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On relations among moduli of continuity in Lorentz spaces

Authors:

Institution:

(1) Department of Mathematics, Faculty of Chemical Technology, Technical University of Volgograd, Prospekt Lenina 28, Volgograd, 400066, Russia

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< agr

(t) and

(t) are so-called phiv

-functions such that agr

_;>

₁

>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 < delta

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.

Keywords:

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