On Dahlberg's Lusin Area Integral Theorem

1995 ◽  
Vol 123 (5) ◽  
pp. 1449
Author(s):  
Marius Mitrea
Keyword(s):  
Integral Theorem ◽  
Area Integral ◽  
Lusin Area Integral

scholarly journals Hardy Spaces Associated to Schrödinger Operators on Product Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-17 ◽  
Author(s):  
Liang Song ◽  
Chaoqiang Tan
Keyword(s):  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Schrödinger Operators ◽  
Nonnegative Function ◽  
Maximal Functions ◽  
Schrodinger Operators ◽  
Schrodinger Operator ◽  
Product Spaces ◽  
Area Integral ◽  
Lusin Area Integral

LetL=−Δ+Vbe a Schrödinger operator onℝn, whereV∈Lloc1(ℝn)is a nonnegative function onℝn. In this article, we show that the Hardy spacesLon product spaces can be characterized in terms of the Lusin area integral, atomic decomposition, and maximal functions.

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

2020 ◽  
Vol 57 (4) ◽  
pp. 465-507
Author(s):  
Hua Wang
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Heat Semigroup ◽  
Schrodinger Operator ◽  
Heisenberg Groups ◽  
Area Integral ◽  
Weak Type Estimates ◽  
Homogeneous Dimension ◽  
Lusin Area Integral ◽  
Reverse Hölder

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

A CHARACTERIZATION OF HARDY SPACE ON STRONGLY LIPSCHITZ DOMAINS OF ℝn BY LITTLEWOOD–PALEY–STEIN FUNCTION

2010 ◽  
Vol 12 (01) ◽  
pp. 71-84 ◽  
Author(s):  
JIZHENG HUANG
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Lipschitz Domain ◽  
Lipschitz Domains ◽  
Area Integral ◽  
Divergence Operator ◽  
Lusin Area Integral ◽  
Strongly Lipschitz Domain

Let Ω be a strongly Lipschitz domain of ℝn and define Hardy spaces on Ω by non-tangential maximal function. In this paper, we will give a characterization of the Hardy spaces on Ω by Littlwood–Paley–Stein function associated to L, where L is an elliptic second-order divergence operator. In order to get our result, we also consider the Lusin area integral characterization of the Hardy spaces on Ω.

scholarly journals SOME CHARACTERIZATIONS OF HARDY SPACES ASSOCIATED WITH TWISTED CONVOLUTION

2009 ◽  
Vol 79 (3) ◽  
pp. 405-417 ◽  
Author(s):  
JIZHENG HUANG
Keyword(s):  
Hardy Space ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Area Integral ◽  
Twisted Convolution ◽  
Lusin Area Integral

AbstractIn this paper, we shall give some characterizations of the Hardy space associated with twisted convolution, including Lusin area integral, Littlewood–Paley g-function and heat maximal function.

scholarly journals The Boundedness of Intrinsic Square Functions on the Weighted Herz Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Hua Wang
Keyword(s):  
Weak Type ◽  
Strong Type ◽  
Herz Spaces ◽  
Square Functions ◽  
Area Integral ◽  
Weak Type Estimates ◽  
Lusin Area Integral ◽  
Intrinsic Square Functions

We will obtain the strong type and weak type estimates of intrinsic square functions including the Lusin area integral, Littlewood-Paley𝒢-function, and𝒢λ*-function on the weighted Herz spacesK˙qα,p(w1,w2)(Kqα,p(w1,w2))with general weights.

Sign in / Sign up

Export Citation Format

Share Document