J Fourier Anal Appl (2014) 20:1020–1049 DOI 10.1007/s00041-014-9343-4 Ulyanov-type Inequalities Between Lorentz–Zygmund Spaces Amiran Gogatishvili · Bohumír Opic · Sergey Tikhonov · Walter Trebels Received: 26 March 2013 / Revised: 19 February 2014 / Published online: 22 July 2014 © Springer Science+Business Media New York 2014 Abstract We establish inequalities of Ulyanov-type for moduli of smoothness relating the source Lorentz–Zygmund space L p,r (log L ) α-γ ,γ> 0, and the target space L p ∗ ,s (log L ) α over R n if 1 < p < p ∗ < ∞ and over T n if 1 < p ≤ p ∗ < ∞. The stronger logarithmic integrability (corresponding to L p ∗ ,s (log L ) α ) is balanced by an additional logarithmic smoothness. Keywords Moduli of smoothness · K -functionals · Lorentz–Zygmund spaces Mathematics Subject Classification Primary 41A17 · 46E30 · Secondary 42B15 · 46E35 Communicated by Hans G. Feichtinger. A. Gogatishvili Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 115 67 Prague 1, Czech Republic e-mail: gogatish@math.cas.cz B. Opic Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic e-mail: opic@karlin.mff.cuni.cz S. Tikhonov (B ) ICREA and Centre de Recerca Matemàtica, Apartat 50, Bellaterra, 08193 Barcelona, Spain e-mail: stikhonov@crm.cat W. Trebels Fb. Mathematik, AG Algebra, TU Darmstadt, 64289 Darmstadt, Germany e-mail: trebels@mathematik.tu-darmstadt.de