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14
Compactness of embeddings of the Trudinger–Strichartz type for rotation invariant functions
 Houston J. Math
"... Abstract. Let H be a close subgroup of the group of rotations of Rn. The subspaces of functions of fractional Sobolev space H n/p p (Rn) and Besov spaces B n/p p,q (Rn) invariant with respect to natural action of H are investigated. We give sufficient and necessary conditions for the compactness of ..."
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Abstract. Let H be a close subgroup of the group of rotations of Rn. The subspaces of functions of fractional Sobolev space H n/p p (Rn) and Besov spaces B n/p p,q (Rn) invariant with respect to natural action of H are investigated. We give sufficient and necessary conditions for the compactness of the TrudingerStrichartz type embeddings of the above spaces into the “exponential ” Orlicz classes. It was noticed in late seventies that symmetry conditions on Rn can be used to obtain compactness of Sobolev embeddings. The phenomenon was observed by several authors for first order Sobolev spaces, we refer to Berestycki and Lions [1], Coleman, Glazer, and Martin [3], Strauss [18], and Lions [9]. The results are important for existence of entire solutions of semilinear equations, cf. [8].The necessary and sufficient condition for the compactness of Sobolev embeddings of rotation invariant subspaces of BesovTriebelLizorkin classes were proved in [14] and [17]. On the other hand much attention has been paid to limiting embeddings, both of TrudingerStrichartz and BrezisWainger 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 TrudingerStrichartz type for rotation invariant functions on Rn.
Decomposition and Moser’s lemma
 Revista Matematica Complutense. 2002. V
"... Using the idea of the optimal decomposition developed in recent papers [EK2] by the same authors and in [CUK] we study the boundedness of the operator Tg(x) = ∫ 1 x g(u) du/u, x ∈ (0, 1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and LorentzOrlicz spaces. These oper ..."
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Using the idea of the optimal decomposition developed in recent papers [EK2] by the same authors and in [CUK] we study the boundedness of the operator Tg(x) = ∫ 1 x g(u) du/u, x ∈ (0, 1), and its logarithmic variant between Lorentz spaces and exponential Orlicz and LorentzOrlicz spaces. These operators are naturally linked with Moser’s lemma, O’Neil’s convolution inequality, and estimates for functions with prescribed rearrangement. We give sufficient conditions for and very simple proofs of uniform boundedness of exponential and double exponential integrals in the spirit of the celebrated lemma due to Moser [Mo]. 1
Recent Developments Concerning Entropy And Approximation Numbers
 International Spring School, Nonlinear Analysis, Function Spaces and Applications V, Prague
, 1994
"... this paper,\Omega is a bounded domain with C ..."
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Limiting Imbeddings  The Case Of Missing Derivatives
, 1995
"... We consider fractional Sobolev spaces with dominating mixed derivatives and prove some generalizations of Trudinger's limiting imbedding theorem. ..."
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We consider fractional Sobolev spaces with dominating mixed derivatives and prove some generalizations of Trudinger's limiting imbedding theorem.
Grand Sobolev spaces and their . . .
, 1996
"... For q> 1, � a bounded open set in Rn, the grand Sobolev space W 1,q)o (�) consists of all functions u ∈ �0<ε≤q−1W 1,q−ε0 (�) such that (1.1) �u�W 1,q)0 = sup0<ε≤q−1 ..."
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For q> 1, � a bounded open set in Rn, the grand Sobolev space W 1,q)o (�) consists of all functions u ∈ �0<ε≤q−1W 1,q−ε0 (�) such that (1.1) �u�W 1,q)0 = sup0<ε≤q−1
Vanishing exponential integrabilities for Riesz
"... Abstract Our aim in this paper is to show the vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, as an improvement of continuity results of Sobolev functions. We also show the vanishing double exponential integrability. 1 Introduction For 0 < ff < n, we de ..."
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Abstract Our aim in this paper is to show the vanishing exponential integrability for Riesz potentials of functions in Orlicz classes, as an improvement of continuity results of Sobolev functions. We also show the vanishing double exponential integrability. 1 Introduction For 0 < ff < n, we define the Riesz potential of order ff for a nonnegative measurable function f on Rn by Rfff (x) = Z x yffnf (y) dy. Here we assume that Rfff 6j 1, or equivalently,Z