Communications of the
Korean Mathematical Society CKMS

We consider a generalization of the $L^q$-spectrum with respect to two Borel probability measures on $\mathbb{R}^n$ having the same compact support, and also study their behavior under orthogonal projections of measures onto an $m$-dimensional subspace. In particular, we try to improve the main result of Bahroun and Bhouri \cite{1}. In addition, we are interested in studying the behavior of the generalized lower and upper $L^q$-spectrum with respect to two measures on ``sliced'' measures in an $(n-m)$-dimensional linear subspace. The results in this article establish relations with the $L^q$-spectrum with respect to two Borel probability measures and its projections and generalize some well-known results.

Keywords: Orthogonal projection, dimension spectra, sections, convolutions

MSC numbers: Primary 28A78, 28A80