scholarly journals Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

2011 ◽  
Vol 203 ◽  
pp. 109-122 ◽  
Author(s):  
Bui The Anh
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Upper Bounds ◽  
Space Of Homogeneous Type ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Suitable Function

AbstractLet L be a nonnegative self-adjoint operator on L2 (X), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e–tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L) is bounded on for 0 < p < 1, the Hardy space associated to operator L, when F is a suitable function.

scholarly journals Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

2011 ◽  
Vol 203 ◽  
pp. 109-122
Author(s):  
Bui The Anh
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Upper Bounds ◽  
Space Of Homogeneous Type ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Suitable Function

AbstractLetLbe a nonnegative self-adjoint operator onL2(X), whereXis a space of homogeneous type. Assume thatLgenerates an analytic semigroupe–tlwhose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplierF(L) is bounded onfor 0&lt; p&lt; 1, the Hardy space associated to operatorL, whenFis a suitable function.

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.

scholarly journals Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications

2020 ◽  
Author(s):  
Peng Chen ◽  
Xuan Thinh Duong ◽  
Liangchuan Wu ◽  
Lixin Yan
Keyword(s):  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Square Function ◽  
Weighted Norm ◽  
Square Functions ◽  
Spectral Multipliers ◽  
Eigenvalue Estimates ◽  
Preservation Condition ◽  
Square Integrability ◽  
The Laplace Operator

Abstract Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder continuity in $x$, but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (1) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

scholarly journals Boundedness of Some Paraproducts on Spaces of Homogeneous Type

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2591
Author(s):  
Xing Fu
Keyword(s):  
Exponential Decay ◽  
Hardy Spaces ◽  
Summation Formula ◽  
Space Of Homogeneous Type ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Doubling Condition ◽  
J 13 ◽  
Abel Summation

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this article, the author develops a partial theory of paraproducts {Πj}j=13 defined via approximations of the identity with exponential decay (and integration 1), which are extensions of paraproducts defined via regular wavelets. Precisely, the author first obtains the boundedness of Π3 on Hardy spaces and then, via the methods of interpolation and the well-known T(1) theorem, establishes the endpoint estimates for {Πj}j=13. The main novelty of this paper is the application of the Abel summation formula to the establishment of some relations among the boundedness of {Πj}j=13, which has independent interests. It is also remarked that, throughout this article, μ is not assumed to satisfy the reverse doubling condition.

scholarly journals Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

2014 ◽  
Vol 216 ◽  
pp. 71-110 ◽  
Author(s):  
Tri Dung Tran
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Orlicz Function ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Divergence Form ◽  
Measurable Coefficients ◽  
Nirenberg Inequality

AbstractLet L be a divergence form elliptic operator with complex bounded measurable coefficients, let ω be a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-type pω ∈ (0, 1], and let ρ(x,t) = t−1/ω−1 (x,t−1) for x ∈ ℝn, t ∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy space Hω,L(ℝn) and its dual space BMOρ,L* (ℝ n), where L* denotes the adjoint operator of L in L2 (ℝ n). The ρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L (ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2 and the Littlewood–Paley g-function gL map Hω,L(ℝn) continuously into L(ω).

scholarly journals Musielak–Orlicz Hardy Spaces Associated with Divergence Form Elliptic Operators Without Weight Assumptions

2014 ◽  
Vol 216 ◽  
pp. 71-110
Author(s):  
Tri Dung Tran
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Carleson Measure ◽  
Dual Space ◽  
Orlicz Function ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Divergence Form ◽  
Measurable Coefficients ◽  
Nirenberg Inequality

AbstractLetLbe a divergence form elliptic operator with complex bounded measurable coefficients, letωbe a positive Musielak-Orlicz function on (0, ∞) of uniformly strictly critical lower-typepω∈ (0, 1], and letρ(x,t) = t−1/ω−1(x,t−1) forx∈ ℝn, t∊ (0, ∞). In this paper, we study the Musielak-Orlicz Hardy spaceHω,L(ℝn) and its dual space BMOρ,L* (ℝn), whereL*denotes the adjoint operator ofLinL2(ℝn). Theρ-Carleson measure characterization and the John-Nirenberg inequality for the space BMOρ,L(ℝn) are also established. Finally, as applications, we show that the Riesz transform ∇L−1/2and the Littlewood–Paleyg-functiongLmapHω,L(ℝn) continuously intoL(ω).

scholarly journals Discrete Paraproduct Operators on Variable Hardy Spaces

2019 ◽  
Vol 63 (2) ◽  
pp. 304-317 ◽  
Author(s):  
Jian Tan
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Variable Exponent ◽  
Discrete Version ◽  
Variable Exponents ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Hölder Continuous ◽  
Analysis Techniques ◽  
Calderón’S Reproducing Formula

AbstractLet$p(\cdot ):\mathbb{R}^{n}\rightarrow (0,\infty )$be a variable exponent function satisfying the globally log-Hölder continuous condition. In this paper, we obtain the boundedness of paraproduct operators$\unicode[STIX]{x1D70B}_{b}$on variable Hardy spaces$H^{p(\cdot )}(\mathbb{R}^{n})$, where$b\in \text{BMO}(\mathbb{R}^{n})$. As an application, we show that non-convolution type Calderón–Zygmund operators$T$are bounded on$H^{p(\cdot )}(\mathbb{R}^{n})$if and only if$T^{\ast }1=0$, where$\frac{n}{n+\unicode[STIX]{x1D716}}<\text{ess inf}_{x\in \mathbb{R}^{n}}p\leqslant \text{ess sup}_{x\in \mathbb{R}^{n}}p\leqslant 1$and$\unicode[STIX]{x1D716}$is the regular exponent of kernel of$T$. Our approach relies on the discrete version of Calderón’s reproducing formula, discrete Littlewood–Paley–Stein theory, almost orthogonal estimates, and variable exponents analysis techniques. These results still hold for variable Hardy space on spaces of homogeneous type by using our methods.

LITTLEWOOD–PALEY CHARACTERIZATION OF WEIGHTED HARDY SPACES ASSOCIATED WITH OPERATORS

2016 ◽  
Vol 103 (2) ◽  
pp. 250-267 ◽  
Author(s):  
GUORONG HU
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Adjoint Operator ◽  
Weighted Hardy Space ◽  
Boundedness Condition ◽  
Weighted Hardy Spaces ◽  
Lusin Area Function ◽  
Extra Assumption ◽  
Type Decomposition

Let$(X,d,\unicode[STIX]{x1D707})$be a metric measure space endowed with a distance$d$and a nonnegative, Borel, doubling measure$\unicode[STIX]{x1D707}$. Let$L$be a nonnegative self-adjoint operator on$L^{2}(X)$. Assume that the (heat) kernel associated to the semigroup$e^{-tL}$satisfies a Gaussian upper bound. In this paper, we prove that for any$p\in (0,\infty )$and$w\in A_{\infty }$, the weighted Hardy space$H_{L,S,w}^{p}(X)$associated with$L$in terms of the Lusin (area) function and the weighted Hardy space$H_{L,G,w}^{p}(X)$associated with$L$in terms of the Littlewood–Paley function coincide and their norms are equivalent. This improves a recent result of Duonget al.[‘A Littlewood–Paley type decomposition and weighted Hardy spaces associated with operators’,J. Geom. Anal.26(2016), 1617–1646], who proved that$H_{L,S,w}^{p}(X)=H_{L,G,w}^{p}(X)$for$p\in (0,1]$and$w\in A_{\infty }$by imposing an extra assumption of a Moser-type boundedness condition on$L$. Our result is new even in the unweighted setting, that is, when$w\equiv 1$.

New variable martingale Hardy spaces

2021 ◽  
pp. 1-29
Author(s):  
Yong Jiao ◽  
Dan Zeng ◽  
Dejian Zhou
Keyword(s):  
Brownian Motion ◽  
Hardy Space ◽  
Hardy Spaces ◽  
Stochastic Integral ◽  
Lebesgue Spaces ◽  
Type Space ◽  
Variable Lebesgue Spaces ◽  
Martingale Inequalities

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

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