On relations among moduli of continuity in Lorentz spaces

Abstract

Estimates are obtained among moduli of continuity of functions in several variables that belong to various Lorentz spaces. The functions considered are periodic with period 1 in each variable. More exactly the following theorem is proved: If 0<α,β<∞ϕ(t) and ψ(t) are so-called ϕ-functions such that α _;ϕ>β_ψ1α_ψ>1, and

$$\int_a^1 {\left( {\frac{{\varphi (t^N )}}{{\psi (t^N )t}}} \right)} ^\beta \frac{{dt}}{t} \leqslant C(\varphi ,\psi ,\beta ,N)\left( {\frac{{\varphi (a^N )}}{{\psi (a^N )a}}} \right)^\beta {\text{ (}}0 < a \leqslant 1),$$

then for any 0 <δ≤1 we have

$$\left( {\int_\delta ^1 {\left( {\frac{{\varphi (t^N )}}{{\psi (t^N )t}}\omega _{\Lambda (\psi ,\alpha )} (f,t)} \right)} ^\beta \frac{{dt}}{t}} \right)^{{1 \mathord{\left/ {\vphantom {1 \beta }} \right. \kern-\nulldelimiterspace} \beta }} \leqslant {\text{ }} \leqslant C_1 (\varphi ,\psi ,\alpha ,\beta ,N)\frac{{\varphi (\delta ^N )}}{{\psi (\delta ^N )^\delta }}\left( {\int_0^\delta {\left( {\frac{{\psi (t^N )}}{{\varphi (t^N )}}\omega _{\Lambda (\varphi ,\beta )} (f,t)} \right)} ^\alpha \frac{{dt}}{t}} \right)^{{1 \mathord{\left/ {\vphantom {1 \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ }}$$

. The exactness of this estimate is also discussed.