scholarly journals Heat Kernels on homogeneous spaces

2005 ◽  
Vol 78 (1) ◽  
pp. 109-147
Author(s):  
C. M. P. A. Smulders
Keyword(s):  
Infinitesimal Generator ◽  
Borel Measure ◽  
Homogeneous Spaces ◽  
Upper Bounds ◽  
Weak Condition ◽  
Weakly Continuous ◽  
Regular Borel Measure ◽  
Lie Subgroup ◽  
Complex Coefficients ◽  
Natural Way

AbstractLet a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

scholarly journals Uniform boundedness principles for regular Borel vector measures

1980 ◽  
Vol 29 (2) ◽  
pp. 206-218 ◽  
Author(s):  
Joseph Kupka
Keyword(s):  
Banach Space ◽  
Borel Measure ◽  
Borel Subset ◽  
Uniform Boundedness ◽  
Vector Measures ◽  
Boundedness Theorem ◽  
Regular Borel Measure ◽  
Pointwise Limit ◽  
Regular Open ◽  
Uniformly Bounded

The setting is a compact Hausfroff space ω. The notion of a Walls class of subsets of Ω is defined via strange axioms—axioms whose justification rests with examples such as the collection of regular open sets or the range of a strong lifting. Avarient of Rosenthal' famous lwmma which applies directly to Banach space-valued measures is esablished, and it is used to obtain, in elementary fashion, the following two uniform boundedness principles: (1)The Nikodym Boundedness Theorem. If K is a family of regular Borel vector measures on Ω which is point-wise bounded on every set of a fixed Wells class, then K is uniformly bounded. (2)The Nikodym Covergence Theorem. If {μn} is a sequence of regular Borel vector measures on Ω which is converguent on every set of a fixed Wells class, then the μn are uniformly countably additive, the sequence {μn} is convergent on every Borel subset of Ω and the pointwise limit constitutes a regular Borel measure.

scholarly journals Lifting, restricting and sifting integral points on affine homogeneous varieties

2012 ◽  
Vol 148 (6) ◽  
pp. 1695-1716 ◽  
Author(s):  
Alexander Gorodnik ◽  
Amos Nevo
Keyword(s):  
Homogeneous Spaces ◽  
Acta Math ◽  
Upper Bounds ◽  
Lattice Points ◽  
List Type ◽  
Integral Points ◽  
Counting Problem ◽  
Symmetric Varieties ◽  
Almost Prime ◽  
Homogeneous Varieties

AbstractIn [Gorodnik and Nevo,Counting lattice points, J. Reine Angew. Math.663(2012), 127–176] an effective solution of the lattice point counting problem in general domains in semisimpleS-algebraic groups and affine symmetric varieties was established. The method relies on the mean ergodic theorem for the action ofGonG/Γ, and implies uniformity in counting over families of lattice subgroups admitting a uniform spectral gap. In the present paper we extend some methods developed in [Nevo and Sarnak,Prime and almost prime integral points on principal homogeneous spaces, Acta Math.205(2010), 361–402] and use them to establish several useful consequences of this property, including:(1)effective upper bounds on lifting for solutions of congruences in affine homogeneous varieties;(2)effective upper bounds on the number of integral points on general subvarieties of semisimple group varieties;(3)effective lower bounds on the number of almost prime points on symmetric varieties;(4)effective upper bounds on almost prime solutions of congruences in homogeneous varieties.

On r*-Invariant Measure on a Locally Compact Semigroup with Recurrence

1980 ◽  
Vol 23 (2) ◽  
pp. 237-239
Author(s):  
Samuel Bourne
Keyword(s):  
Invariant Measure ◽  
Borel Measure ◽  
Compact Semigroup ◽  
Locally Compact ◽  
Borel Sets ◽  
Left Group ◽  
Locally Compact Semigroup ◽  
Regular Borel Measure

A regular Borel measure μ is said to be r*-invariant on a locally compact semigroup if μ(Ba-1) = μ(B) for all Borel sets B and points a of S, where Ba-1 ={xϵS, xaϵB}. In [1] Argabright conjectured that the support of an r*-invariant measure on a locally compact semigroup is a left group, Mukherjea and Tserpes [4] proved this conjecture in the case that the measure is finite; however their method of proof fails when the measure is infinite.

On the Ideal-Triangularizability of Semigroups of Quasinilpotent Positive Operators on C(K)

1998 ◽  
Vol 41 (3) ◽  
pp. 298-305
Author(s):  
M. T. Jahandideh
Keyword(s):  
Banach Lattice ◽  
Compact Hausdorff Space ◽  
Borel Measure ◽  
Hausdorff Space ◽  
Lower Semicontinuous ◽  
Integral Operators ◽  
Positive Operators ◽  
Locally Compact ◽  
Regular Borel Measure ◽  
The Ideal

AbstractIt is known that a semigroup of quasinilpotent integral operators, with positive lower semicontinuous kernels, on L2(X, μ), where X is a locally compact Hausdorff-Lindelöf space and μ is a σ-finite regular Borel measure on X, is triangularizable. In this article we use the Banach lattice version of triangularizability to establish the ideal-triangularizability of a semigroup of positive quasinilpotent integral operators on C(K) where K is a compact Hausdorff space.

Boundary trace of the solutions of the prescribed Gaussian curvature equation

2000 ◽  
Vol 130 (3) ◽  
pp. 527-560 ◽  
Author(s):  
Michele Grillot ◽  
Laurent Véron
Keyword(s):  
Gaussian Curvature ◽  
Radon Measure ◽  
Borel Measure ◽  
Sufficient Conditions ◽  
Open Ball ◽  
Curvature Equation ◽  
Borel Measures ◽  
Boundary Trace ◽  
Regular Borel Measure ◽  
Prescribed Gaussian Curvature

We study the existence of a boundary trace for minorized solutions of the equation Δu + K (x) e2u = 0 in the unit open ball B2 of R2. We prove that this trace is an outer regular Borel measure on ∂B2, not necessarily a Radon measure. We give sufficient conditions on Borel measures on ∂B2 so that they are the boundary trace of a solution of the above equation. We also give boundary removability results in terms of generalized Bessel capacities.

Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups

2006 ◽  
Vol 58 (4) ◽  
pp. 691-725 ◽  
Author(s):  
A. Bendikov ◽  
L. Saloff-Coste
Keyword(s):  
Continuous Function ◽  
Smooth Function ◽  
Infinitesimal Generator ◽  
Borel Measure ◽  
Compact Groups ◽  
Absolutely Continuous ◽  
Infinite Dimensional ◽  
Continuous Density ◽  
Open Set ◽  
Gaussian Convolution

AbstractOn a compact connected group G, consider the infinitesimal generator –L of a central symmetric Gaussian convolution semigroup (μt)t>0. Using appropriate notions of distribution and smooth function spaces, we prove that L is hypoelliptic if and only if (μt)t>0 is absolutely continuous with respect to Haar measure and admits a continuous density x ⟼ μt (x), t > 0, such that limt!0t log μt(e) = 0. In particular, this condition holds if and only if any Borel measure u which is solution of Lu = 0 in an open set Ω can be represented by a continuous function in Ω. Examples are discussed.

scholarly journals On Lebesgue-type decompositions for Banach algebras

1970 ◽  
Vol 3 (1) ◽  
pp. 39-47
Author(s):  
Howard Anton
Keyword(s):  
Banach Algebra ◽  
Maximal Ideal ◽  
Borel Measure ◽  
Banach Algebras ◽  
Maximal Ideal Space ◽  
Ideal Space ◽  
Borel Sets ◽  
Direct Sum ◽  
Gelfand Transform ◽  
Regular Borel Measure

If the maximal ideal space of a commutative complex unitary Banach algebra, X, is equipped with a nonnegative, finite, regular Borel measure, m, then for each element, x, in X, a. complex regular Borel measure, mx, can be obtained by integrating the Gelfand transform of x with respect to m over the Borel sets. This paper considers the possibility of direct sum decompositions of the form X = Ax ⊕ Px where Ax = {z ε X: mz ≪ mx} and Px = {z ε X: mz ┴ mx}.

scholarly journals COMPACT TOEPLITZ OPERATORS WITH CONTINUOUS SYMBOLS

2009 ◽  
Vol 51 (2) ◽  
pp. 257-261 ◽  
Author(s):  
TRIEU LE
Keyword(s):  
Hardy Space ◽  
Unit Ball ◽  
Unit Sphere ◽  
Orthogonal Projection ◽  
Toeplitz Operators ◽  
Borel Measure ◽  
Borel Function ◽  
Closed Unit Ball ◽  
Rotation Invariant ◽  
Regular Borel Measure

AbstractFor any rotation-invariant positive regular Borel measure ν on the closed unit ball $\overline{\mathbb{B}}_n$ whose support contains the unit sphere $\mathbb{S}_n$, let L2a be the closure in L2 = L2($\overline{\mathbb{B}}_n, dν) of all analytic polynomials. For a bounded Borel function f on $\overline{\mathbb{B}}_n$, the Toeplitz operator Tf is defined by Tf(ϕ) = P(fϕ) for ϕ ∈ L2a, where P is the orthogonal projection from L2 onto L2a. We show that if f is continuous on $\overline{\mathbb{B}}_n$, then Tf is compact if and only if f(z) = 0 for all z on the unit sphere. This is well known when L2a is replaced by the classical Bergman or Hardy space.

On the L1-algebras of some compact totally ordered spaces

1997 ◽  
Vol 122 (1) ◽  
pp. 173-184 ◽  
Author(s):  
ANAHITA SAGHAFI
Keyword(s):  
Banach Algebra ◽  
Borel Measure ◽  
Integrable Functions ◽  
Regular Borel Measure ◽  
Ordered Space ◽  
Second Dual

Let X be a compact totally ordered space made into a semigroup by the multiplication xy=max{x, y}. Suppose that there is a continuous regular Borel measure μ on X with supp μ=X. Then the space L1(μ) of μ-integrable functions becomes a Banach algebra when provided with convolution as multiplication. The second dual L1(μ)** therefore has two Arens multiplications, each making it a Banach algebra. We shall always consider L1(μ)** to have the first of these: if F, G∈L1(μ)** and F=w*−limi ϕi, G=w*−limj ψj, where (ϕi), (ψj) are bounded nets in L1(μ), thenformula here

Compact embedding results of Sobolev spaces and positive solutions to an elliptic equation

2016 ◽  
Vol 146 (4) ◽  
pp. 693-721 ◽  
Author(s):  
Qi Han
Keyword(s):  
Elliptic Equation ◽  
Sobolev Space ◽  
Positive Solutions ◽  
Borel Measure ◽  
Nonlinear Term ◽  
Compact Embedding ◽  
Proper Subspace ◽  
Superlinear Growth ◽  
Regular Borel Measure ◽  
The Laplace Operator

Using a regular Borel measure μ ⩾ 0 we derive a proper subspace of the commonly used Sobolev space D1(ℝN) when N ⩾ 3. The space resembles the standard Sobolev space H1(Ω) when Ω is a bounded region with a compact Lipschitz boundary ∂Ω. An equivalence characterization and an example are provided that guarantee that is compactly embedded into L1(RN). In addition, as an application we prove an existence result of positive solutions to an elliptic equation in ℝN that involves the Laplace operator with the critical Sobolev nonlinearity, or with a general nonlinear term that has a subcritical and superlinear growth. We also briefly discuss the compact embedding of to Lp(ℝN) when N ⩾ 2 and 2 ⩽ p ⩽ N.

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