### A Characterization of Pontryagin Duality.

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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 $\mu \ast {L}^{p}\subseteq {L}^{p+\epsilon}$ 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 $\mu \ast {L}^{p}\subseteq {L}^{q}$ for any measure μ in our class.

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 ${L}^{p}\left(\right)$ to ${L}^{q}\left(\right)$. We also give a condition on p which is necessary if this operator maps ${L}^{p}\left(\right)$ into L²().

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 $q\in [2,\mathrm{\infty})$ and is unbounded on ${L}^{p}(G)$ if $p\in [1,2)$.