scholarly journals A problem on rough parametric Marcinkiewicz functions

2002 ◽  
Vol 72 (1) ◽  
pp. 13-22 ◽  
Author(s):  
Yong Ding ◽  
Shanzhen Lu ◽  
Kôzô Yabuta
Keyword(s):  
Kernel Function ◽  
Radial Function ◽  
Marcinkiewicz Integral ◽  
Area Function ◽  
Lusin Area Function ◽  
Parametric Marcinkiewicz Integral

AbstractIn this note the authors give the L2(n) boundedness of a class of parametric Marcinkiewicz integral with kernel function Ω in L log+L(Sn−1) and radial function h(|x|) ∈ l ∞ l(Lq)(+) for 1 < q ≦.As its corollary, the Lp (n)(2 < p < ∞) boundedness of and and with Ω in L log+L (Sn-1) and h(|x|) ∈ l∞ (Lq)(+) are also obtained. Here and are parametric Marcinkiewicz functions corresponding to the Littlewood-Paley g*λ-function and the Lusin area function S, respectively.

scholarly journals Existence and Boundedness of Parametrized Marcinkiewicz Integral with Rough Kernel on Campanato Spaces

2006 ◽  
Vol 181 ◽  
pp. 103-148 ◽  
Author(s):  
Yong Ding ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Single Point ◽  
Morrey Spaces ◽  
Rough Kernel ◽  
Marcinkiewicz Integral ◽  
Type Condition ◽  
Lipschitz Spaces ◽  
Area Function ◽  
Bmo Function ◽  
Lusin Area Function ◽  
Lipschitz Kernel

AbstractLet g(f), S(f), g*λ(f) be the Littlewood-Paley g function, Lusin area function, and Littlewood-Paley g*λ(f) function of f, respectively. In 1990 Chen Jiecheng and Wang Silei showed that if, for a BMO function f, one of the above functions is finite for a single point in ℝn, then it is finite a.e. on ℝn, and BMO boundedness holds. Recently, Sun Yongzhong extended this result to the case of Campanato spaces (i.e. Morrey spaces, BMO, and Lipschitz spaces). One of us improved his g*λ(f) result further, and treated parametrized Marcinkiewicz functions with Lipschitz kernel μρ(f), μρs(f) and μλ*,ρ(f). In this paper, we show that the same results hold also in the case of rough kernel satisfying Lp-Dini type condition.

BOUNDS FOR NONISOTROPIC MARCINKIEWICZ INTEGRALS ASSOCIATED TO SURFACES

2015 ◽  
Vol 99 (3) ◽  
pp. 380-398 ◽  
Author(s):  
FENG LIU ◽  
SUZHEN MAO
Keyword(s):  
Radial Function ◽  
Integral Operators ◽  
Marcinkiewicz Integral ◽  
Marcinkiewicz Integrals ◽  
Parametric Marcinkiewicz Integral ◽  
Marcinkiewicz Operators ◽  
L Log L

In an extrapolation argument, we prove certain $L^{p}\,(1<p<\infty )$ estimates for nonisotropic Marcinkiewicz operators associated to surfaces under the integral kernels given by the elliptic sphere functions ${\rm\Omega}\in L(\log ^{+}L)^{{\it\alpha}}({\rm\Sigma})$ and the radial function $h\in {\mathcal{N}}_{{\it\beta}}(\mathbb{R}^{+})$. As applications, the corresponding results for parametric Marcinkiewicz integral operators related to area integrals and Littlewood–Paley $g_{{\it\lambda}}^{\ast }$-functions are given.

scholarly journals Boundedness of Generalized Parametric Marcinkiewicz Integrals Associated to Surfaces

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 886 ◽  
Author(s):  
Mohammed Ali ◽  
Oqlah Al-Refai
Keyword(s):  
Kernel Function ◽  
Integral Operators ◽  
Marcinkiewicz Integral ◽  
Marcinkiewicz Integrals ◽  
Block Space ◽  
Parametric Marcinkiewicz Integral

In this article, the boundedness of the generalized parametric Marcinkiewicz integral operators M Ω , ϕ , h , ρ ( r ) is considered. Under the condition that Ω is a function in L q ( S n - 1 ) with q ∈ ( 1 , 2 ] , appropriate estimates of the aforementioned operators from Triebel–Lizorkin spaces to L p spaces are obtained. By these estimates and an extrapolation argument, we establish the boundedness of such operators when the kernel function Ω belongs to the block space B q 0 , ν - 1 ( S n - 1 ) or in the space L ( l o g L ) ν ( S n - 1 ) . Our results represent improvements and extensions of some known results in generalized parametric Marcinkiewicz integrals.

scholarly journals Generalized parabolic Marcinkiewicz integrals associated with polynomial compound curves with rough kernels

2020 ◽  
Vol 53 (1) ◽  
pp. 44-57
Author(s):  
Mohammed Ali ◽  
Qutaibeh Katatbeh
Keyword(s):  
Integral Operators ◽  
Marcinkiewicz Integral ◽  
Rough Kernels ◽  
Marcinkiewicz Integrals ◽  
Natural Extensions ◽  
Parametric Marcinkiewicz Integral ◽  
Weaker Conditions

AbstractIn this article, we study the generalized parabolic parametric Marcinkiewicz integral operators { {\mathcal M} }_{{\Omega },h,{\Phi },\lambda }^{(r)} related to polynomial compound curves. Under some weak conditions on the kernels, we establish appropriate estimates of these operators. By the virtue of the obtained estimates along with an extrapolation argument, we give the boundedness of the aforementioned operators from Triebel-Lizorkin spaces to Lp spaces under weaker conditions on Ω and h. Our results represent significant improvements and natural extensions of what was known previously.

Generalized Littlewood–Paley characterizations of fractional Sobolev spaces

2018 ◽  
Vol 20 (07) ◽  
pp. 1750077 ◽  
Author(s):  
Shuichi Sato ◽  
Fan Wang ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Sobolev Spaces ◽  
Weak Type ◽  
Metric Measure Spaces ◽  
Fractional Sobolev Spaces ◽  
Area Function ◽  
Weak Type Estimates ◽  
Measure Spaces ◽  
Lusin Area Function

In this paper, the authors characterize the Sobolev spaces [Formula: see text] with [Formula: see text] and [Formula: see text] via a generalized Lusin area function and its corresponding Littlewood–Paley [Formula: see text]-function. The range [Formula: see text] is also proved to be nearly sharp in the sense that these new characterizations are not true when [Formula: see text] and [Formula: see text]. Moreover, in the endpoint case [Formula: see text], the authors also obtain some weak type estimates. Since these generalized Littlewood–Paley functions are of wide generality, these results provide some new choices for introducing the notions of fractional Sobolev spaces on metric measure spaces.

Littlewood–Paley Characterizations of Second-Order Sobolev Spaces via Averages on Balls

2016 ◽  
Vol 59 (01) ◽  
pp. 104-118 ◽  
Author(s):  
Ziyi He ◽  
Dachun Yang ◽  
Wen Yuan
Keyword(s):  
Sobolev Spaces ◽  
Second Order ◽  
Area Function ◽  
Lusin Area Function

Abstract In this paper, the authors characterize second-order Sobolev spaces W2,p(ℝn), with p ∊ [2,∞) and n ∊ N or p ∊ (1, 2) and n ∊ {1, 2, 3}, via the Lusin area function and the Littlewood–Paley g*λ -function in terms of ball means.

scholarly journals ORLICZ–HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES–GAFFNEY ESTIMATES

2011 ◽  
Vol 13 (02) ◽  
pp. 331-373 ◽  
Author(s):  
RENJIN JIANG ◽  
DACHUN YANG
Keyword(s):  
Hardy Space ◽  
Adjoint Operator ◽  
Riesz Transform ◽  
Concave Function ◽  
Maximal Functions ◽  
Heat Semigroup ◽  
Area Function ◽  
Poisson Semigroup ◽  
Function Characterization ◽  
Lusin Area Function

Let [Formula: see text] be a metric space with doubling measure, L a nonnegative self-adjoint operator in [Formula: see text] satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type pω∈(0, 1] and ρ(t) = t-1/ω-1(t-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space [Formula: see text] via the Lusin area function associated to the heat semigroup, and the BMO-type space [Formula: see text]. The authors then establish the duality between [Formula: see text] and [Formula: see text]; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space [Formula: see text]. Characterizations of [Formula: see text], including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let [Formula: see text] and L = -Δ+V be a Schrödinger operator, where [Formula: see text] is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L-1/2 is bounded from Hω, L(ℝn) to L(ω). Moreover, if there exist q1, q2∈(0, ∞) such that q1<1<q2 and [ω(tq2)]q1 is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space Hω, L(ℝn), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = tp for all t ∈ (0, ∞) and p ∈ (0, 1).

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