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...(x) = ∫ ∞ 0 χ∗{|f |>t}(x) dt . In the following proposition, we collect without proofs the main features of the rearrangement f ∗ (for a complete presentation and more details, we refer the reader to =-=[29, 32, 46]-=- and the references therein). Proposition 3.10. Under the above notation, the following properties hold: • The rearrangement f ∗ of a function f is a nonnegative radially symmetric and nonincreasing f...

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... Ω of R2 . Precisely, if f ∈ H1 0(Ω) ∩ ˙ Cα (Ω) then, its corresponding symmetric non -increasing function, usually denoted by f ⋆ , is in f ⋆ ∈ H1 0(BR) ∩ ˙ Cα √ |Ω| (BR), where R = . We 2π refer to =-=[12]-=-, [2] for the definition, the properties and applications of rearrangements of functions. Applying Theorem 1.2 and Theorem 1.3 results to f ⋆ and using the fact that ‖f ⋆ ‖L∞ = ‖f‖L∞ ‖∇f ⋆ ‖ L 2 ≤ ‖∇f...

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... it is almost verbally the same as the one for inequality (6.11) given in [24] when [24, Theorem 2.1] is replaced by Theorem 1. Finally, we come to the case p > n. The sharp Morrey–Sobolev inequality =-=[66]-=- states that if f ∈ W 1,p (R n ), p > n, such that V (sprtf) < ∞, then (6.13) ‖f‖∞ ≤ αn,p V (sprtf) (p−n)/np ‖∇f‖p, where the optimal constant αn,p is given by αn,p = n −1/p κ −1/n n ( p − 1 p − n ) (...