Given a measure on a regular planar domain D, the Gaussian multiplicative chaos measure of studied in this paper is the random measure e obtained as the limit of the exponential of the -parameter circle averages of the Gaussian free field on D weighted by . We investigate the dimensional and geometric properties of these random measures. We first show that if is a finite Borel measure on D with exact dimension > 0, then the associated GMC measure e is non-degenerate and is almost surely exact dimensional with dimension 􀀀 22 , provided 22 < . We then show that if t is a Holder-continuously parameterized family of measures then the total mass of et varies Holder-continuously with t, provided that is sufficiently small. As an application we show that if < 0:28, then, almost surely, the orthogonal projections of the -Liouville quantum gravity measure e on a rotund convex domain D in all directions are simultaneously absolutely continuous with respect to Lebesgue measure with Holder continuous densities. Furthermore, e has positive Fourier dimension almost surely.
|Number of pages||37|
|Journal||Transactions of the American Mathematical Society|
|Early online date||23 May 2019|
|Publication status||Published - 2019|