Some embedding theorems for generalized Nikol'skii classes from Lorentz spaces

Quasi-normed Lorentz spaces Λ_{ψ, q} of 2π-periodic functions with quasinorms $$left| f right|_{psi ,q} = left{ {intlimits_0^{2pi } {psi ^q (t)left[ {frac{1}{t}intlimits_0^t {f * (x)} dx} right]} ^q frac{{dt}}{t}} right}^{{1 mathord{left/ {vphantom {1 q}} right. kern-nulldelimiterspace} q}} $$ (0<q<∞,ω(t): [0,2π]→R is a continuous concave function with finite derivative everywhere on (0, 2gp)) and classes of functions $$H_{psi ,q}^omega equiv { f(x):f(x) in Lambda _{psi ,q} ;mathop {sup }limits_{0 leqq h leqq delta } left| {f(x + h) - f(x)} right|_{psi ,q} = O{ omega (delta )} , delta to + 0} $$ (ω(δ) — modulus of continuity) are studied. Precise embedding conditions of classes H _{ψ, q} ^ω into Lorentz spaces and into each other are obtained: $$begin{array}{*{20}c} {H_{psi ,q_1 }^omega subset Lambda _{psi ,q_2 } ;} & {H_{psi ,q_1 }^omega subset {rm H}_{psi ,q_2 }^{omega * } ,} & {0< q_2< q_1< infty ,} end{array} $$ under conditions (mathop {lim }limits_{t to infty } frac{{psi (2t)}}{{psi (t)}} > 1,mathop {overline {lim } }limits_{x to infty } frac{{psi (2t)}}{{psi (t)}}< 2) andω(δ)=O{ω(δ ²)},δ→+0, andω ^* (δ) is an arbitrary modulus of continuity.