发表机构
Università di Napoli “Federico II”（那不勒斯费德里科二世大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种模多项式稀疏控制理论，通过多项式检验条件刻画Calderón-Zygmund算子，并导出加权Hardy-Sobolev不等式及Riesz势点态界，揭示了BMO到L^∞检验条件的定量转变。
AI 中文摘要
我们发展了一种稀疏控制理论，该理论同时捕捉输入函数的局部多项式逼近以及每个稀疏立方体尺度以下的相消性。其系数是输入减去其局部多项式逼近后的局部光滑极大函数的百分位数，结合了平均振荡稀疏界和Conde-Alonso、Lorist和Rey的相消稀疏界的特征。利用小波表示，我们通过显式的多项式检验条件刻画了其导数满足这些界的Calderón-Zygmund算子。在齐次情形下，相同的条件刻画了加权Hardy-Sobolev不等式，其依赖于$A_\infty$特征的线性关系。这种定量依赖检测到了Lerner发现的从BMO到$L^\infty$检验条件的转变，该转变对应于Riesz势的点态界，而定性加权估计无法恢复这一区别。在非齐次情形下，我们证明了相应的加权不等式，并表明从二阶导数阶起，它们严格弱于点态稀疏控制。推论包括Banach范围以下的Hardy-Sobolev估计、尖锐的加权Sobolev不等式以及Riesz势的点态界。
英文摘要
We develop a sparse domination theory that captures both local polynomial approximation and cancellation of the input below the scale of each sparse cube. Its coefficients are percentiles of localized smooth maximal functions of the input minus its local polynomial approximation, combining features of mean-oscillation sparse bounds and of the cancellative sparse bounds of Conde-Alonso, Lorist and Rey. Using wavelet representations, we characterize the Calderón-Zygmund operators whose derivatives admit these bounds by explicit polynomial testing conditions. In the homogeneous setting, the same conditions characterize weighted Hardy-Sobolev inequalities with linear dependence on the $A_\infty$ characteristic. This quantitative dependence detects the transition from BMO to $L^\infty$ testing conditions that Lerner found for pointwise bounds by Riesz potentials, a distinction that qualitative weighted estimates cannot recover. In the inhomogeneous setting, we prove corresponding weighted inequalities and show that they are strictly weaker than pointwise sparse domination from the second derivative order onward. Consequences include Hardy-Sobolev estimates below the Banach range, sharp weighted Sobolev inequalities, and pointwise bounds by Riesz potentials.
Comments64 pages. Submitted
