Through a geometric capacitary analysis based on space dualities, this paper addresses several fundamental aspects of functional analysis and potential theory for the Morrey spaces in harmonic analysis over the Euclidean spaces.

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For a compact connected set$X$⊆$ℓ$^{∞}, we define a quantity$β$′($x$,$r$) that measures how close$X$may be approximated in a ball$B$($x$,$r$) by a geodesic curve. We then show that there is$c$>0 so that if$β$′($x$,$r$)>$β$>0 for all$x$∈$X$and$r$<$r$_{0}, then $\operatorname{dim}X>1+c\beta^{2}$ . This generalizes a theorem of Bishop and Jones and answers a question posed by Bishop and Tyson.