MAXIMAL HARDY SPACES ASSOCIATED TO NONNEGATIVE SELF-ADJOINT OPERATORS

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.

scholarly journals Spectral Multipliers of Self-Adjoint Operators on Besov and Triebel–Lizorkin Spaces Associated to Operators

2019 ◽  
Author(s):  
The Anh Bui ◽  
Xuan Thinh Duong
Keyword(s):  
Heat Kernel ◽  
Hardy Spaces ◽  
Adjoint Operator ◽  
Space Of Homogeneous Type ◽  
Multiplier Theorem ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Gaussian Estimate ◽  
Adjoint Operators

Abstract Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS

2020 ◽  
pp. 1-25
Author(s):  
HONG CHUONG DOAN
Keyword(s):  
Heat Kernel ◽  
Maximal Function ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Weak Type ◽  
Adjoint Operator ◽  
Beltrami Operator ◽  
Maximal Functions ◽  
Heat Semigroup ◽  
Doubling Condition

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$ . Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

Hardy–Orlicz Spaces of Dirichlet Series: An Interpolation Problem on Abscissae of Convergence

2019 ◽  
Author(s):  
Maxime Bailleul ◽  
Pascal Lefèvre ◽  
Luis Rodríguez-Piazza
Keyword(s):  
Dirichlet Series ◽  
Hardy Spaces ◽  
Upper Bound ◽  
Interpolation Problem ◽  
Orlicz Spaces ◽  
Elementary Method ◽  
Abscissa Of Convergence ◽  
Upper Bound Estimate

Abstract The study of Hardy spaces of Dirichlet series denoted by $\mathscr{H}^p$ ($p\geq 1$) was initiated in [7] when $p=2$ and $p=\infty $, and in [2] for the general case. In this paper we introduce the Orlicz version of spaces of Dirichlet series $\mathscr{H}^\psi $. We focus on the case $\psi =\psi _q(t)=\exp (t^q)-1,$ and we compute the abscissa of convergence for these spaces. It turns out that its value is $\min \{1/q\,,1/2\}$ filling the gap between the case $\mathscr{H}^\infty $, where the abscissa is equal to $0$, and the case $\mathscr{H}^p$ for $p$ finite, where the abscissa is equal to $1/2$. The upper-bound estimate relies on an elementary method that applies to many spaces of Dirichlet series. This answers a question raised by Hedenmalm in [6].

Heat kernel and Lipschitz–Besov spaces

2015 ◽  
Vol 27 (6) ◽  
Author(s):  
Alexander Grigor'yan ◽  
Liguang Liu
Keyword(s):  
Heat Kernel ◽  
Besov Spaces ◽  
Measure Space ◽  
Metric Measure Space

AbstractOn a metric measure space (

scholarly journals The Davies method revisited for heat kernel upper bounds of regular Dirichlet forms on metric measure spaces

2018 ◽  
Vol 30 (5) ◽  
pp. 1129-1155 ◽  
Author(s):  
Jiaxin Hu ◽  
Xuliang Li
Keyword(s):  
Heat Kernel ◽  
Upper Bound ◽  
Dirichlet Form ◽  
Measure Space ◽  
Dirichlet Forms ◽  
Upper Bounds ◽  
Metric Measure Spaces ◽  
Unified Approach ◽  
Measure Spaces ◽  
Non Local

AbstractWe apply the Davies method to prove that for any regular Dirichlet form on a metric measure space, an off-diagonal stable-like upper bound of the heat kernel is equivalent to the conjunction of the on-diagonal upper bound, a cutoff inequality on any two concentric balls, and the jump kernel upper bound, for any walk dimension. If in addition the jump kernel vanishes, that is, if the Dirichlet form is strongly local, we obtain a sub-Gaussian upper bound. This gives a unified approach to obtaining heat kernel upper bounds for both the non-local and the local Dirichlet forms.

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

2020 ◽  
Vol 8 (1) ◽  
pp. 418-429
Author(s):  
Athanasios G. Georgiadis ◽  
George Kyriazis
Keyword(s):  
Heat Kernel ◽  
Riemannian Manifolds ◽  
Besov Spaces ◽  
Measure Space ◽  
General Framework ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Embedding Theorems ◽  
Euclidean Case ◽  
Volume Property

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.

M. A. Krasnosel'skii Theorem and Iterative Methods for Solving Ill-Posed Linear Problems with a Self-Adjoint Operator

2015 ◽  
Vol 15 (3) ◽  
pp. 373-389
Author(s):  
Oleg Matysik ◽  
Petr Zabreiko
Keyword(s):  
Hilbert Space ◽  
Iterative Methods ◽  
Critical Case ◽  
Adjoint Operator ◽  
Successive Approximations ◽  
Operator Equations ◽  
Ill Posed ◽  
Linear Problems ◽  
Adjoint Operators ◽  
Linear Operator Equations

AbstractThe paper deals with iterative methods for solving linear operator equations ${x = Bx + f}$ and ${Ax = f}$ with self-adjoint operators in Hilbert space X in the critical case when ${\rho (B) = 1}$ and ${0 \in \operatorname{Sp} A}$. The results obtained are based on a theorem by M. A. Krasnosel'skii on the convergence of the successive approximations, their modifications and refinements.

scholarly journals Second Hankel determinant for certain class of bi-univalent functions defined by Chebyshev polynomials

2019 ◽  
Vol 12 (02) ◽  
pp. 1950017
Author(s):  
H. Orhan ◽  
N. Magesh ◽  
V. K. Balaji
Keyword(s):  
Unit Disk ◽  
Upper Bound ◽  
Chebyshev Polynomials ◽  
Univalent Functions ◽  
Function Class ◽  
Open Unit ◽  
Open Unit Disk ◽  
Hankel Determinant ◽  
Second Hankel Determinant ◽  
Upper Bound Estimate

In this work, we obtain an upper bound estimate for the second Hankel determinant of a subclass [Formula: see text] of analytic bi-univalent function class [Formula: see text] which is associated with Chebyshev polynomials in the open unit disk.

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