Boundedness of Marcinkiewicz functions.

• Autorzy: Minako Sakamoto , Kôzô Yabuta
• Języki publikacji: EN

Abstrakty

EN

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 $ʃ_{S^{n-1}} Ω(x')dσ(x') = 0$. 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.

Słowa kluczowe

EN

Marcinkiewicz function; Littlewood-Paley function; area function

Kategorie tematyczne

• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 135
• Numer: 2
• Strony: 103-142

Daty

wydano

1999

otrzymano

1995-05-04

poprawiono

1999-01-25

Twórcy

autor

Minako Sakamoto

• Department of Mathematics, Nara Women's University, Kitauoya-Nishimachi, Nara 630-8506, Japan

autor

Kôzô Yabuta

• School of Science, Kwansei Gakuin University, Uegahara 1-1-155, Nishinomiya, Hyogo 662-8501, Japan

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