粗糙奇异积分极大截断的稀疏逐点界与Sobolev型不等式

中文摘要 AI 辅助

设 $1 < \rho < n$，且 $\Omega$ 属于 $L^\rho(S^{(n-1)})$ 并具有消失均值。我们证明粗糙奇异积分 $T_\Omega$ 的极大截断 $T^*_\Omega$ 在逐点意义下被有限多个形如 $\sum_{Q \in S} l(Q) * ( (1/|Q|) * \int_Q |\nabla f|^p )^{1/p}$ 的稀疏势所控制，其中 $1/\rho~ = 1/\rho' + 1/n$ 且 $\rho~ \leq p < n$。该估计关于截断参数一致成立，并将 Hoang、Moen 和 Perez 关于 $T_\Omega$ 的次临界界推广到 $T^*_\Omega$。由于论证不要求 $T^*_\Omega$ 在目标空间上的有界性，它导出了双权 Sobolev 不等式：$\parallel T^*_\Omega f \parallel L^q(u) \leq C * \parallel \nabla f \parallel L^p(v)$，在联合双权条件下成立，而目标权 $u$ 本身仅需属于 $A_\infty$。此外，我们证明了一个涉及梯度 Morrey 范数的 Hedberg 型估计，并将其应用于加权 grand Lebesgue 空间。在加权 Lebesgue、Orlicz 和变指数 Lebesgue 空间中获得了进一步的结果。

英文摘要

Let $1 < ρ< n$ and let Omega be in $L^ρ(S^{(n-1)})$ with vanishing mean. We prove that the maximal truncation $T^*_Ω$ of the rough singular integral $T_Ω$ is pointwise dominated by finitely many sparse potentials of the form: $\sum_{Q \in S} l(Q) * ( (1/|Q|) * \int_Q |\nabla f|^p )^{1/p}$, where $1/ρ~ = 1/ρ' + 1/n$ and $ρ~ \leq p < n$. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for $T_Ω$ to $T^*_Ω$. Since the argument does not require the boundedness of $T^*_Ω$ on the target space, it yields two-weight Sobolev inequalities: $\parallel T^*_Ωf \parallel L^q(u) \leq C * \parallel \nabla f \parallel L^p(v)$, under joint two-weight conditions, while the target weight u itself is only required to belong to $A_\infty$. Additionally, we prove a Hedberg-type estimate involving a Morrey norm of the gradient and apply it to weighted grand Lebesgue spaces. Further consequences are obtained in weighted Lebesgue, Orlicz, and variable Lebesgue spaces.

发表机构

• Technical University of Civil Engineering Bucharest（布加勒斯特土木工程学院）

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