A sublinear version of the Schur Test and weighted norm inequalities
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In this thesis, we study the sublinear elliptic equations of the form (-?u - u qs = 0 s - a.e. on ? u = 0 on ??, where 0 greater than q less than 1 and s is a positive Borel measure on ?. Our results show that, under restrictions on the kernel G, the associated integral equation u - G(uqs) equals 0 has positive solutions in Lq (?, s) if and only if the weighted norm inequality kG?kLq(s) = ?k?k, ?? ? M+(?), holds for some finite constant ? independent of ?. Results characterizing the associated (1, q)-weak type weighted norm inequality kG?kLq,8(s) = ?k?k, ?? ? M+(?), in terms of the capacity of sets is provided. These results apply to elliptic equations for which the kernel satisfies a quasimetric property, is quasimetrically modifiable, or satisfies a complete maximum principle. Analogous results are proved for the fractional maximal operator Ma?.