Variable martingale Hardy-Lorentz-Karamata spaces and their applications in Fourier Analysis
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913244164128768 |
|---|---|
| author | Hao, Zhiwei Ding, Xinru Li, Libo Weisz, Ferenc |
| author_facet | Hao, Zhiwei Ding, Xinru Li, Libo Weisz, Ferenc |
| contents | In this paper, we introduce a new class of function spaces, which unify and generalize Lorentz-Karamata spaces, variable Lorentz spaces and other several classical function spaces. Based on the new spaces, we develop the theory of variable martingale Hardy-Lorentz-Karamata spaces and apply it to Fourier Analysis. To be precise, we discuss the basic properties of Lorentz-Karamata spaces with variable exponents. We introduce five variable martingale Hardy-Lorentz-Karamata spaces and characterize them via simple atoms as well as via atoms. As applications of the atomic decompositions, dual theorems and the generalized John-Nirenberg theorem for the new framework are presented. Moreover, we obtain the boundedness of $σ$-sublinear operator defined on variable martingale Hardy-Lorentz-Karamata spaces, which leads to martingale inequalities and the relation of the five variable martingale Hardy-Lorentz-Karamata spaces. Also, we investigate the boundedness of fractional integral operators in this new framework. Finally, we deal with the applications of variable martingale Hardy-Lorentz-Karamata spaces in Fourier analysis by using the previous results. More precisely, we show that the partial sums of the Walsh-Fourier series converge to the function in norm if $f\in L_{p(\cdot),q,b}$ with $1<p_-\le p_+<\infty$. The Fejér summability method is also studied and it is proved that the maximal Fejér operator is bounded from variable martingale Hardy-Lorentz-Karamata spaces to variable Lorentz-Karamata spaces. As a consequence, we get conclusions about almost everywhere and norm convergence of Fejér means. The results obtained in this paper generalize the results for martingale Hardy-Lorentz-Karamata spaces and variable martingale Hardy-Lorentz spaces. Especially, we remove the condition that $b$ is nondecreasing in previous literature. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_16133 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Variable martingale Hardy-Lorentz-Karamata spaces and their applications in Fourier Analysis Hao, Zhiwei Ding, Xinru Li, Libo Weisz, Ferenc Functional Analysis Probability In this paper, we introduce a new class of function spaces, which unify and generalize Lorentz-Karamata spaces, variable Lorentz spaces and other several classical function spaces. Based on the new spaces, we develop the theory of variable martingale Hardy-Lorentz-Karamata spaces and apply it to Fourier Analysis. To be precise, we discuss the basic properties of Lorentz-Karamata spaces with variable exponents. We introduce five variable martingale Hardy-Lorentz-Karamata spaces and characterize them via simple atoms as well as via atoms. As applications of the atomic decompositions, dual theorems and the generalized John-Nirenberg theorem for the new framework are presented. Moreover, we obtain the boundedness of $σ$-sublinear operator defined on variable martingale Hardy-Lorentz-Karamata spaces, which leads to martingale inequalities and the relation of the five variable martingale Hardy-Lorentz-Karamata spaces. Also, we investigate the boundedness of fractional integral operators in this new framework. Finally, we deal with the applications of variable martingale Hardy-Lorentz-Karamata spaces in Fourier analysis by using the previous results. More precisely, we show that the partial sums of the Walsh-Fourier series converge to the function in norm if $f\in L_{p(\cdot),q,b}$ with $1<p_-\le p_+<\infty$. The Fejér summability method is also studied and it is proved that the maximal Fejér operator is bounded from variable martingale Hardy-Lorentz-Karamata spaces to variable Lorentz-Karamata spaces. As a consequence, we get conclusions about almost everywhere and norm convergence of Fejér means. The results obtained in this paper generalize the results for martingale Hardy-Lorentz-Karamata spaces and variable martingale Hardy-Lorentz spaces. Especially, we remove the condition that $b$ is nondecreasing in previous literature. |
| title | Variable martingale Hardy-Lorentz-Karamata spaces and their applications in Fourier Analysis |
| topic | Functional Analysis Probability |
| url | https://arxiv.org/abs/2402.16133 |