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Using interpolation methods, new results on the boundedness of quasilinear joint weak type operators on Lorentz-Karamata (LK) spaces are established. LK spaces generalize many function spaces introduced before in literature, for example, the generalized Lorentz- Zygmund spaces, the Zygmund spaces, the Lorentz spaces and, of course, the Lebesgue spaces. The focus is mainly on the limiting cases of interpolation, where the spaces involved are, in certain sense, very close to the endpoint spaces. The results contain both necessary and sufficient conditions for the boundedness of the given operator on LK spaces. The complete characterization of embeddings of LK spaces is also included and the optimality of achieved results is then discussed. Finally, we apply our results to the conjugate function operator, which is known to be bounded on $L_p$ only if $1<p<\infty.$ Powered by TCPDF (www.tcpdf.org)