Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators

Abstract We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces are proved as well. Our result generalize the Euclidean case and are new for many settings of independent interest such as the ball, the interval and Riemannian manifolds.

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Heat kernel and Lipschitz–Besov spaces

Forum Mathematicum ◽

10.1515/forum-2014-0034 ◽

2015 ◽

Vol 27 (6) ◽

Cited By ~ 4

Author(s):

Alexander Grigor'yan ◽

Liguang Liu

Keyword(s):

Heat Kernel ◽

Besov Spaces ◽

Measure Space ◽

Metric Measure Space

AbstractOn a metric measure space (

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MAXIMAL HARDY SPACES ASSOCIATED TO NONNEGATIVE SELF-ADJOINT OPERATORS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972714001105 ◽

2014 ◽

Vol 91 (2) ◽

pp. 286-302

Author(s):

GUORONG HU

Keyword(s):

Heat Kernel ◽

Hardy Spaces ◽

Upper Bound ◽

Measure Space ◽

Adjoint Operator ◽

Maximal Functions ◽

Metric Measure Space ◽

Holder Continuity ◽

Upper Bound Estimate ◽

Adjoint Operators

AbstractLet $(X,d,{\it\mu})$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions. Let $\mathscr{L}$ be a nonnegative self-adjoint operator on $L^{2}(X,d{\it\mu})$ satisfying a pointwise Gaussian upper bound estimate and Hölder continuity for its heat kernel. In this paper, we introduce the Hardy spaces $H_{\mathscr{L}}^{p}(X)$, $0<p\leq 1$, associated to $\mathscr{L}$ in terms of grand maximal functions and show that these spaces are equivalently characterised by radial and nontangential maximal functions.

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DIAGONALIZATION MODULO NORM IDEALS AND HAUSDORFF DIMENSIONALITY

International Journal of Mathematics ◽

10.1142/s0129167x00000520 ◽

2000 ◽

Vol 11 (08) ◽

pp. 1057-1078

Author(s):

JINGBO XIA

Keyword(s):

Hausdorff Measure ◽

Metric Spaces ◽

Adjoint Operator ◽

Trace Class ◽

Multiplication Operators ◽

Von Neumann ◽

Dimensional Hausdorff Measure ◽

One Dimensional ◽

Compact Metric Spaces ◽

The One

Kuroda's version of the Weyl-von Neumann theorem asserts that, given any norm ideal [Formula: see text] not contained in the trace class [Formula: see text], every self-adjoint operator A admits the decomposition A=D+K, where D is a self-adjoint diagonal operator and [Formula: see text]. We extend this theorem to the setting of multiplication operators on compact metric spaces (X, d). We show that if μ is a regular Borel measure on X which has a σ-finite one-dimensional Hausdorff measure, then the family {Mf:f∈ Lip (X)} of multiplication operators on T2(X, μ) can be simultaneously diagonalized modulo any [Formula: see text]. Because the condition [Formula: see text] in general cannot be dropped (Kato-Rosenblum theorem), this establishes a special relation between [Formula: see text] and the one-dimensional Hausdorff measure. The main result of the paper is that such a relation breaks down in Hausdorff dimensions p>1.

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UNIQUENESS THEOREMS FOR HARMONIC MAPS INTO METRIC SPACES

Communications in Contemporary Mathematics ◽

10.1142/s0219199702000828 ◽

2002 ◽

Vol 04 (04) ◽

pp. 725-750 ◽

Cited By ~ 2

Author(s):

CHIKAKO MESE

Keyword(s):

Riemannian Manifolds ◽

Metric Space ◽

Metric Spaces ◽

Harmonic Maps ◽

Uniqueness Theorems ◽

Singular Spaces ◽

Domain Space ◽

Recent Developments

Recent developments extend much of the known theory of classical harmonic maps between smooth Riemannian manifolds to the case when the target is a metric space of curvature bounded from above. In particular, the existence and regularity theorems for harmonic maps into these singular spaces have been successfully generalized. Furthermore, the uniqueness of harmonic maps is known when the domain has a boundary (with a smallness of image condition if the target curvature is bounded from above by a positive number). In this paper, we will address the question of uniqueness when the domain space is without a boundary in two cases: one, when the curvature of the target is strictly negative and two, for a map between surfaces with nonpositive target curvature.

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Riesz triple probabilisitic of almost lacunary ces$\acute{A}$ro $C_{111}$ statistical convergence of $\chi^{3}$ defined by a Musielak Orlicz function

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v36i4.32870 ◽

2018 ◽

Vol 36 (4) ◽

pp. 23-32 ◽

Cited By ~ 1

Author(s):

Dr Vandana ◽

_ Deepmala ◽

N. Subramanian ◽

Vishnu Narayan Mishra

Keyword(s):

Functional Analysis ◽

General Framework ◽

Statistical Convergence ◽

Metric Spaces ◽

Orlicz Function ◽

The Subject

In this paper we study the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{3}$ over probabilistic $p-$ metric spaces defined by Musielak Orlicz function. Since the study of convergence in PP-spaces is fundamental to probabilistic functional analysis, we feel that the concept of almost lacunary statistical Ces$\acute{a}$ro of $\chi^{2}$ over probabilistic $p-$ metric spaces defined by Musielak in a PP-space would provide a more general framework for the subject.

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