Boundedness of Marcinkiewicz functions.

Sakamoto, Minako, and Yabuta, Kôzô. "Boundedness of Marcinkiewicz functions.." Studia Mathematica 135.2 (1999): 103-142. <http://eudml.org/doc/216646>.

@article{Sakamoto1999,
abstract = {The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g*_λ$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts $μ_\{S\}^\{ϱ\}$ and $μ_\{λ\}^\{*,ϱ\}$ to S and $g*_λ$. The definition of $μ_\{S\}^\{ϱ\}(f)$ is as follows: $μ_\{S\}^\{ϱ\}(f)(x) = (ʃ_\{|y-x| < t\}| 1/t^\{ϱ\} ʃ_\{|z|≤ t\} Ω(z)/(|z|^\{n-ϱ\}) f(y-z) dz|^2 (dydt)/(t^\{n+1\}) )^\{1/2\}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^\{n-1\}$, and $ʃ_\{S^\{n-1\}\} Ω(x^\{\prime \})dσ(x^\{\prime \}) = 0$. We show that if σ = Reϱ > 0, then $μ_\{S\}^\{ϱ\}$ is $L^p$ bounded for max(1,2n/(n+2σ) < p < ∞, and for 0 < ϱ ≤ n/2 and 1 ≤ p ≤ 2n/(n+2ϱ), then $L^p$ boundedness does not hold in general, in contrast to the case of the S function. Similar results hold for $μ_\{λ\}^\{*,ϱ\}$. Their boundedness in the Campanato space $ε^\{α,p\}$ is also considered.},
author = {Sakamoto, Minako, Yabuta, Kôzô},
journal = {Studia Mathematica},
keywords = {Marcinkiewicz function; Littlewood-Paley function; area function; boundedness; Littlewood-Paley’s -function; Campanato space},
language = {eng},
number = {2},
pages = {103-142},
title = {Boundedness of Marcinkiewicz functions.},
url = {http://eudml.org/doc/216646},
volume = {135},
year = {1999},
}

TY - JOUR
AU - Sakamoto, Minako
AU - Yabuta, Kôzô
TI - Boundedness of Marcinkiewicz functions.
JO - Studia Mathematica
PY - 1999
VL - 135
IS - 2
SP - 103
EP - 142
AB - The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley’s g-function, Lusin’s S function, Littlewood-Paley’s $g*_λ$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley’s g-function. In this note, we treat counterparts $μ_{S}^{ϱ}$ and $μ_{λ}^{*,ϱ}$ to S and $g*_λ$. The definition of $μ_{S}^{ϱ}(f)$ is as follows: $μ_{S}^{ϱ}(f)(x) = (ʃ_{|y-x| < t}| 1/t^{ϱ} ʃ_{|z|≤ t} Ω(z)/(|z|^{n-ϱ}) f(y-z) dz|^2 (dydt)/(t^{n+1}) )^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and $ʃ_{S^{n-1}} Ω(x^{\prime })dσ(x^{\prime }) = 0$. We show that if σ = Reϱ > 0, then $μ_{S}^{ϱ}$ is $L^p$ bounded for max(1,2n/(n+2σ) < p < ∞, and for 0 < ϱ ≤ n/2 and 1 ≤ p ≤ 2n/(n+2ϱ), then $L^p$ boundedness does not hold in general, in contrast to the case of the S function. Similar results hold for $μ_{λ}^{*,ϱ}$. Their boundedness in the Campanato space $ε^{α,p}$ is also considered.
LA - eng
KW - Marcinkiewicz function; Littlewood-Paley function; area function; boundedness; Littlewood-Paley’s -function; Campanato space
UR - http://eudml.org/doc/216646
ER -