Abstract: The median of a function on Euclidean space was introduced by F. John in 1965, and can be regarded as a type of average. Unlike the integral average, even for non-integrable functions, a median always exists. However, the median is not unique, in general. In fact, it is well-known that the set of all medians for a function is a closed interval. With the aid of a result due to Poelhuis and Torchinsky (2012), we can see that the endpoints of the closed interval are two distinct rearrangements. We introduce a fractional version of medians and give a similar expression for the set of all fractional medians. We introduce the maximal operator defined via medians instead of integral averages, and establish smoothing properties for the fractional maximal operator. Finally, we give a short proof of Alvino's embedding,$\mathrm{ B}V\to L^{n/(n-1),1}$ by using properties of medians and the coarea formula. Our estimate is covered by a result by Spector (2020).

 

Link Information: https://us02web.zoom.us/j/86263081187?pwd=j8PEZdvRs30anTkXfqSpLR6VtDEEbC.1

Meeting ID: 862 6308 1187

Passcode: 912919

 

Organizer: Daniel Spector (NTNU)