Sharp $\ell^p$ inequalities for discrete singular integrals on the lattice $\mathbb{Z}^d$
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866929482805280768 |
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| author | Bañuelos, Rodrigo Kim, Daesung Kwaśnicki, Mateusz |
| author_facet | Bañuelos, Rodrigo Kim, Daesung Kwaśnicki, Mateusz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_09737 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Sharp $\ell^p$ inequalities for discrete singular integrals on the lattice $\mathbb{Z}^d$ Bañuelos, Rodrigo Kim, Daesung Kwaśnicki, Mateusz Probability Classical Analysis and ODEs Functional Analysis Primary 60G44 42A50, 42B20, Secondary 60J70, 39A12 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. |
| title | Sharp $\ell^p$ inequalities for discrete singular integrals on the lattice $\mathbb{Z}^d$ |
| topic | Probability Classical Analysis and ODEs Functional Analysis Primary 60G44 42A50, 42B20, Secondary 60J70, 39A12 |
| url | https://arxiv.org/abs/2209.09737 |