A note on certain integrals along polynomial compound curves
Abstract In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves {x=P(φ(|y|))y′} $\{x=P(\varphi (|y|))y'\}$, where P is a real polynomial satisfying P(0)=0 $P(0)=0$...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2019-03-01
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| Series: | Journal of Inequalities and Applications |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13660-019-2023-9 |
| _version_ | 1818133227798265856 |
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| author | Feng Liu Pu Zhang |
| author_facet | Feng Liu Pu Zhang |
| author_sort | Feng Liu |
| collection | DOAJ |
| description | Abstract In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves {x=P(φ(|y|))y′} $\{x=P(\varphi (|y|))y'\}$, where P is a real polynomial satisfying P(0)=0 $P(0)=0$ and φ satisfies certain smooth conditions. Under the conditions that Ω∈H1(Sn−1) $\varOmega \in H^{1} (\mathbf{S}^{n-1})$ and h∈Δγ(R+) $h\in \Delta _{\gamma }(\mathbf{R}_{+})$ for some γ>1 $\gamma >1$, we prove that the above operators are bounded on the Lebesgue space L2(Rn) $L^{2}( \mathbf{R}^{n})$. Moreover, the L2 $L^{2}$-bounds of the maximal functions related to the above integrals are also established. Particularly, the bounds are independent of the coefficients of the polynomial P. In addition, we also present certain Hardy type inequalities related to these operators. |
| first_indexed | 2024-12-11T08:49:23Z |
| format | Article |
| id | doaj.art-5b65907c2291402daeefcb383e2d4fba |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-11T08:49:23Z |
| publishDate | 2019-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-5b65907c2291402daeefcb383e2d4fba2022-12-22T01:14:03ZengSpringerOpenJournal of Inequalities and Applications1029-242X2019-03-012019111610.1186/s13660-019-2023-9A note on certain integrals along polynomial compound curvesFeng Liu0Pu Zhang1College of Mathematics and Systems Sciences, Shandong University of Science and TechnologyDepartment of Mathematics, Mudanjiang Normal UniversityAbstract In this paper we consider a singular integral operator and a parametric Marcinkiewicz integral operator with rough kernel. These operators have singularity along sets of the form curves {x=P(φ(|y|))y′} $\{x=P(\varphi (|y|))y'\}$, where P is a real polynomial satisfying P(0)=0 $P(0)=0$ and φ satisfies certain smooth conditions. Under the conditions that Ω∈H1(Sn−1) $\varOmega \in H^{1} (\mathbf{S}^{n-1})$ and h∈Δγ(R+) $h\in \Delta _{\gamma }(\mathbf{R}_{+})$ for some γ>1 $\gamma >1$, we prove that the above operators are bounded on the Lebesgue space L2(Rn) $L^{2}( \mathbf{R}^{n})$. Moreover, the L2 $L^{2}$-bounds of the maximal functions related to the above integrals are also established. Particularly, the bounds are independent of the coefficients of the polynomial P. In addition, we also present certain Hardy type inequalities related to these operators.http://link.springer.com/article/10.1186/s13660-019-2023-9Singular integralsMarcinkiewicz integralsMaximal operatorsPolynomial compound curvesHardy spaces |
| spellingShingle | Feng Liu Pu Zhang A note on certain integrals along polynomial compound curves Journal of Inequalities and Applications Singular integrals Marcinkiewicz integrals Maximal operators Polynomial compound curves Hardy spaces |
| title | A note on certain integrals along polynomial compound curves |
| title_full | A note on certain integrals along polynomial compound curves |
| title_fullStr | A note on certain integrals along polynomial compound curves |
| title_full_unstemmed | A note on certain integrals along polynomial compound curves |
| title_short | A note on certain integrals along polynomial compound curves |
| title_sort | note on certain integrals along polynomial compound curves |
| topic | Singular integrals Marcinkiewicz integrals Maximal operators Polynomial compound curves Hardy spaces |
| url | http://link.springer.com/article/10.1186/s13660-019-2023-9 |
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