scholarly journals Sharp endpoint estimates for some operators associated with the Laplacian with drift in Euclidean space

2020 ◽  
pp. 1-27
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Euclidean Space ◽  
Exponential Growth ◽  
Euclidean Distance ◽  
Weak Type ◽  
Riesz Transforms ◽  
Poisson Semigroups

Abstract Let $v \ne 0$ be a vector in ${\mathbb {R}}^n$ . Consider the Laplacian on ${\mathbb {R}}^n$ with drift $\Delta _{v} = \Delta + 2v\cdot \nabla $ and the measure $d\mu (x) = e^{2 \langle v, x \rangle } dx$ , with respect to which $\Delta _{v}$ is self-adjoint. This measure has exponential growth with respect to the Euclidean distance. We study weak type $(1, 1)$ and other sharp endpoint estimates for the Riesz transforms of any order, and also for the vertical and horizontal Littlewood–Paley–Stein functions associated with the heat and the Poisson semigroups.

scholarly journals Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Heisenberg Group ◽  
Weak Type ◽  
Riesz Transforms ◽  
Heisenberg Groups ◽  
Related Measure

AbstractIn the Heisenberg group of dimension $$2n+1$$ 2 n + 1 , we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$ L p bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

A NOTE ON SLq(2) COVARIANCE AND QUANTUM EUCLIDEAN PLANE

2007 ◽  
Vol 22 (14) ◽  
pp. 1031-1037 ◽  
Author(s):  
Z. BENTALHA ◽  
M. TAHIRI
Keyword(s):  
Euclidean Space ◽  
Quantum Group ◽  
Bilinear Form ◽  
Algebraic Structure ◽  
Euclidean Distance ◽  
Euclidean Plane ◽  
Three Dimensional ◽  
Central Element ◽  
The Other ◽  
Euclidean Metric

A quantum analogue of SL(2) covariance is given. The method consists of defining a bilinear form on SL q(2) co-module, then we ask the defined form to be invariant under SL q(2) co-actions. We showed that the required invariance leads to the standard algebraic structure of SL q(2) ideal. On the other hand, the geometry of the three-dimensional quantum Euclidean space has been evoked by computing the quantum Euclidean metric, the quantum Euclidean "distance" (the central element) and the relation of orthogonality of SO q(3) quantum group.

scholarly journals SCENE CLASSIFICATION BASED ON THE INTRINSIC MEAN OF LIE GROUP

2020 ◽  
Vol V-3-2020 ◽  
pp. 75-82
Author(s):  
C. Xu ◽  
G. Zhu ◽  
K. Yang
Keyword(s):  
Remote Sensing ◽  
Euclidean Space ◽  
Lie Group ◽  
Euclidean Distance ◽  
Geodesic Distance ◽  
Scene Classification ◽  
Satisfactory Performance ◽  
Group Manifold ◽  
Intrinsic Mean ◽  
High Resolution Images

Abstract. Remote Sensing scene classification aims to identify semantic objects with similar characteristics from high resolution images. Even though existing methods have achieved satisfactory performance, the features used for classification modeling are still limited to some kinds of vector representation within a Euclidean space. As a result, their models are not robust to reflect the essential scene characteristics, hardly to promote classification accuracy higher. In this study, we propose a novel scene classification method based on the intrinsic mean on a Lie Group manifold. By introducing Lie Group machine learning into scene classification, the new method uses the geodesic distance on the Lie Group manifold, instead of Euclidean distance, solving the problem that non-euclidean space samples could not be calculated by Euclidean distance directly. The experiments show that our method produces satisfactory performance on two public and challenging remote sensing scene datasets, UC Merced and SIRI-WHU, respectively.

The Hardy Space H1 on Manifolds and Submanifolds

1972 ◽  
Vol 24 (5) ◽  
pp. 915-925 ◽  
Author(s):  
Robert S. Strichartz
Keyword(s):  
Partial Differential Equations ◽  
Hardy Space ◽  
Euclidean Space ◽  
Differential Equations ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Riesz Transforms ◽  
Partial Differential ◽  
Integrable Functions

It is well-known that the space L1(Rn) of integrable functions on Euclidean space fails to be preserved by singular integral operators. As a result the rather large Lp theory of partial differential equations also fails for p = 1. Since L1 is such a natural space, many substitute spaces have been considered. One of the most interesting of these is the space we will denote by H1(Rn) of integrable functions whose Riesz transforms are integrable.

SISTEM INFORMASI GEOGRAFIS PEMETAAN TEMPAT KOST BERBASIS WEB MENGGUNAKAN METODE EUCLIDEAN DISTANCE

2021 ◽  
Vol 5 (1) ◽  
pp. 105
Author(s):  
Suparmi Suparmi ◽  
Soeheri Soeheri
Keyword(s):  
Euclidean Space ◽  
Euclidean Distance ◽  
The Common ◽  
Common Era

<p><em>Ketersediaan informasi yang sangat terbatas mengenai lokasi tempat kost menyebabkan masyarakat cenderung tidak memiliki informasi yang akurat dan relevan sehingga tidak mengetahui lokasi mana yang memiliki tempat kost dengan biaya terjangkau, dekat tempat kerja, lembaga pendidikan. Tujuan penelitian menghasilkan sistem informasi geografis berbasis web mengenai informasi lokasi tempat (rumah) kost di Medan. Informasi dalam bentuk google map sehingga memberikan kemudahan kepada pengguna dalam mencari lokasi tempat kost. Metode pengembangan sistem perangkat lunak menggunakan metode Euclidean distance. Euclidean distance adalah perhitungan jarak dari 2 buah titik dalam Euclidean space. Euclidean space diperkenalkan oleh Euclid, seorang matematikawan dari Yunani sekitar tahun 300 B.C.E. (Before the Common Era) untuk mempelajari hubungan antara sudut dan jarak. Euclidean ini berkaitan dengan Teorema Phytagoras dan biasanya diterapkan pada 1, 2, dan 3 dimensi. Hasil yang diperoleh dari penelitian ini akan menghasilkan informasi lokasi/tempat, nama, gambar dan jarak terdekat dari kampus.</em></p>

Oscillation and variation of the laguerre heat and poisson semigroups and riesz transforms

2012 ◽  
Vol 32 (3) ◽  
pp. 907-928 ◽  
Author(s):  
J.J. Betancor ◽  
R. Crescimbeni ◽  
J.L. Torrea
Keyword(s):  
Riesz Transforms ◽  
Poisson Semigroups

scholarly journals CONSTRUCTING DEGREE-3 SPANNERS WITH OTHER SPARSENESS PROPERTIES

1996 ◽  
Vol 07 (02) ◽  
pp. 121-135 ◽  
Author(s):  
GAUTAM DAS ◽  
PAUL J. HEFFERNAN
Keyword(s):  
Euclidean Space ◽  
Shortest Path ◽  
Spanning Tree ◽  
Maximum Degree ◽  
Euclidean Distance ◽  
Minimum Spanning Tree ◽  
Dimensional Euclidean Space ◽  
Edge Weight

Let V be any set of n points in k-dimensional Euclidean space. A subgraph of the complete Euclidean graph is a t-spanner if for all u and υ in V, the length of the shortest path from u to υ in the spanner is at most t times the Euclidean distance between u and υ. We show that for any δ>1, there exists a t-spanner (where t is a constant that depends only on δ and k) with the following properties: its maximum degree is 3, it has at most n·δ edges, its total edge weight is at most O(1) times the weight of the minimum spanning tree of V, and it can be constructed in O(n log n) time. The constants implicit in the O-notation depend on δ and k.

scholarly journals Riesz transforms of a general Ornstein–Uhlenbeck semigroup

2021 ◽  
Vol 60 (4) ◽  
Author(s):  
Valentina Casarino ◽  
Paolo Ciatti ◽  
Peter Sjögren
Keyword(s):  
Invariant Measure ◽  
Weak Type ◽  
Riesz Transform ◽  
Positive Definite Matrix ◽  
Positive Definite ◽  
Riesz Transforms ◽  
Real Matrix

AbstractWe consider Riesz transforms of any order associated to an Ornstein–Uhlenbeck operator with covariance given by a real, symmetric and positive definite matrix, and with drift given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type (1, 1) with respect to the invariant measure if and only if its order is at most 2.

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