论文标题
Fefferman-Stein类型的混合不平等，用于单数积分运算符
Mixed inequalities of Fefferman-Stein type for singular integral operators
论文作者
论文摘要
我们给出了与Calderón-Zygmund操作员混合估计相关的Feffermain-Stein类型不平等。更准确地说，给定$δ> 0 $，$ q> 1 $，$φ（z）= z（1+ \ log^+z）^δ$，一种非负和本地可集成的函数$ u $和$ v \ in \ mathrm {rh} _ \ infty \ infty \ infty \ cap a_q a_q a_q a_q $，我们证明是必需的。 \ [uv \ left（\ left \ {x \ in \ mathbb {r}^n：\ frac {| t（fv）（x）|} {v（x）}> t \ right \} \ right）\ leq \ frac {c} {t} \ int _ {\ mathbb {r}^n} | f | \ left（m_ {φ，v^{1-q'}} u \ right）m（ψ（ψ（v））\] 保留$ψ（z）= z^{p'+1- q'} \ Mathcal {x} _ {[0,1]}（z）+z^{p'} \ Mathcal {x} _ {[x} _ {[1，\ infty）}（z）}（z）}（z）$，每一个$ t> 0 $ p> $ p> $ p>} $ c>+\ c}这种不平等为Calderón-Zygmund操作员提供了更通用的混合估计，以\ cite {Cruzuribe-Martell-Perez}证明。它还概括了同一运算符中\ cite {p94}给出的fefferman-stein估计值。 我们进一步获得了卷积类型运算符的类似估计，其中内核满足了$ l^φ-$Hörmander条件，从而概括了一些以前已知的结果，这些结果涉及混合估计值和Fefferman-Stein不平等。
We give Feffermain-Stein type inequalities related to mixed estimates for Calderón-Zygmund operators. More precisely, given $δ>0$, $q>1$, $φ(z)=z(1+\log^+z)^δ$, a nonnegative and locally integrable function $u$ and $v\in \mathrm{RH}_\infty\cap A_q$, we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T(fv)(x)|}{v(x)}>t\right\}\right)\leq \frac{C}{t}\int_{\mathbb{R}^n}|f|\left(M_{φ, v^{1-q'}}u\right)M(Ψ(v))\] holds with $Ψ(z)=z^{p'+1-q'}\mathcal{X}_{[0,1]}(z)+z^{p'}\mathcal{X}_{[1,\infty)}(z)$, for every $t>0$ and every $p>\max\{q,1+1/δ\}$. This inequality provides a more general version of mixed estimates for Calderón-Zygmund operators proved in \cite{CruzUribe-Martell-Perez}. It also generalizes the Fefferman-Stein estimates given in \cite{P94} for the same operators. We further get similar estimates for operators of convolution type with kernels satisfying an $L^Φ-$Hörmander condition, generalizing some previously known results which involve mixed estimates and Fefferman-Stein inequalities for these operators.
