Abstract

We suggest an elementary harmonic analysis approach to canceling and weakly canceling differential operators, which allows us to extend these notions to the anisotropic setting and replace differential operators with Fourier multiplies with mild smoothness regularity. In this more general setting of anisotropic Fourier multipliers, we prove the inequality \(\|f\|_{L_\infty} \lesssim \|Af\|_{L_1}\) if \(A\) is a weakly canceling operator of order \(d\) and the inequality \(\|f\|_{L_2} \lesssim \|Af\|_{L_1}\) if \(A\) is a canceling operator of order \(d/2\), provided \(f\) is a function of \(d\) variables.

中文翻译：

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弱抵消算子和奇异积分

摘要

我们建议使用基本谐波分析方法来消除和弱消除微分算子，这使我们可以将这些概念扩展到各向异性设置，并用具有柔和平滑度规则性的傅立叶乘法替换微分算子。在各向异性傅里叶乘子的这个更一般的设置中，如果\（A \）是弱抵消，我们证明不等式\（\ | f \ | _ {L_ \ infty} \ lesssim \ | Af \ | _ {L_1} \）订单\（d \）和不等式\（\ | f \ | _ {L_2} \ lesssim \ | Af \ | __L {L_1} \）的运算符，如果\（A \）是订单\（d / 2 \），前提是\（f \）是\（d \）变量的函数。