We find a real analytic Levi-flat hypersurface in$C$^{2}containing a bounded contractible domain which is a determining set for pluriharmonic functions.

We give sharp estimates for the fractional maximal function in terms of Hausdorff capacity. At the same time we identify the real interpolation spaces between$L$_{1}and the Morrey space $\mathcal{L}^{1,\lambda}$ . The result can be viewed as an analogue of the Hardy–Littlewood maximal theorem for the fractional maximal function.

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