##
**The size of \(L^ p\)-improving measures.**
*(English)*
Zbl 0678.43001

A measure \(\mu\) on a locally compact abelian group G is \(L^ p\)- improving if there is \(r>p\) with \(\mu *L^ p(G)\subseteq L^ r(G)\). (If such an r exists for one p with \(p>1\), then an r exists for any other p with \(1<p<\infty.)\) The present paper continues the study of such measures in three different ways. First, the relationship with absolute continuity is considered. If G is compact and \(\mu\) is non-negative, then for each bounded Borel function f the measure \(f\mu\) is also \(L^ p\)- improving. On the other hand, there exist \(L^ p\)-improving measures \(\mu\) on G for which \(| \mu |\) is not \(L^ p\)-improving. Also, for any \(L^ p\)-improving \(\mu\), there is a probability measure \(\nu\) equivalent to \(\mu\) but for which \(\nu\) is not \(L^ p\)-improving.

The second collection of results relates only to the case in which G is the circle group. The measure \(\mu\) is said to belong to the class Lip(\(\alpha)\) if the distribution function of \(\mu\) satisfies a Lipschitz condition of order \(\alpha\). If, for some \(p<2\), \(\mu *L^ p(G)\subseteq L^ 2(G)\), then \(\mu\) belongs to Lip(1/p-1/2). On the other hand, there are measures in every Lip(\(\alpha)\) class for \(0<\alpha <1\) which are not \(L^ p\)-improving. Moreover, if \(f\in L^ 1(G)\) is such that \(| \hat f(n)|\) decreases monotonically in both directions then f (considered as a measure) is \(L^ p\)-improving if and only if it belongs to some Lipschitz class.

The remaining results are principally concerned with the Fourier- Stieltjes transform \({\hat \mu}\) of an \(L^ p\)-improving measure \(\mu\) on a compact abelian group G. For example, lim sup\(| {\hat \mu}| \leq (2-2/p)^{1/2}\| \mu \|\). If \(sp_ p(\mu)\) denotes the spectrum of \(\mu\) as an operator on \(L^ p(G)\) and \(\Gamma\) the dual group of G, then for each p with \(1<p<\infty\), \(sp_ p(\mu)=cl {\hat \mu}(\Gamma)\) (and this conclusion also holds if \(\mu\) is absolutely continuous with respect to an \(L^ p\)-improving measure). The paper concludes with a list of open questions.

The second collection of results relates only to the case in which G is the circle group. The measure \(\mu\) is said to belong to the class Lip(\(\alpha)\) if the distribution function of \(\mu\) satisfies a Lipschitz condition of order \(\alpha\). If, for some \(p<2\), \(\mu *L^ p(G)\subseteq L^ 2(G)\), then \(\mu\) belongs to Lip(1/p-1/2). On the other hand, there are measures in every Lip(\(\alpha)\) class for \(0<\alpha <1\) which are not \(L^ p\)-improving. Moreover, if \(f\in L^ 1(G)\) is such that \(| \hat f(n)|\) decreases monotonically in both directions then f (considered as a measure) is \(L^ p\)-improving if and only if it belongs to some Lipschitz class.

The remaining results are principally concerned with the Fourier- Stieltjes transform \({\hat \mu}\) of an \(L^ p\)-improving measure \(\mu\) on a compact abelian group G. For example, lim sup\(| {\hat \mu}| \leq (2-2/p)^{1/2}\| \mu \|\). If \(sp_ p(\mu)\) denotes the spectrum of \(\mu\) as an operator on \(L^ p(G)\) and \(\Gamma\) the dual group of G, then for each p with \(1<p<\infty\), \(sp_ p(\mu)=cl {\hat \mu}(\Gamma)\) (and this conclusion also holds if \(\mu\) is absolutely continuous with respect to an \(L^ p\)-improving measure). The paper concludes with a list of open questions.

Reviewer: J.S.Pym

### MSC:

43A05 | Measures on groups and semigroups, etc. |

43A25 | Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups |

43A46 | Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) |

### Keywords:

convolution; locally compact abelian group; absolute continuity; \(L^ p\)-improving measures; distribution function; Lipschitz class; Fourier- Stieltjes transform; dual group
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\textit{C. C. Graham} et al., J. Funct. Anal. 84, No. 2, 472--495 (1989; Zbl 0678.43001)

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### References:

[1] | Bary, N, Trigonometric series, (1964), Macmillan Co New York, two volumes · Zbl 0129.28002 |

[2] | Beckner, W; Janson, S; Jerison, J, Convolution inequalities on the circle, (), 32-43 |

[3] | Boas, R.P, Integrability theorems for trigonometric transforms, (1967), Springer-Verlag New York · Zbl 0145.06804 |

[4] | Bonami, A, Étude des coefficients de Fourier des fonctions de Lp(G), J. inst. Fourier (Grenoble), 20, 335-402, (1970), fasc. 2 · Zbl 0195.42501 |

[5] | Christ, M, A convolution inequality concerning Cantor-Lebesgue measures, Rev. mat. iberoamericana, 1, 79-83, (1985) · Zbl 0644.42011 |

[6] | Duren, P, Theory of Hp spaces, (1970), Academic Press New York · Zbl 0215.20203 |

[7] | Edwards, R.E, Fourier series, (1967), Springer-Verlag New York, two volumes · Zbl 0189.06602 |

[8] | Enflo, P; Starbird, T, Subspaces of L1 containing L1, Studia math., 42, 203-225, (1979) · Zbl 0433.46027 |

[9] | {\scJ. J. F. Fournier}, private communication. |

[10] | {\scC. C. Graham}, Summary of some results on Lp-improving measures, in “Proceedings, SLU-GTE Conference” (D. Colella, Ed.), American Mathematics Society, Providence, RI, in press. · Zbl 0684.42009 |

[11] | Graham, C.C; McGehee, O.C, Essays in commutative harmonic analysis, (1979), Springer-Verlag New York · Zbl 0439.43001 |

[12] | Hare, K, A characterization of Lp-improving measures, (), 295-299 · Zbl 0664.43001 |

[13] | {\scK. Hare}, private communication. |

[14] | Igari, S, Functions of Lp-multipliers, Tôhoku math. J., 21, 304-320, (1969) · Zbl 0183.14802 |

[15] | López, J; Ross, K, Sidon sets, (1975), Dekker New York · Zbl 0351.43008 |

[16] | Oberlin, D, A convolution property of the Cantor-Lebesgue measure, (), 113-117 · Zbl 0501.42007 |

[17] | Ritter, D, Some singular measures on the circle which improve Lp spaces, (), 133-144 · Zbl 0637.43002 |

[18] | Ritter, D, Most Riesz product are Lp-improving, (), 291-295 · Zbl 0593.43002 |

[19] | Rudin, W, Fourier analysis on groups, (1962), Wiley New York · Zbl 0107.09603 |

[20] | Sarnak, P, Spectra of singular measures as multipliers on Lp, J. funct. anal., 37, 302-317, (1980) · Zbl 0449.47025 |

[21] | Stein, E.M, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501 |

[22] | Stein, E.M, Harmonic analysis on rn, (), 97-135 |

[23] | Stein, E.M; Weiss, G, Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton Univ. Press Princeton · Zbl 0232.42007 |

[24] | Taylor, J.L, Measure algebras, () · Zbl 0186.20001 |

[25] | Zafran, M, Spectra of multipliers, Pacific J. math., 47, 609-626, (1973) · Zbl 0242.43006 |

[26] | Zygmund, A, Trigonometric series, (1959), Cambridge Univ. Press Cambridge, England, two volumes · JFM 58.0280.01 |

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