Search results for "28A12"

showing 2 items of 12 documents

Density of Lipschitz functions in energy

2020

In this paper, we show that the density in energy of Lipschitz functions in a Sobolev space $N^{1,p}(X)$ holds for all $p\in [1,\infty)$ whenever the space $X$ is complete and separable and the measure is Radon and finite on balls. Emphatically, $p=1$ is allowed. We also give a few corollaries and pose questions for future work. The proof is direct and does not involve the usual flow techniques from prior work. It also yields a new approximation technique, which has not appeared in prior work. Notable with all of this is that we do not use any form of Poincar\'e inequality or doubling assumption. The techniques are flexible and suggest a unification of a variety of existing literature on th…

Primary 46E35 Secondary 30L99 26B30 28A12Mathematics - Classical Analysis and ODEsApplied MathematicsClassical Analysis and ODEs (math.CA)FOS: MathematicsfunktionaalianalyysiAnalysis
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A note on the dimensions of Assouad and Aikawa

2013

We show that in Euclidean space and other regular metric spaces, the notions of dimensions defined by Assouad and Aikawa coincide. In addition, in more general metric spaces, we study the relationship between these two dimensions and a related codimension and give an application of the Aikawa (co)dimension for the Hardy inequalities.

Pure mathematicsAssouad dimensionEuclidean spaceGeneral Mathematicsmetric spaceDimension (graph theory)Mathematical analysista111CodimensionAikawa dimension54F4554E35Metric space26D15Hardy inequalitydoubling measureMathematics::Metric Geometry28A12MathematicsJournal of the Mathematical Society of Japan
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