Abstract
We prove that for α ∈ (d − 1,d), one has the trace inequality
for all solenoidal vector measures F, i.e., \(F\in M_{b}(\mathbb {R}^{d};\mathbb {R}^{d})\) and divF = 0. Here Iα denotes the Riesz potential of order α and \(\mathcal M^{d-\alpha }(\mathbb {R}^{d})\) the Morrey space of (d − α)-dimensional measures on \(\mathbb {R}^{d}\).
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Acknowledgments
The authors would like to thank the referee for his or her comments, which have improved the readability of the paper. D. Stolyarov is partially supported by RFBR grant 20-01-00209 and by “Native towns”, a social investment program of PJSC “Gazprom Neft”. D. Spector is supported by the Taiwan Ministry of Science and Technology under research grant number 110-2115-M-003-020-MY3 and the Taiwan Ministry of Education under the Yushan Fellow Program. Part of this work was undertaken while D. Spector was visiting the National Center for Theoretical Sciences in Taiwan. He would like to thank the NCTS for its support and warm hospitality during the visit.
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Raiţă, B., Spector, D. & Stolyarov, D. A Trace Inequality for Solenoidal Charges. Potential Anal 59, 2093–2104 (2023). https://doi.org/10.1007/s11118-022-10008-x
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DOI: https://doi.org/10.1007/s11118-022-10008-x