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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.