The logarithmic convolution energy sup over E of the double integral of log(1/|x-y|) G(u(x))G(u(y)) is finite exactly when G is subcritical with exponent gamma >= 1, and maximizers exist in the strictly subcritical case.
Baernstein, A unified approach to symmetrization, Partial Differential equations of elliptic type, eds, Symposia matematica 35, Cambridge University Press 1995, 47-91
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Fractional Trudinger-Moser type inequalities with logarithmic convolution potentials
The logarithmic convolution energy sup over E of the double integral of log(1/|x-y|) G(u(x))G(u(y)) is finite exactly when G is subcritical with exponent gamma >= 1, and maximizers exist in the strictly subcritical case.