论文标题
在晶格$ \ mathbb {z}^d $上的离散奇异积分的敏锐$ \ ell^p $不平等现象
Sharp $\ell^p$ inequalities for discrete singular integrals on the lattice $\mathbb{Z}^d$
论文作者
论文摘要
本文调查了在[Bañuelos和Kwaśnicki，Duke Math中验证的长期猜想的较高尺寸版本。 J.（2019）]，整数上离散的希尔伯特变换的$ \ ell^p $ - 与真实线上希尔伯特变换的$ l^p $ - norm相同。它计算晶格$ \ m artbb {z}^{d} $，$ d \ geq 1的$ \ ell^p $ - norms。离散运算符具有与经典Riesz在$ \ Mathbb {r}^d $上转换的相同$ p $ norms。它们被构造为DOOB H-PROCESSES的Martingale Transforms的有条件期望，这些条件是退出上半部空间$ \ MATHBB {R}^D \ TIMES \ MATHBB {r} _ {+} _ {+} $仅在lattice $ \ Mathbb {z}^d $上。该论文还提供了经典旋转方法的离散类似物，该方法给出了离散riesz的不同变体在$ \ mathbb {z}^d $上转换的规范。一路走来，基于用于识别[Bañuelos和Kwaśnicki，duke Math中的离散希尔伯特变换的规范的傅立叶变换技术，用于确定离散的希尔伯特变换规范。 J.（2019）]。陈述了开放问题。
This paper investigates higher dimensional versions of the longstanding conjecture verified in [Bañuelos and Kwaśnicki, Duke Math. J. (2019)] that the $\ell^p$-norm of the discrete Hilbert transform on the integers is the same as the $L^p$-norm of the Hilbert transform on the real line. It computes the $\ell^p$-norms of a family of discrete operators on the lattice $\mathbb{Z}^{d}$, $d\geq 1.$ They are discretizations of a new class of singular integrals on $\mathbb{R}^d$ that have the same kernels as the classical Riesz transforms near zero and similar behavior at infinity. The discrete operators have the same $p$-norms as the classical Riesz transforms on $\mathbb{R}^d$. They are constructed as conditional expectations of martingale transforms of Doob h-processes conditioned to exit the upper--half space $\mathbb{R}^d\times \mathbb{R}_{+}$ only on the lattice $\mathbb{Z}^d$. The paper also presents a discrete analogue of the classical method of rotations which gives the norm of a different variant of discrete Riesz transforms on $\mathbb{Z}^d$. Along the way a new proof is given based on Fourier transform techniques of the key identity used to identify the norm of the discrete Hilbert transform in [Bañuelos and Kwaśnicki, Duke Math. J. (2019)]. Open problems are stated.
