Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics
In this article we obtain sharp Kolmogorov-type inequalities that estimate the uniform norm of a hypersingular integral operator $$ D^{w,\Omega}_K f(x): = \int_{C} w(|t|_K) (f(x+t) - f(x))\Omega(t)dt, x\in C, $$ using the uniform norm of the function $f$ and either the norm $\|f\|_{H^\omega_K(...
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Oles Honchar Dnipro National University
2024-12-01
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| Series: | Researches in Mathematics |
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| Online Access: | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/430/430 |
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| author | V.F. Babenko O.V. Kovalenko N.V. Parfinovych |
| author_facet | V.F. Babenko O.V. Kovalenko N.V. Parfinovych |
| author_sort | V.F. Babenko |
| collection | DOAJ |
| description | In this article we obtain sharp Kolmogorov-type inequalities that estimate the uniform norm of a hypersingular integral operator
$$
D^{w,\Omega}_K f(x): = \int_{C} w(|t|_K) (f(x+t) - f(x))\Omega(t)dt, x\in C,
$$
using the uniform norm of the function $f$ and either the norm $\|f\|_{H^\omega_K(C)}$ determined by a modulus of continuity $\omega$, or the weighted integral norm $\| \Omega^{\frac 1p} \cdot |\nabla f|_{K^\circ}\|_{L_p(C)}$ of the gradient $\nabla f$. Here $C$ is a convex cone in ${\mathbb R}^d$, $d\geq 2$, $\Omega\colon C\to\mathbb R$ is a non-negative homogeneous of degree 0 locally integrable function, $w\colon (0,\infty)\to [0,\infty)$ is some weight function, $|\cdot|_K$ is an arbitrary norm in ${\mathbb R}^d$, $|\cdot|_{K^\circ}$ is its polar norm, and $p\in (d,\infty]$. |
| format | Article |
| id | doaj-art-f63f34e1acb642d3b224d441be9470ed |
| institution | Kabale University |
| issn | 2664-4991 2664-5009 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | Oles Honchar Dnipro National University |
| record_format | Article |
| series | Researches in Mathematics |
| spelling | doaj-art-f63f34e1acb642d3b224d441be9470ed2025-01-05T19:26:10ZengOles Honchar Dnipro National UniversityResearches in Mathematics2664-49912664-50092024-12-01322213910.15421/242417Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristicsV.F. Babenko0https://orcid.org/0000-0001-6677-1914O.V. Kovalenko1https://orcid.org/0000-0002-0446-1125N.V. Parfinovych2https://orcid.org/0000-0002-3448-3798Oles Honchar Dnipro National UniversityOles Honchar Dnipro National UniversityOles Honchar Dnipro National UniversityIn this article we obtain sharp Kolmogorov-type inequalities that estimate the uniform norm of a hypersingular integral operator $$ D^{w,\Omega}_K f(x): = \int_{C} w(|t|_K) (f(x+t) - f(x))\Omega(t)dt, x\in C, $$ using the uniform norm of the function $f$ and either the norm $\|f\|_{H^\omega_K(C)}$ determined by a modulus of continuity $\omega$, or the weighted integral norm $\| \Omega^{\frac 1p} \cdot |\nabla f|_{K^\circ}\|_{L_p(C)}$ of the gradient $\nabla f$. Here $C$ is a convex cone in ${\mathbb R}^d$, $d\geq 2$, $\Omega\colon C\to\mathbb R$ is a non-negative homogeneous of degree 0 locally integrable function, $w\colon (0,\infty)\to [0,\infty)$ is some weight function, $|\cdot|_K$ is an arbitrary norm in ${\mathbb R}^d$, $|\cdot|_{K^\circ}$ is its polar norm, and $p\in (d,\infty]$.https://vestnmath.dnu.dp.ua/index.php/rim/article/view/430/430kolmogorov-type inequalityhypersingular integral operatormodulus of continuity |
| spellingShingle | V.F. Babenko O.V. Kovalenko N.V. Parfinovych Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics Researches in Mathematics kolmogorov-type inequality hypersingular integral operator modulus of continuity |
| title | Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics |
| title_full | Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics |
| title_fullStr | Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics |
| title_full_unstemmed | Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics |
| title_short | Kolmogorov-type inequalities for hypersingular integrals with homogeneous characteristics |
| title_sort | kolmogorov type inequalities for hypersingular integrals with homogeneous characteristics |
| topic | kolmogorov-type inequality hypersingular integral operator modulus of continuity |
| url | https://vestnmath.dnu.dp.ua/index.php/rim/article/view/430/430 |
| work_keys_str_mv | AT vfbabenko kolmogorovtypeinequalitiesforhypersingularintegralswithhomogeneouscharacteristics AT ovkovalenko kolmogorovtypeinequalitiesforhypersingularintegralswithhomogeneouscharacteristics AT nvparfinovych kolmogorovtypeinequalitiesforhypersingularintegralswithhomogeneouscharacteristics |