论文标题
$ l^{p} $估算和加权估计值的最大粗糙奇异积分的均匀组的加权估计值
$L^{p}$ estimates and weighted estimates of fractional maximal rough singular integrals on homogeneous groups
论文作者
论文摘要
在本文中，我们研究了$ l^{p} $有界和$ l^{p}（w）$界（$ 1 <$ 1 <p <p <\ infty $和$ w $ a muckenhoupt $ a_ {p} $ a_ {p} $重量）分数最大值的奇数集成运算符$ t_， $ω（x）$在任意同质组$ \ mathbb h $ dimension $ \ mathbb {q} $的$。我们表明，如果$ 0 <α<\ mathbb {q} $，$ω\ in l^{1}（σ）$，并且满足订单$ [α] $的取消条件，那么对于任何$ 1 <p <\ p <\ infty $，\ \ | t_ {ω，α}^{\＃} f \ | _ {l^{l^{p}（\ Mathbb {h}）} \ simsim \ | | | | | _ | _ {l^{l^{l^{1}（σ）}（σ）}} \ end {align*}在情况下$α= 0 $，tao（\ cite {tao}）和sato（\ cite {sato}）研究了粗糙单数积分运算符及其最大运算符的$ l^p $界限。 我们还为这些操作员获得了定量加权绑定。具体而言，如果$ 0 \leqα<\ mathbb {q} $和$ω$满足相同的取消条件，但是对于某些$ q>> \ mathbb {q} Q}/α$的$ q>>>> \ qu>> \ mathbb {q}/α$的$ω\ in l^{q}（σ）$都更强\ begin {align*} \ | t_ {ω，α}^{\＃} f \ | _ {l^{p}}（w）} \ lyseSim \ | |ω\ | ___ {l^{l^{ q}（σ）} \ {w \} _ {a_p}（w）_ {a_p} \ | f \ | _ {l_α^{p} {p}（w）}，\ \ \ 1 <p <\ infty。 \ end {align*}
In this paper, we study the $L^{p}$ boundedness and $L^{p}(w)$ boundedness ($1<p<\infty$ and $w$ a Muckenhoupt $A_{p}$ weight) of fractional maximal singular integral operators $T_{Ω,α}^{\#}$ with homogeneous convolution kernel $Ω(x)$ on an arbitrary homogeneous group $\mathbb H$ of dimension $\mathbb{Q}$. We show that if $0<α<\mathbb{Q}$, $Ω\in L^{1}(Σ)$ and satisfies the cancellation condition of order $[α]$, then for any $1<p<\infty$, \begin{align*} \|T_{Ω,α}^{\#}f\|_{L^{p}(\mathbb{H})}\lesssim\|Ω\|_{L^{1}(Σ)}\|f\|_{L_α^{p}(\mathbb{H})}, \end{align*} where for the case $α=0$, the $L^p$ boundedness of rough singular integral operator and its maximal operator were studied by Tao (\cite{Tao}) and Sato (\cite{sato}), respectively. We also obtain a quantitative weighted bound for these operators. To be specific, if $0\leqα<\mathbb{Q}$ and $Ω$ satisfies the same cancellation condition but a stronger condition that $Ω\in L^{q}(Σ)$ for some $q>\mathbb{Q}/α$, then for any $1<p<\infty$ and $w\in A_{p}$, \begin{align*} \|T_{Ω,α}^{\#}f\|_{L^{p}(w)}\lesssim\|Ω\|_{L^{q}(Σ)}\{w\}_{A_p}(w)_{A_p}\|f\|_{L_α^{p}(w)},\ \ 1<p<\infty. \end{align*}
