Hardy Spaces, Besov Spaces, and Triebel–Lizorkin Spaces

2014 ◽  
pp. 55-152
Author(s):  
Loukas Grafakos
Keyword(s):  
Hardy Spaces ◽  
Besov Spaces

scholarly journals A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-250 ◽  
Author(s):  
Yongsheng Han ◽  
Detlef Müller ◽  
Dachun Yang
Keyword(s):  
Hardy Spaces ◽  
Besov Spaces ◽  
Metric Measure Spaces ◽  
Order Zero ◽  
Wide Range ◽  
Measure Spaces ◽  
Real Method ◽  
Local Hardy Spaces ◽  
Approximation Of The Identity

We work on RD-spaces𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces whenp≥1. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains inℂN. Our theory applies in a wide range of settings.

On Hankel Forms of Higher Weights: The Case of Hardy Spaces

2010 ◽  
Vol 62 (2) ◽  
pp. 439-455
Author(s):  
Marcus Sundhäll ◽  
Edgar Tchoundja
Keyword(s):  
Hardy Spaces ◽  
Besov Spaces ◽  
Carleson Measure ◽  
Bergman Spaces ◽  
One Dimension ◽  
Weighted Bergman Spaces ◽  
Schatten Class ◽  
Full Characterization ◽  
Higher Weights

AbstractIn this paper we study bilinear Hankel forms of higher weights on Hardy spaces in several dimensions. (The Schatten class Hankel forms of higher weights on weighted Bergman spaces have already been studied by Janson and Peetre for one dimension and by Sundh¨all for several dimensions). We get a full characterization of Schatten class Hankel forms in terms of conditions for the symbols to be in certain Besov spaces. Also, the Hankel forms are bounded and compact if and only if the symbols satisfy certain Carleson measure criteria and vanishing Carleson measure criteria, respectively.

HARDY SPACES, APPROXIMATION ISSUES AND BOUNDARY VALUE PROBLEMS

2018 ◽  
Vol 9 (3) ◽  
pp. 85-94
Author(s):  
V.I. Vlasov ◽  
◽  
◽  
Keyword(s):  
Boundary Value Problems ◽  
Hardy Spaces ◽  
Boundary Value

scholarly journals Variation inequalities for rough singular integrals and their commutators on Morrey spaces and Besov spaces

2021 ◽  
Vol 11 (1) ◽  
pp. 72-95
Author(s):  
Xiao Zhang ◽  
Feng Liu ◽  
Huiyun Zhang
Keyword(s):  
Hilbert Transform ◽  
Besov Spaces ◽  
Singular Integrals ◽  
Riesz Transform ◽  
Morrey Spaces ◽  
Riesz Transforms ◽  
Rough Kernels ◽  
Lebesgue Spaces ◽  
Variation Operators

Abstract This paper is devoted to investigating the boundedness, continuity and compactness for variation operators of singular integrals and their commutators on Morrey spaces and Besov spaces. More precisely, we establish the boundedness for the variation operators of singular integrals with rough kernels Ω ∈ Lq (S n−1) (q > 1) and their commutators on Morrey spaces as well as the compactness for the above commutators on Lebesgue spaces and Morrey spaces. In addition, we present a criterion on the boundedness and continuity for a class of variation operators of singular integrals and their commutators on Besov spaces. As applications, we obtain the boundedness and continuity for the variation operators of Hilbert transform, Hermit Riesz transform, Riesz transforms and rough singular integrals as well as their commutators on Besov spaces.

Fourier transform of anisotropic mixed-norm Hardy spaces

2021 ◽  
Vol 16 (1) ◽  
pp. 119-139
Author(s):  
Long Huang ◽  
Der-Chen Chang ◽  
Dachun Yang
Keyword(s):  
Fourier Transform ◽  
Hardy Spaces ◽  
Mixed Norm
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