scholarly journals Local Riesz Transform and Local Hardy Spaces on Riemannian Manifolds with Bounded Geometry

2022 ◽  
Vol 32 (2) ◽  
Author(s):  
Stefano Meda ◽  
Giona Veronelli
Keyword(s):  
Riemannian Manifolds ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Bounded Geometry ◽  
Local Hardy Spaces

scholarly journals Local Hardy Spaces of Differential Forms on Riemannian Manifolds

2011 ◽  
Vol 23 (1) ◽  
pp. 106-169 ◽  
Author(s):  
Andrea Carbonaro ◽  
Alan McIntosh ◽  
Andrew J. Morris
Keyword(s):  
Riemannian Manifolds ◽  
Hardy Spaces ◽  
Differential Forms ◽  
Local Hardy Spaces

scholarly journals BOUNDEDNESS OF FOURIER INTEGRAL OPERATORS ON HARDY SPACES

2008 ◽  
Vol 51 (2) ◽  
pp. 443-463 ◽  
Author(s):  
Marco M. Peloso ◽  
Silvia Secco
Keyword(s):  
Hardy Spaces ◽  
Bounded Operator ◽  
Integral Operators ◽  
Fourier Integral ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Compact Set ◽  
Radon Transforms ◽  
Bounded Geometry ◽  
Local Hardy Spaces

AbstractFor $0\ltp\le1$, let $h^p(\mathbb{R}^n)$ denote the local Hardy space. Let $\mathcal{F}$ be a Fourier integral operator defined by the oscillatory integral$$ \mathcal{F}f(x)=\iint_{\mathbb{R}^{2n}}\exp(2\pi\mathrm{i}(\phi(x,\xi)-y\cdot\xi))b(x,y,\xi)f(y)\,\mathrm{d} y\,\mathrm{d}\xi, $$where $\phi$ is a $\mathcal{C}^\infty$ non-degenerate real phase function, and $b$ is a symbol of order $\mu$ and type $(\rho,1-\rho)$, $\sfrac12\lt\rho\le1$, vanishing for $x$ outside a compact set of $\mathbb{R}^n$. We show that when $p\le1$ and $\mu\le-(n-1)(1/p-1/2)$ then $\mathcal{F}$ initially defined on Schwartz functions in $h^p(\mathbb{R}^n)$ extends to a bounded operator $\mathcal{F}:h^p(\mathbb{R}^n)\rightarrow h^p(\mathbb{R}^n)$. The range of $p$ and $\mu$ is sharp. This result extends to the local Hardy spaces the seminal result of Seeger \et for the $L^p$ spaces. As immediate applications we prove the boundedness of smooth Radon transforms on hypersurfaces with non-vanishing Gaussian curvature on the local Hardy spaces.Finally, we prove a local version for the boundedness of Fourier integral operators on local Hardy spaces on smooth Riemannian manifolds of bounded geometry.

scholarly journals Riesz Transform Characterization of Weighted Hardy Spaces Associated with Schrödinger Operators

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Hua Zhu
Keyword(s):  
Schrödinger Operator ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Nonnegative Function ◽  
Riesz Transforms ◽  
Schrodinger Operator ◽  
Weighted Hardy Spaces ◽  
Local Hardy Spaces ◽  
Reverse Hölder

We characterize the weighted local Hardy spaceshρ1(ω)related to the critical radius functionρand weightsω∈A1ρ,∞(Rn)by localized Riesz transformsR^j; in addition, we give a characterization of weighted Hardy spacesHL1(ω)via Riesz transforms associated with Schrödinger operatorL, whereL=-Δ+Vis a Schrödinger operator onRn(n≥3) andVis a nonnegative function satisfying the reverse Hölder inequality.

Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

2020 ◽  
Vol 32 (4) ◽  
pp. 919-936 ◽  
Author(s):  
Jiao Chen ◽  
Wei Ding ◽  
Guozhen Lu
Keyword(s):  
Hardy Spaces ◽  
Differential Operators ◽  
Integral Operators ◽  
Fourier Multipliers ◽  
Regularity Of Solutions ◽  
Local Version ◽  
Fourier Integral Operators ◽  
Pseudo Differential Operators ◽  
Local Hardy Spaces ◽  
The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators

2015 ◽  
Vol 67 (5) ◽  
pp. 1161-1200 ◽  
Author(s):  
Junqiang Zhang ◽  
Jun Cao ◽  
Renjin Jiang ◽  
Dachun Yang
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Riesz Transform ◽  
Degenerate Elliptic Operators ◽  
Degenerate Elliptic Operator ◽  
Muckenhoupt Class ◽  
Degenerate Elliptic ◽  
Function Characterization

AbstractLet w be either in the Muckenhoupt class of A2(ℝn) weights or in the class of QC(ℝn) weights, and let be the degenerate elliptic operator on the Euclidean space ℝn, n ≥ 2. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space associated with , and when with , the authors prove that the associated Riesz transform is bounded from to the weighted classical Hardy space .

scholarly journals New Riemannian manifolds with Lp-unbounded Riesz transform for p > 2

2020 ◽  
Author(s):  
Alex Amenta
Keyword(s):  
Large Class ◽  
Riemannian Manifolds ◽  
Fractal Structure ◽  
Arbitrary Dimension ◽  
Riesz Transform

Abstract We construct a large class of Riemannian manifolds of arbitrary dimension with Riesz transform unbounded on $$L^p(M)$$ L p ( M ) for all $$p > 2$$ p > 2 . This extends recent results for Vicsek manifolds, and in particular shows that fractal structure is not necessary for this property.

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