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In this paper we establish new optimal bounds for the derivative of some discrete maximal functions, in both the centred and uncentred versions. In particular, we solve a question originally posed by Bober et al. [‘On a discrete version of Tanaka’s theorem for maximal functions’, Proc. Amer. Math. Soc.140 (2012), 1669–1680].

Consider the Sobolev embedding operator from the space of functions in $W^{1,p}(I)$ with average zero into $L^p$, where $I$ is a finite interval and $p\,{>}\,1$. This operator plays an important role in recent work. The operator norm and its approximation numbers in closed form are calculated. The closed form of the norm and approximation numbers of several similar Sobolev embedding operators on a finite interval have recently been found. It is proved in the paper that most of these operator norms and approximation numbers on a finite interval are the same.

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SHARP INEQUALITIES FOR THE VARIATION OF THE DISCRETE MAXIMAL FUNCTION

APPROXIMATION NUMBERS OF SOBOLEV EMBEDDING OPERATORS ON AN INTERVAL

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2 results

SHARP INEQUALITIES FOR THE VARIATION OF THE DISCRETE MAXIMAL FUNCTION

APPROXIMATION NUMBERS OF SOBOLEV EMBEDDING OPERATORS ON AN INTERVAL