Radial subspaces of Besov and Lizorkin-Triebel classes: Extended strauss lemma and compactness of embeddings

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Sobolev inequalities with symmetry (2009, background): “The inhomogeneous version of (2) has been given in [8] whose proof is based on the atomic decomposition.”
On fractional Schr\"odinger equations in (\mathbb{R}^N) without the Ambrosetti-Rabinowitz condition (2012, background): “ly symmetric H1 functions, this is true by a theorem of Strauss ([24]). In fractional Sobolev spaces, the situation is more complicated. The following theorem is proved (in a more general setting) in [21]. See also [22]. Theorem 7.4. Let 0 <p ≤+∞. (i) Let either s >1/p and 0 <q ≤ +∞or s =1/”