed Applicata》1977,112(1):33-48
Summary Under study is the existence of averaging operators determined by measurable maps φ from a measure space (S, Σ, μ) into an
arbitrary Hausdorff topological space T. The map φ induces a continuous map φe from the space Cb(T) into the normed (Banach) function space Lϱ = Lϱ(S, Σ, μ) defined by φe(f)=foφ for all f ε Cb(T). An integral representation for such operators is first studied. The existence is then determined by the existence of
an averaging operator U1 for the restriction of φ to a certain measurable subset B1 of S. Utilizing a representation of Lϱ(S, Σ, μ) as a Banach function space over a compact extremally disconnected Hausdorff space Ŝ, we are able to give a definition
for the concept of plural points and irreducible map. A significant upper bound is given for the operator U1. Finally conditions are considered under which no bounded projection from Lϱ onto the range of φe may exist. From a topological point of view the development is pursued in a general setting. Averaging operators have recently
been used for the study of injective Banach spaces of the type Cb(T) and in non-linear prediction and approximation theory relative to Tshebyshev subspaces of Lϱ.
Entrata in Redazione l’ll settembre 1975. 相似文献
20.
Let (ℋ
t
)
t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ
d
with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ
∞ the invariant measure for (ℋ
t
)
t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L
2(γ
∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ*
f(x)=sup
t≥o
|ℋ
t
f(x)| is of weak type 1 with respect to the invariant measure γ
∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L
p
(γ
∞) if and only if 1<p≤∞.
相似文献