Extremal function for Moser–Trudinger type inequality with logarithmic weight. (April 2016) - Roy, Prosenjit
Abstract: On the space of weighted radial Sobolev space, the following generalization of Moser–Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If β ∈ [ 0, 1 ) and w 0 ( x ) = | log | x | | β then sup ∫ B ∣ ∇ u ∣ 2 w 0 ≤ 1, u ∈ H 0, r a d 1 ( w 0, B ) ∫ B e α u 2 1 − β d x < ∞, if and only if α ≤ α β = 2 [ 2 π ( 1 − β ) ] 1 1 − β . We prove the existence of an extremal function for the above inequality for the critical case when α = α β thereby generalizing the result of Carleson–Chang who proved the case when β = 0 .