Geodesic
A boundedness criterion for averaging operators in variable exponent spaces of periodic functions
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 52, Tome 537 (2024), pp. 40-63

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A criterion for the uniform boundedness of Steklov averaging operators in variable exponent spaces of periodic functions is obtained. This criterion coincides with the known local analog of the Muckenhoupt condition. The boundedness of Steklov averages was previously known under the Dini–Lipschitz condition. The norms of averaging operators are estimated explicitly. 
@article{ZNSL_2024_537_a1,
     author = {O. L. Vinogradov},
     title = {A boundedness criterion for averaging operators in variable exponent spaces of periodic functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {40--63},
     publisher = {mathdoc},
     volume = {537},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2024_537_a1/}
}
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O. L. Vinogradov. A boundedness criterion for averaging operators in variable exponent spaces of periodic functions. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 52, Tome 537 (2024), pp. 40-63. http://geodesic.mathdoc.fr/item/ZNSL_2024_537_a1/