# On convolution operators with small support which are far from being convolution by a bounded measure

Colloquium Mathematicae (1994)

- Volume: 67, Issue: 1, page 33-60
- ISSN: 0010-1354

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topGranirer, Edmond. "On convolution operators with small support which are far from being convolution by a bounded measure." Colloquium Mathematicae 67.1 (1994): 33-60. <http://eudml.org/doc/210262>.

@article{Granirer1994,

abstract = {Let $CV_p(F)$ be the left convolution operators on $L^p(G)$ with support included in F and $M_p(F)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $CV_p(F)$, $CV_p(F)/M_p(F)$ and $CV_p(F)/W$ are as big as they can be, namely have $l^∞$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_p(F)$. Other subspaces of $CV_p(F)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.},

author = {Granirer, Edmond},

journal = {Colloquium Mathematicae},

keywords = {convolution operators; locally compact groups; bounded measure},

language = {eng},

number = {1},

pages = {33-60},

title = {On convolution operators with small support which are far from being convolution by a bounded measure},

url = {http://eudml.org/doc/210262},

volume = {67},

year = {1994},

}

TY - JOUR

AU - Granirer, Edmond

TI - On convolution operators with small support which are far from being convolution by a bounded measure

JO - Colloquium Mathematicae

PY - 1994

VL - 67

IS - 1

SP - 33

EP - 60

AB - Let $CV_p(F)$ be the left convolution operators on $L^p(G)$ with support included in F and $M_p(F)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $CV_p(F)$, $CV_p(F)/M_p(F)$ and $CV_p(F)/W$ are as big as they can be, namely have $l^∞$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_p(F)$. Other subspaces of $CV_p(F)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

LA - eng

KW - convolution operators; locally compact groups; bounded measure

UR - http://eudml.org/doc/210262

ER -

## References

top- [BL] Y. Benyamini and P. K. Lin, Norm one multipliers on ${L}^{p}\left(G\right)$, Ark. Mat. 24 (1986), 159-173.
- [BE] B. Brainerd and R. E. Edwards, Linear operators which commute with translations. Part I: Representation theorems, J. Austral. Math. Soc. 6 (1966), 289-327. Zbl0154.39202
- [Ch1] C. Chou, Weakly almost periodic functions and Fourier-Stieltjes algebras of locally compact groups, Trans. Amer. Math. Soc. 274 (1982), 141-157. Zbl0505.43004
- [Ch2] C. Chou, Topological invariant means on the von Neumann algebra VN(G), ibid. 273 (1982), 207-229. Zbl0507.22007
- [Co] H. S. Collins, Strict, weighted, and mixed topologies and applications, Adv. in Math. 19 (1976), 207-237. Zbl0347.46023
- [Cow] M. Cowling, An application of Littlewood-Paley theory in harmonic analysis, Math. Ann. 241 (1979), 83-96. Zbl0399.43004
- [CF] M. Cowling and J. J. F. Fournier, Inclusions and noninclusion of spaces of convolution operators, Trans. Amer. Math. Soc. 221 (1976), 56-95. Zbl0331.43007
- [De1] J. Delaporte, Convoluteurs continus et topologie stricte, thèse, Université Lausanne, 1989.
- [De2] J. Delaporte, Convoluteurs continus et groupes quotients, C. R. Math. Rep. Acad. Sci. Canada 14 (1992), 167-172.
- [Der] A. Derighetti, A propos des convoluteurs d'un groupe quotient, Bull. Sci. Math. 107 (1983), 3-23. Zbl0522.43003
- [DU] J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys 15, Amer. Math. Soc., 1977.
- [Do] Y. Domar, Harmonic analysis based on certain commutative Banach algebras, Acta Math. 96 (1956), 1-66. Zbl0071.11302
- [DR1] C. F. Dunkl and D. E. Ramirez, ${L}^{p}$ multipliers on compact groups, preprint.
- [DR2] C. F. Dunkl and D. E. Ramirez, ${C}^{*}$-algebras generated by Fourier-Stieltjes transforms, Trans. Amer. Math. Soc. 164 (1972), 435-441. Zbl0211.15903
- [EP] R. E. Edwards and J. F. Price, A naively constructive approach to boundedness principles with applications to harmonic analysis, Enseign. Math. 16 (1970), 255-296. Zbl0208.15503
- [Ey] P. Eymard, Algèbres ${A}_{p}$ et convoluteurs de ${L}^{p}$, Séminaire Bourbaki, 22e année, 1969/70, no. 367.
- [Fe] G. Fendler, An ${L}^{p}$-version of a theorem of D. A. Raikov, Ann. Inst. Fourier (Grenoble) 35 (1) (1985), 125-135.
- [FG] A. Figà-Talamanca and G. I. Gaudry, Multipliers and sets of uniqueness of ${L}^{p}$, Michigan Math. J. 17 (1970), 179-191. Zbl0197.40103
- [GI] G. I. Gaudry and I. R. Inglis, Approximation of multipliers, Proc. Amer. Math. Soc. 44 (1974), 381-384. Zbl0287.42007
- [GMc] C. C. Graham and O. C. McGehee, Essays in Commutative Harmonic Analysis, Springer, New York, 1979. Zbl0439.43001
- [Gr1] E. E. Granirer, On some spaces of linear functionals on the algebras ${A}_{p}\left(G\right)$ for locally compact groups, Colloq. Math. 52 (1987), 119-132. Zbl0649.43004
- [Gr2] E. E. Granirer, Geometric and topological properties of certain ${w}^{*}$ compact convex subsets of double duals of Banach spaces, which arise from the study of invariant means, Illinois J. Math. 30 (1986), 148-174. Zbl0606.46006
- [Gr3] E. E. Granirer, On Baire measures on D-topological spaces, Fund. Math. 60 (1967), 1-22. Zbl0146.12204
- [Gr4] E. E. Granirer, On convolution operators which are far from being convolution by a bounded measure. Expository memoir, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 187-204. Zbl0791.43003
- [Ha] R. Haydon, A non-reflexive Grothendieck space that does not contain ${l}_{\infty}$, Israel J. Math. 40 (1981), 65-73.
- [Hz1] C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (3) (1973), 91-123. Zbl0257.43007
- [Hz2] C. Herz, Une généralisation de la notion de transformée de Fourier-Stieltjes, ibid. 24 (3) (1974), 145-157. Zbl0287.43006
- [Hz3] C. Herz, The theory of p-spaces with an application to convolution operators, Trans. Amer. Math. Soc. 154 (1971), 69-82. Zbl0216.15606
- [HR] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vols. I, II, Springer, 1970.
- [Ka1] J.-P. Kahane et R. Salem, Sur les ensembles linéaires ne portant pas de pseudomesures, C. R. Acad. Sci. Paris 243 (1956), 1185-1187.
- [Ka2] J.-P. Kahane, Sur les réarrangements de fonctions de la classe A, Studia Math. 31 (1968), 287-293. Zbl0177.42202
- [Ka3] J.-P. Kahane, Séries de Fourier Absolument Convergentes, Springer, 1970. Zbl0195.07602
- [Ko] T. W. Körner, A pseudofunction on a Helson set. I, Astérisque 5 (1973), 3-224. Zbl0281.43004
- [La] R. Larsen, An Introduction to the Theory of Multipliers, Springer, 1971.
- [LT] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. I, Springer, 1977. Zbl0362.46013
- [LR] T. S. Liu and A. van Rooij, Invariant means on a locally compact group, Monatsh. Math. 78 (1974), 356-359.
- [Lo] L. H. Loomis, The spectral characterization of a class of almost periodic functions, Ann. of Math. 72 (1960), 362-368. Zbl0094.05801
- [P1] F. Lust-Piquard, Produits tensoriels projectifs d'espaces de Banach faiblement sequentiellement complets, Colloq. Math. 36 (1976), 255-267. Zbl0356.46058
- [P2] F. Lust-Piquard, Means on $C{V}_{p}\left(G\right)$-subspaces of $C{V}_{p}\left(G\right)$ with RNP and Schur property, Ann. Inst. Fourier (Grenoble) 39 (1989), 969-1006. Zbl0675.43001
- [P3] F. Lust-Piquard, Eléments ergodiques et totalement ergodiques dans ${L}^{\infty}\left(\Gamma \right)$, Studia Math. 69 (1981), 191-225. Zbl0476.43001
- [Mc] O. C. McGehee, Helson sets in ${T}^{n}$, in: Conference on Harmonic Analysis, College Park, Maryland, 1971, Lecture Notes in Math. 266, Springer, 1972, 229-237.
- [Me] Y. Meyer, Recent advances in spectral synthesis, ibid., 239-253.
- [Ne] C. Nebbia, Convolution operators on the group of isometries of a homogeneous tree, Boll. Un. Mat. Ital. C (6) 2 (1983), 277-292. Zbl0544.43003
- [Pa] A. L. T. Paterson, Amenability, Math. Surveys Monographs 29, Amer. Math. Soc., 1988.
- [Pi] J. P. Pier, Amenable Locally Compact Groups, Wiley, 1984. Zbl0597.43001
- [Pr] J. F. Price, Some strict inclusions between spaces of ${L}^{p}$-multipliers, Trans. Amer. Math. Soc. 152 (1970), 321-330. Zbl0216.14802
- [Ro] H. P. Rosenthal, Some recent discoveries in the isomorphic theory of Banach spaces, Bull. Amer. Math. Soc. 84 (1978), 803-831. Zbl0391.46016
- [Ru1] W. Rudin, Fourier Analysis on Groups, Wiley, 1960.
- [Ru2] W. Rudin, Functional Analysis, McGraw-Hill, 1973.
- [Sa] E. Saab, Some characterizations of weak Radon-Nikodym sets, Proc. Amer. Math. Soc. 86 (1982), 307-311. Zbl0494.46047
- [S] S. Saeki, Helson sets which disobey spectral synthesis, ibid. 47 (1975), 371-377. Zbl0297.43007
- [St] E. Stein, On limits of sequences of operators, Ann. of Math. 74 (1961), 140-170. Zbl0103.08903
- [T] M. Talagrand, Un nouveau C(K) qui possède la propriété de Grothendieck, Israel J. Math. 37 (1980), 181-191.
- [Wo1] G. S. Woodward, Une classe d'ensembles épars, C. R. Acad. Sci. Paris 274 (1972), 221-223. Zbl0228.43009
- [Wo2] G. S. Woodward, Invariant means and ergodic sets in Fourier analysis, Pacific J. Math. 54 (1974), 281-299. Zbl0307.43006
- [Wo3] G. S. Woodward, The generalized almost periodic part of an ergodic function, Studia Math. 50 (1974), 103-116. Zbl0283.42019

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