## 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.
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.)
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### References:

  Bary, N, Trigonometric series, (1964), Macmillan Co New York, two volumes · Zbl 0129.28002  Beckner, W; Janson, S; Jerison, J, Convolution inequalities on the circle, (), 32-43  Boas, R.P, Integrability theorems for trigonometric transforms, (1967), Springer-Verlag New York · Zbl 0145.06804  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  Christ, M, A convolution inequality concerning Cantor-Lebesgue measures, Rev. mat. iberoamericana, 1, 79-83, (1985) · Zbl 0644.42011  Duren, P, Theory of Hp spaces, (1970), Academic Press New York · Zbl 0215.20203  Edwards, R.E, Fourier series, (1967), Springer-Verlag New York, two volumes · Zbl 0189.06602  Enflo, P; Starbird, T, Subspaces of L1 containing L1, Studia math., 42, 203-225, (1979) · Zbl 0433.46027  {\scJ. J. F. Fournier}, private communication.  {\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  Graham, C.C; McGehee, O.C, Essays in commutative harmonic analysis, (1979), Springer-Verlag New York · Zbl 0439.43001  Hare, K, A characterization of Lp-improving measures, (), 295-299 · Zbl 0664.43001  {\scK. Hare}, private communication.  Igari, S, Functions of Lp-multipliers, Tôhoku math. J., 21, 304-320, (1969) · Zbl 0183.14802  López, J; Ross, K, Sidon sets, (1975), Dekker New York · Zbl 0351.43008  Oberlin, D, A convolution property of the Cantor-Lebesgue measure, (), 113-117 · Zbl 0501.42007  Ritter, D, Some singular measures on the circle which improve Lp spaces, (), 133-144 · Zbl 0637.43002  Ritter, D, Most Riesz product are Lp-improving, (), 291-295 · Zbl 0593.43002  Rudin, W, Fourier analysis on groups, (1962), Wiley New York · Zbl 0107.09603  Sarnak, P, Spectra of singular measures as multipliers on Lp, J. funct. anal., 37, 302-317, (1980) · Zbl 0449.47025  Stein, E.M, Singular integrals and differentiability properties of functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501  Stein, E.M, Harmonic analysis on rn, (), 97-135  Stein, E.M; Weiss, G, Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton Univ. Press Princeton · Zbl 0232.42007  Taylor, J.L, Measure algebras, () · Zbl 0186.20001  Zafran, M, Spectra of multipliers, Pacific J. math., 47, 609-626, (1973) · Zbl 0242.43006  Zygmund, A, Trigonometric series, (1959), Cambridge Univ. Press Cambridge, England, two volumes · JFM 58.0280.01
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