engleski

Mixed means over balls and annuli and lower bounds for operator norms of maximal functions

We prove mixed-means inequalities for integral means of arbitrary real order, where one of the means is taken over the ball in R^n centered at x and of radius delta*x, delta>0. From this result we deduce the operator norm of the operator S_delta which averages a function |f| from L^p(R^n) over the same balls, introduced by M. Christ and L. Grafakos (Hardy type inequality). We also obtain the operator norm of the related geometric mean operator (Carleman type inequality). Moreover, we indicate analogous results for annuli and discuss estimations related to Hardy-Littlewood and spherical maximal functions.

mixed means; integral means; balls and annuli; potential weights; Hardy's inequality; Hardy-Littlewood maximal function; spherical maximal function

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