By Irving Kaplansky

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This justifies the fact that the log definition of the logarithmic dyadic blocks j does not take into account the low frequencies. log • Clearly there is j0 ∈ Z such that for any function u in S (Rd ), we have j u ≡ 0 for j ≤ j0 . 6. This obviously ensures log that the sequence ( j u) j≥ j0 is (2 j ) j≥ j0 log-oscillating. • Since for applications to the Orlicz space, the logarithmic Littlewood-Paley theory is mostly relevant in 2ND case, we shall limit ourselves in what follows to this case. 23) 1 · Indeed, by with K N = √ 2N (2π)2N 3 Where obviously ξ = |ξ| · ω, with ω ∈ S2N −1 .

Chemin, R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, Grundlehren der mathematischen Wissenschaften (Springer, New York, 2011) 7. H. Bahouri, M. Majdoub, N. Masmoudi, On the lack of compactness in the 2D critical Sobolev embedding. J. Funct. Anal. 260, 208–252 (2011) 8. H. Bahouri, C. Fermanian, I. Gallagher, Refined inequalities on graded lie groups. Notes aux Comptes-Rendus de l’Académie des Sciences de Paris, Série 350(350), 393–397 (2012) 9. H. Bahouri, M. Majdoub, N.

We start with || f ||2 − ||g||2 = 1 on Q(n − l, l). We write f = π A f ⊕ (1 − π A f ) and similarly for g to obtain (9) 1 = ||π A f ||2 + ||(1 − π A ) f ||2 − ||π B g||2 − ||(1 − π B )g||2 on the set Q(n − l, l). We multiply the first and third terms on the right-hand side of (9) by ||z||2 − ||w||2 , which is 1 on Q(n − l, l). P. D’Angelo We then use the two formal identities ||h ⊗ H ||2 = ||h||2 ||H ||2 ||h ⊕ H ||2 = ||h||2 + ||H ||2 each several times to obtain the result. 1 allows us to construct many polynomial and rational maps via iterating the tensor product operation.