scholarly journals Restricted averaging operators to cones over finite fields

2018 ◽  
Vol 30 (6) ◽  
pp. 1345-1361
Author(s):  
Doowon Koh ◽  
Chun-Yen Shen ◽  
Seongjun Yeom
Keyword(s):  
Finite Fields ◽  
Dimensional Subspace ◽  
Surface Measure ◽  
Averaging Operators ◽  
Counting Measure ◽  
Dimensional Vector Space ◽  
Averaging Operator ◽  
Optimal Estimates ◽  
Complex Valued ◽  
Odd Dimensions

AbstractWe investigate the sharp{L^{p}\to L^{r}}estimates for the restricted averaging operator{A_{C}}over the coneCof thed-dimensional vector space{\mathbb{F}_{q}^{d}}over the finite field{\mathbb{F}_{q}}withqelements. The restricted averaging operator{A_{C}}for the coneCis defined by the relation{A_{C}f=f\ast\sigma|_{C}}, where σ denotes the normalized surface measure on the coneC, andfis a complex-valued function on the space{\mathbb{F}_{q}^{d}}with the normalized counting measuredx. In the previous work [D. Koh, C.-Y. Shen and I. Shparlinski, Averaging operators over homogeneous varieties over finite fields, J. Geom. Anal. 26 2016, 2, 1415–1441], the sharp boundedness of{A_{C}}was obtained in odd dimensions{d\geq 3}, but only partial results were given in even dimensions{d\geq 4}. In this paper we prove the optimal estimates in even dimensions{d\geq 6}in the case when the cone{C\subset\mathbb{F}_{q}^{d}}contains a{{d/2}}-dimensional subspace.

scholarly journals Extension and averaging operators for finite fields

2013 ◽  
Vol 56 (2) ◽  
pp. 599-614 ◽  
Author(s):  
Doowon Koh ◽  
Chun-Yen Shen
Keyword(s):  
Quadratic Form ◽  
Finite Field ◽  
Decay Estimate ◽  
Sharp Estimate ◽  
Dimensional Subspace ◽  
Extension Operator ◽  
Averaging Operators ◽  
Averaging Operator ◽  
Extension Result ◽  
Odd Dimensions

AbstractIn this paper we study Lp−Lr estimates of both the extension operator and the averaging operator associated with the algebraic variety S = {x ∈ : Q(x) = 0}, where Q(x) is a non-degenerate quadratic form over the finite field with q elements. We show that the Fourier decay estimate on S is good enough to establish the sharp averaging estimates in odd dimensions. In addition, the Fourier decay estimate enables us to simply extend the sharp L2−L4 conical extension result in , due to Mockenhaupt and Tao, to the L2−L2(d+1)/(d−1) estimate in all odd dimensions d ≥ 3. We also establish a sharp estimate of the mapping properties of the average operators in the case when the variety S in even dimensions d ≥ 4 contains a d/2-dimensional subspace.

scholarly journals Averaging operators over nondegenerate quadratic surfaces in finite fields

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Doowon Koh
Keyword(s):  
Finite Fields ◽  
Averaging Operators ◽  
Averaging Operator ◽  
Quadratic Surfaces

AbstractWe study mapping properties of the averaging operator related to the variety

THE REGULAR PART OF A SEMIGROUP OF LINEAR TRANSFORMATIONS WITH RESTRICTED RANGE

2017 ◽  
Vol 103 (3) ◽  
pp. 402-419 ◽  
Author(s):  
WORACHEAD SOMMANEE ◽  
KRITSADA SANGKHANAN
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Dimensional Subspace ◽  
Finite Dimensional Subspace ◽  
Restricted Range ◽  
Linear Transformations ◽  
Dimensional Vector Space ◽  
Finite Dimensional ◽  
Idempotent Rank ◽  
Image Position

Let$V$be a vector space and let$T(V)$denote the semigroup (under composition) of all linear transformations from$V$into$V$. For a fixed subspace$W$of$V$, let$T(V,W)$be the semigroup consisting of all linear transformations from$V$into$W$. In 2008, Sullivan [‘Semigroups of linear transformations with restricted range’,Bull. Aust. Math. Soc.77(3) (2008), 441–453] proved that$$\begin{eqnarray}\displaystyle Q=\{\unicode[STIX]{x1D6FC}\in T(V,W):V\unicode[STIX]{x1D6FC}\subseteq W\unicode[STIX]{x1D6FC}\} & & \displaystyle \nonumber\end{eqnarray}$$is the largest regular subsemigroup of$T(V,W)$and characterized Green’s relations on$T(V,W)$. In this paper, we determine all the maximal regular subsemigroups of$Q$when$W$is a finite-dimensional subspace of$V$over a finite field. Moreover, we compute the rank and idempotent rank of$Q$when$W$is an$n$-dimensional subspace of an$m$-dimensional vector space$V$over a finite field$F$.

scholarly journals On $k$-simplexes in $(2k-1)$-dimensional vector spaces over finite fields

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Nous Montrons ◽  
International Audience

International audience We show that if the cardinality of a subset of the $(2k-1)$-dimensional vector space over a finite field with $q$ elements is $\gg q^{2k-1-\frac{1}{ 2k}}$, then it contains a positive proportional of all $k$-simplexes up to congruence. Nous montrons que si la cardinalité d'un sous-ensemble de l'espace vectoriel à $(2k-1)$ dimensions sur un corps fini à $q$ éléments est $\gg q^{2k-1-\frac{1}{ 2k}}$, alors il contient une proportion non-nulle de tous les $k$-simplexes de congruence.

ON THE POSSIBILITY OF GROUP-THEORETIC DESCRIPTION OF AN EQUIVALENCE RELATION CONNECTED TO THE PROBLEM OF COVERING SUBSETS IN FINITE FIELDS WITH COSETS OF LINEAR SUBSPACES

2019 ◽  
Vol 53 (1 (248)) ◽  
pp. 23-27
Author(s):  
D.S. Sargsyan
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Equivalence Relation ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Linear Subspaces ◽  
Theoretic Description

Let $ F^{n}_{q} $ be an $ n $-dimensional vector space over a finite field $ F_q $ . Let $ C(F^{n}_{q} ) $ denote the set of all cosets of linear subspaces in $ F^{n}_{q} $. Cosets $ H_1, H_2, \ldots H_s $ are called exclusive if $ H_i \nsubseteq H_j $, $ 1 \mathclose{\leq} i \mathclose{<} j \mathclose{\leq} s $. A permutation $ f $ of $ C(F^{n}_{q} ) $ is called a $ C $-permutation, if for any exclusive cosets $ H, H_1, H_2, \ldots H_s $ such that $ H \subseteq H_1 \cup H_2 \cup \cdots \cup H_s $ we have:i) cosets $ f(H), f(H_1), f(H_2), \ldots, f(H_s) $ are exclusive;ii) cosets $ f^{-1}(H), f^{-1}(H_1), f^{-1}(H_2), \ldots, f^{-1}(H_s) $ are exclusive;iii) $ f(H) \subseteq f(H_1) \cup f(H_2) \cup \cdots \cup f(H_s) $;vi) $ f^{-1}(H) \subseteq f^{-1}(H_1) \cup f^{-1}(H_2) \cup \cdots \cup f^{-1}(H_s) $.In this paper we show that the set of all $ C $-permutations of $ C(F^{n}_{q} ) $ is the General Semiaffine Group of degree $ n $ over $ F_q $.

Cesàro averaging operators on Hardy spaces

1996 ◽  
Vol 126 (3) ◽  
pp. 617-624 ◽  
Author(s):  
Kenneth F. Andersen
Keyword(s):  
Hardy Spaces ◽  
Averaging Operators ◽  
Averaging Operator ◽  
Cesaro Averaging ◽  
Open Question

It is shown that the Cesàro averaging operatorℜα > – 1, satisfies an inequality which immediately implies that it is bounded on certain Hardy spaces including Hp, 0 < p < ∞. This answers an open question of Stempak, who introduced these operators and obtained their boundedness on Hp, 0 < p ≦ 2, for ℜα ≧ 0. The operator which is conjugate to on H2 is also shown to be bounded on Hp for 1 < p < ∞ and ℜα = – 1. This extends a result of Stempak who obtained this boundedness for 2 ≦ p≦ ∞ and ℜα ≧:0.

scholarly journals Averaging operators on the ring of continuous functions on a compact space

1964 ◽  
Vol 4 (3) ◽  
pp. 293-298 ◽  
Author(s):  
Barron Brainerd
Keyword(s):  
Compact Space ◽  
Linear Transformation ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Continuous Functions ◽  
Topological Product ◽  
Averaging Operators ◽  
Averaging Operator ◽  
Ring Of Continuous Functions ◽  
Image Position

In this note we answer the following question: Given C(X) the latticeordered ring of real continuous functions on the compact Hausdorff space X and T an averaging operator on C(X), under what circumstances can X be decomposed into a topological product such that supports a measure m and Tf = h where By an averaging operator we mean a linear transformation T on C(X) such that: 1. T is positive, that is, if f>0 (f(x) ≧ 0 for all x ∈ and f(x) > 0 for some a ∈ X), then Tf>0. 2. T(fTg) = (Tf)(Tg). 3. T l = 1 where l(x) = 1 for all x ∈ X.

scholarly journals On the Number of Orthogonal Systems in Vector Spaces over Finite Fields

10.37236/907 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Le Anh Vinh
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Finite Fields ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Graph Theoretic ◽  
Theoretic Proof ◽  
Orthogonal Systems

Iosevich and Senger (2008) showed that if a subset of the $d$-dimensional vector space over a finite field is large enough, then it contains many $k$-tuples of mutually orthogonal vectors. In this note, we provide a graph theoretic proof of this result.

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