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...n has been paid to limiting embeddings, both of Trudinger-Strichartz and Brezis-Wainger types. We refer to Strichartz [13], Brezis and Wainger [2], Edmunds, Edmunds and Triebel [4], Edmunds and Krbec =-=[5]-=-,Triebel [19], Edmunds and Triebel [7]. The aim of the paper is to prove the sufficient and necessary conditions for compactness of limiting embeddings of Trudinger-Strichartz type for rotation invari...

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... In the last ten years further tuning of the scale of the Sobolev type spaces imbedded into exponential spaces has been tackled, leading to target spaces of multiple exponential type (see e.g. [FLS], =-=[EK1]-=-). The decomposition technique was used in [KSchm] for an easy proof of a characterization of the double exponential spaces analogous to those in Theorem 2.1 (see also [EGO] for the “classical” global...

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...back certainly to Trudinger [68] and Strichartz [59]; but in [1] references to earlier Russian work will be found. (For very recent variants of this result and for related work, see [26], [27], [28], =-=[29]-=- and [39].) After various upper estimates had been obtained for the approximation and entropy numbers of the embedding map I of the limiting space H n=p p (\Omega\Gamma in L1 (log L) \Gammaa(\Omega\Ga...

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...n (n− 1)γ > 0 and Kn,α = B 1 B nω γ n n−1 . By the results from [26], Orlicz-Sobolev spaces with Φ satisfying (1.5) can be identified as the Lorentz-Sobolev spaces for which we have the embedding results from [15]–[19]. In particular, the space W0L Φ(Ω) with Φ satisfying (1.5) is embedded into an Orlicz space with the Young function that behaves like exp(tγ) for large t. Moreover in the limiting case α = n− 1 we have the embedding into a double exponential space and for α > n−1 we have the embedding into L∞(Ω). For further information we refer the reader to [13], [15], [16], [17], [18], [19], [20], [21] and [26]. The following theorem summarizes known versions of Moser-type inequality for embedding into single exponential spaces (see [9], [15] and [22]). For an information concerning the Moser-type inequalities for the spaces embedded into double and other multiple exponential spaces see [12]. Theorem 1.1. Let n ≥ 2, α < n−1 and K ≥ 0. Suppose that Ω ⊂ Rn is an open bounded set. Let Φ be a Young function satisfying (1.5). (i) If u ∈ W0L Φ(Ω), then ∫ Ω exp ( K|u(x)|γ ) < ∞ . (ii) If K 6= Kn,α, then sup u∈W0LΦ(Ω),‖∇u‖LΦ(Ω)≤1 ∫ Ω exp ( K|u|γ ) { ≤ C(n, α,Φ, |Ω|,K) when K < Kn,α = ∞ when K...

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...functions in W 1,n satisfy finite exponential integrability (see also [1], [3], [13], [17]). Recently great progress has been made for Riesz potentials in the limiting case αp = n (see e.g. [4], [5], =-=[6]-=-, [12], [14]). In this paper, we are concerned with continuity (or differentiability) property for Riesz potentials and aim to show vanishing exponential integrabilities, as an improvement of the resu...

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...unctions in W 1,n satisfy finite exponential integrability (see also [1], [3], [13], [17]). Recently great progress has been made for Riesz potentials in the limiting case ffp = n (see e.g. [4], [5], =-=[6]-=-, [12], [14]). In this paper, we are concerned with continuity (or differentiability) property for Riesz potentials and aim to show vanishing exponential integrabilities, as an improvement of the resu...

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... (log(e + |x0 − x| −1 )) −1 }, then so that � f(y) p(y) ≥ Cf(y) n/α (log(e + f(y))) aα E f(y) n/α (log(e + f(y))) aα dy < ∞, which proves (5.8), since 0 <a<(n − α)/α 2 . With the aid of Edmunds-Krbec =-=[7]-=- and the authors [20] we also obtain Theorem 5.6 in the Euclidean case. Finally we are concerned with the case a =(s − α)/α 2 . Lemma 5.8. Let f be a nonnegative measurable function on B0 with �f�p(·)...