A finiteness theorem in the Bruhat-Tits building: an application of Landvogt's embedding theorem

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine hyperplane, an affine span, an affine map, and an affine transformation. It then considers a notation stating that the convex closure of a subset a of X is the intersection of all convex sets containing a and another notation that denotes by AGL(X) the group of all affine transformations of X and by Trans(X) the set of all translations of X. It also describes Euclidean spaces and assumes that the real vector space X is of finite dimension n and that d is a Euclidean metric on X. Finally, it discusses Euclidean representations and the standard metric.

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Embedding Theorems and Area Operators on Bergman Spaces with Doubling Measure

Complex Analysis and Operator Theory ◽

10.1007/s11785-021-01089-4 ◽

2021 ◽

Vol 15 (3) ◽

Author(s):

Changbao Pang ◽

Antti Perälä ◽

Maofa Wang

Keyword(s):

Borel Measure ◽

Bergman Spaces ◽

Embedding Theorem ◽

Weighted Bergman Spaces ◽

Embedding Theorems ◽

Positive Borel Measure ◽

Doubling Property ◽

Area Operator ◽

Special Cases ◽

Upper Half Plane

AbstractWe establish an embedding theorem for the weighted Bergman spaces induced by a positive Borel measure $$d\omega (y)dx$$ d ω ( y ) d x with the doubling property $$\omega (0,2t)\le C\omega (0,t)$$ ω ( 0 , 2 t ) ≤ C ω ( 0 , t ) . The characterization is given in terms of Carleson squares on the upper half-plane. As special cases, our result covers the standard weights and logarithmic weights. As an application, we also establish the boundedness of the area operator.

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$L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

Journal of Inequalities and Applications ◽

10.1186/s13660-021-02616-9 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Jing Li ◽

Shuxiang Feng ◽

Peibiao Zhao

Keyword(s):

Riemannian Manifold ◽

Riemannian Manifolds ◽

Schrödinger Operators ◽

Schrodinger Operators ◽

Finiteness Theorem ◽

Conformally Flat ◽

Locally Conformally Flat ◽

Flat Riemannian Manifold ◽

Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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Interpretability over peano arithmetic

Journal of Symbolic Logic ◽

10.2307/2586787 ◽

1999 ◽

Vol 64 (4) ◽

pp. 1407-1425

Author(s):

Claes Strannegård

Keyword(s):

Modal Logic ◽

Completeness Theorem ◽

Peano Arithmetic ◽

Embedding Theorem ◽

Compactness Theorem ◽

Partial Answer ◽

Recursively Enumerable ◽

Recursively Enumerable Sets ◽

Interpretability Logic

AbstractWe investigate the modal logic of interpretability over Peano arithmetic. Our main result is a compactness theorem that extends the arithmetical completeness theorem for the interpretability logic ILMω. This extension concerns recursively enumerable sets of formulas of interpretability logic (rather than single formulas). As corollaries we obtain a uniform arithmetical completeness theorem for the interpretability logic ILM and a partial answer to a question of Orey from 1961. After some simplifications, we also obtain Shavrukov's embedding theorem for Magari algebras (a.k.a. diagonalizable algebras).

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A note on a space Hp, a of holomorphic functions

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700013447 ◽

1987 ◽

Vol 35 (3) ◽

pp. 471-479

Author(s):

H. O. Kim ◽

S. M. Kim ◽

E. G. Kwon

Keyword(s):

Hardy Space ◽

Complex Plane ◽

Unit Disc ◽

Weak Type ◽

Holomorphic Functions ◽

Embedding Theorem ◽

Type Operator ◽

Taylor Coefficients ◽

Bounded Holomorphic

For 0 < p < ∞ and 0 ≤a; ≤ 1, we define a space Hp, a of holomorphic functions on the unit disc of the complex plane, for which Hp, 0 = H∞, the space of all bounded holomorphic functions, and Hp, 1 = Hp, the usual Hardy space. We introduce a weak type operator whose boundedness extends the well-known Hardy-Littlewood embedding theorem to Hp, a, give some results on the Taylor coefficients of the functions of Hp, a and show by an example that the inner factor cannot be divisible in Hp, a.

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