Symmetrization and second-order Sobolev inequalities

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Optimal Sobolev imbeddings (2006, background): “Theorem A will have a number of important applications, as is clear, for example, from [5] or [2].”
Adams’ Inequality and Limiting Sobolev Embeddings into Zygmund Spaces (2012, methodology): “Then, we will consider the non limiting embeddings for second order Sobolev space into Lorentz space, which are known (see [11]), apart from the embedding constant: here we give a new proof, based upon symmetrization argument and one dimensional estimates, and we exhibit the optimal constant.”
Higher-order Sobolev embeddings and isoperimetric inequalities (2013, background): “3) was proved in [26], as a consequence of a non-standard rearrangement inequality for second-order derivatives (see also [24] for a related one-dimensional second-order rearrangement inequality).”