A remark on the asymmetry of convolution operators

Saverio Giulini

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti (1989)

  • Volume: 83, Issue: 1, page 85-88
  • ISSN: 0392-7881

Abstract

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A convolution operator, bounded on L q ⁢ ( ℝ n ) , is bounded on L p ⁢ ( ℝ n ) , with the same operator norm, if p and q are conjugate exponents. It is well known that this fact is false if we replace ℝ n with a general non-commutative locally compact group G . In this paper we give a simple construction of a convolution operator on a suitable compact group G , wich is bounded on L q ⁢ ( G ) for every q ∈ [ 2 , ∞ ) and is unbounded on L p ⁢ ( G ) if p ∈ [ 1 , 2 ) .

How to cite

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Giulini, Saverio. "A remark on the asymmetry of convolution operators." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti 83.1 (1989): 85-88. <http://eudml.org/doc/289333>.

@article{Giulini1989,
abstract = {A convolution operator, bounded on $L^\{q\}(\mathbb\{R\}^\{n\})$, is bounded on $L^\{p\}(\mathbb\{R\}^\{n\})$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace $\mathbb\{R\}^\{n\}$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^\{q\}(G)$ for every $q \in [2,\infty)$ and is unbounded on $L^\{p\}(G)$ if $p \in [1,2)$.},
author = {Giulini, Saverio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti},
keywords = {Non-commutative groups; Convolution operators; Asymmetry},
language = {eng},
month = {12},
number = {1},
pages = {85-88},
publisher = {Accademia Nazionale dei Lincei},
title = {A remark on the asymmetry of convolution operators},
url = {http://eudml.org/doc/289333},
volume = {83},
year = {1989},
}

TY - JOUR
AU - Giulini, Saverio
TI - A remark on the asymmetry of convolution operators
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti
DA - 1989/12//
PB - Accademia Nazionale dei Lincei
VL - 83
IS - 1
SP - 85
EP - 88
AB - A convolution operator, bounded on $L^{q}(\mathbb{R}^{n})$, is bounded on $L^{p}(\mathbb{R}^{n})$, with the same operator norm, if $p$ and $q$ are conjugate exponents. It is well known that this fact is false if we replace $\mathbb{R}^{n}$ with a general non-commutative locally compact group $G$. In this paper we give a simple construction of a convolution operator on a suitable compact group $G$, wich is bounded on $L^{q}(G)$ for every $q \in [2,\infty)$ and is unbounded on $L^{p}(G)$ if $p \in [1,2)$.
LA - eng
KW - Non-commutative groups; Convolution operators; Asymmetry
UR - http://eudml.org/doc/289333
ER -

References

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  1. BARONTI, M. and FORESTI, G., 1982. An example of asymmetry of convolution operators. Rend. Circ. Mat. Palermo, (2), 31: 341-350. Zbl0505.43005MR693581DOI10.1007/BF02851145
  2. CLARKSON, J.A., 1936. Uniformly convex spaces. Trans. Amer. Mat. Soc., 40: 396-414. Zbl0015.35604MR1501880DOI10.2307/1989630
  3. HERZ, C., 1976. On the asymmetry of norms of convolution operators. J. Functional Anal., 23: 11-22. Zbl0332.43005MR420138
  4. HEWITT, E. and ROSS, K., 1970. Abstract Harmonic Analysis. II. Springer Verlag, New York. Zbl0213.40103
  5. MANTERO, A.M., 1982. Asymmetry of twisted convolution operators. J. Functional Analysis, 47: 145-158. Zbl0533.43007MR663831DOI10.1016/0022-1236(82)90098-2
  6. MANTERO, A.M., 1985. Asymmetry of convolution operators on the Heisenberg group. Boll. Un. Mat. Ital., (6), 4-A: 19-27. Zbl0561.43004MR781790
  7. OBERLIN, D., 1975. M p ( G ) ≠ M p ( G ) ( p - 1 + q - 1 = 1 ) . Israel J. Math., 22: 175-179. MR387956DOI10.1007/BF02760165

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