Summary
In 1997 Juha Kinnunen proved that maximal operators satisfy a Sobolev bound if the Sobolev exponent p is strictly larger than 1. His article initiated the study of regularity of maximal functions, a field which has attracted several dozens of authors to this day. Geometric techniques have recently lead to a series of breakthrough endpoint regularity bounds for maximal operators in higher dimensions. This project pursues the novel strategy of combining these new geometric tools with already established extremization techniques in order to solve a wide range of open questions in the field.
The goals of the project are organized around two themes: gradient bounds and sharp constants. The main goal from the first theme is to prove that the variation of the non-centered Hardy-Littlewood maximal function can be controlled by the variation of the function in all dimensions. This is the endpoint p=1 of Juha Kinnunen's original bound and is one of the main long standing open questions in the field. This project also aims to prove this variation bound for further maximal operators, along with the operator continuity of their gradient and bounds for higher derivatives.
The main goal from the second theme is to prove that the centered Hardy-Littlewood maximal operator in one dimension does not increase the variation of a function. This bound would be sharp because examples show that in general, maximal operators do not strictly decrease the variation of a function. This project further aims to prove this sharp bound for convolution type maximal operators and to find the sharp constant in the variation bound for the dyadic maximal operator in all dimensions.
The goals of the project are organized around two themes: gradient bounds and sharp constants. The main goal from the first theme is to prove that the variation of the non-centered Hardy-Littlewood maximal function can be controlled by the variation of the function in all dimensions. This is the endpoint p=1 of Juha Kinnunen's original bound and is one of the main long standing open questions in the field. This project also aims to prove this variation bound for further maximal operators, along with the operator continuity of their gradient and bounds for higher derivatives.
The main goal from the second theme is to prove that the centered Hardy-Littlewood maximal operator in one dimension does not increase the variation of a function. This bound would be sharp because examples show that in general, maximal operators do not strictly decrease the variation of a function. This project further aims to prove this sharp bound for convolution type maximal operators and to find the sharp constant in the variation bound for the dyadic maximal operator in all dimensions.
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| Web resources: | https://cordis.europa.eu/project/id/101151034 |
| Start date: | 01-09-2025 |
| End date: | 31-08-2027 |
| Total budget - Public funding: | - 172 750,00 Euro |
Cordis data
Original description
In 1997 Juha Kinnunen proved that maximal operators satisfy a Sobolev bound if the Sobolev exponent p is strictly larger than 1. His article initiated the study of regularity of maximal functions, a field which has attracted several dozens of authors to this day. Geometric techniques have recently lead to a series of breakthrough endpoint regularity bounds for maximal operators in higher dimensions. This project pursues the novel strategy of combining these new geometric tools with already established extremization techniques in order to solve a wide range of open questions in the field.The goals of the project are organized around two themes: gradient bounds and sharp constants. The main goal from the first theme is to prove that the variation of the non-centered Hardy-Littlewood maximal function can be controlled by the variation of the function in all dimensions. This is the endpoint p=1 of Juha Kinnunen's original bound and is one of the main long standing open questions in the field. This project also aims to prove this variation bound for further maximal operators, along with the operator continuity of their gradient and bounds for higher derivatives.
The main goal from the second theme is to prove that the centered Hardy-Littlewood maximal operator in one dimension does not increase the variation of a function. This bound would be sharp because examples show that in general, maximal operators do not strictly decrease the variation of a function. This project further aims to prove this sharp bound for convolution type maximal operators and to find the sharp constant in the variation bound for the dyadic maximal operator in all dimensions.
Status
SIGNEDCall topic
HORIZON-MSCA-2023-PF-01-01Update Date
25-11-2024
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