Trudinger–Moser type inequalities with logarithmic weights in dimension NN

We consider borderline embeddings of Trudinger–Moser type for weighted Sobolev spaces in bounded domains in RNRN. The embeddings go into Orlicz spaces with exponential growth functions. It turns out that the most interesting weights are powers of the logarithm, for which an explicit dependence of the maximal growth functions can be established. Corresponding Moser type results are also proved, with explicit sharp exponents. In the particular case of a logarithmic weight with the limiting exponent N−1N−1, a maximal growth of double exponential type is obtained, while for any larger exponent the embedding goes into L∞L∞.