N.Yu. Antonov, A.N. Lukoyanov. Order estimates for Lebesgue constants of Fourier sums in Orlicz spaces ... P. 35-47

We consider the problem of order estimates for partial sums of trigonometric Fourier series as operators from Orlicz spaces $L^{\varphi}_{2\pi}$ to the space of $2\pi$-periodic continuous functions $C_{2\pi}$. It is established that an arbitrary function $\varphi$ generating an Orlicz class satisfies the estimate
$$
||S_n(f)||_{C_{2\pi}} \le C \varphi ^{-1} (n) \ln (n+1) ||f||_{L^{\varphi}_{2\pi}}, \qquad \qquad \qquad \qquad \qquad \qquad (\ast )
$$
where $f \in L^{\varphi}_{2\pi}$, $n \in \mathbb{N}$, $S_n(f)$ is the $n$th partial sum of the trigonometric Fourier series of $f$, and the constant $C>0$ is independent of $f$ and $n$. In addition, it is shown that if the function $\varphi$ satisfies the $\Delta_2$-condition, then the estimate can be improved. More exactly,
$$
||S_n(f)||_{C_{2\pi}} \le C \varphi ^{-1} (n) ||f||_{L^{\varphi}_{2\pi}}, \qquad f \in L^{\varphi}_{2\pi}, \, n \in \mathbb{N}, \, C=C(\varphi ). \qquad \qquad (\ast \ast )
$$
Counterexamples are constructed, which show that if $\varphi$ satisfies the $\Delta_2$-condition, then estimate ($\ast \ast $) is unimprovable in order on the space $L^{\varphi}_{2\pi}$ and, if $\varphi$ satisfies the $\Delta^2$-condition, then estimate ($\ast $) is unimprovable in order on the space $ L^{\varphi}_{2\pi}$.

Keywords: Fourier series, Orlicz space, Lebesgue constants

REFERENCES

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Received July 28, 2021

Revised October 25, 2021

Accepted October 27, 2021

Nikolay Yur’evich Antonov, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: Nikolai.Antonov@imm.uran.ru

Alexander Nikolaevich Lukoyanov, graduate student, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: LiableFish@yandex.ru

Cite this article as: N.Yu.Antonov, A.N.Lukoyanov. Order estimates for Lebesgue constants of Fourier sums in Orlicz spaces, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 4, pp. 35–47.