A left inverse approach to sharp Sobolev inequalities

The thesis is devoted to sharp embeddings from Soblolev spaces into Lebesque spaces in one dimension. The paper included in the thesis concerns the embedding from Sobolev spaces of functions that have square integrable derivatives of integer order k into the space of Lebesgue integrable functions (p=1). The main result is the sharp constant and the explicit extremal functions given by the Landau kernels. The result improves on the two-sided estimates obtained by Kalyabin in 2014. We survey the literature of the sharp Sobolev inequalities and discuss discrete analogues and possible extensions.