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A convolution property of some measures with self-similar fractal support

Denise Szecsei (2007)

Colloquium Mathematicae

We define a class of measures having the following properties: (1) the measures are supported on self-similar fractal subsets of the unit cube I M = [ 0 , 1 ) M , with 0 and 1 identified as necessary; (2) the measures are singular with respect to normalized Lebesgue measure m on I M ; (3) the measures have the convolution property that μ L p L p + ε for some ε = ε(p) > 0 and all p ∈ (1,∞). We will show that if (1/p,1/q) lies in the triangle with vertices (0,0), (1,1) and (1/2,1/3), then μ L p L q for any measure μ in our class.

A convolution property of the Cantor-Lebesgue measure, II

Daniel M. Oberlin (2003)

Colloquium Mathematicae

For 1 ≤ p,q ≤ ∞, we prove that the convolution operator generated by the Cantor-Lebesgue measure on the circle is a contraction whenever it is bounded from L p ( ) to L q ( ) . We also give a condition on p which is necessary if this operator maps L p ( ) into L²().

A remark on the asymmetry of convolution operators

Saverio Giulini (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

A convolution operator, bounded on L q ( n ) , is bounded on L p ( 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 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 ) .

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