A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on ({mathbb{R}^d}) for which there exist some constants C, η > 0 such that ({mu(B(x,varepsilon))leq Cvarepsilon^eta}) for all ε > 0 and all ({xinmathbb{R}^d}) . For such measures μ, we prove that the upper quantization dimension ({overline{D}(mu)}) of μ is bounded from above by its upper packing dimension and the lower one ({underline{D}(mu)}) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.