Boundedness of truncated Havin–Maz'ya potentials in Herz spaces with applications to $\mathrm{p}(\cdot)$-Laplace equations

Zareen A. Khan; Waqar Afzal; Mujahid Abbas; Daniel Breaz; Luminiţa-Ioana Cotîrlă; Zareen A. Khan; Waqar Afzal; Mujahid Abbas; Daniel Breaz; Luminiţa-Ioana Cotîrlă

doi:10.3934/math.2026841

AIMS Mathematics

2026, Volume 11, Issue 7: 20677-20710. doi: 10.3934/math.2026841

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Boundedness of truncated Havin–Maz'ya potentials in Herz spaces with applications to $\mathrm{p}(\cdot)$-Laplace equations

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Abdus Salam School of Mathematical Sciences, Government College University, Lahore 54000, Pakistan
3.
Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku, AZ1096, Azerbaijan
4.
International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore 54792, Pakistan
5.
Department of Mechanical Engineering Science, Faculty of Engineering and the Built Environment, University of Johannesburg, Johannesburg 2092, South Africa
6.
Department of Medical Research, China Medical University, Taichung 40447, Taiwan
7.
Department of Mathematics, "1 Decembrie 1918" University of Alba Iulia, Alba Iulia, 510009, Romania
8.
Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, 400114, Romania

Received: 23 January 2026 Revised: 30 March 2026 Accepted: 07 April 2026 Published: 14 July 2026
MSC : 42B35, 26D15, 46B25, 47G10

In this paper, we developed a variational framework based on Dirichlet-type energy integrals for the $ \mathrm{p}(\cdot) $-Laplace equation, establishing coercivity and weak lower semicontinuity of the associated functional to guarantee existence and uniqueness of minimizers. The central contribution is a pointwise estimation of weak solutions via fractional truncated potentials, employing Havin–Maz'ya-type nonlinear operators in Herz spaces. Our approach subsumes several earlier results, with classical linear potential estimates recovered as limiting cases.
Citation: Zareen A. Khan, Waqar Afzal, Mujahid Abbas, Daniel Breaz, Luminiţa-Ioana Cotîrlă. Boundedness of truncated Havin–Maz'ya potentials in Herz spaces with applications to $\mathrm{p}(\cdot)$-Laplace equations[J]. AIMS Mathematics, 2026, 11(7): 20677-20710. doi: 10.3934/math.2026841

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Abstract

In this paper, we developed a variational framework based on Dirichlet-type energy integrals for the $ \mathrm{p}(\cdot) $-Laplace equation, establishing coercivity and weak lower semicontinuity of the associated functional to guarantee existence and uniqueness of minimizers. The central contribution is a pointwise estimation of weak solutions via fractional truncated potentials, employing Havin–Maz'ya-type nonlinear operators in Herz spaces. Our approach subsumes several earlier results, with classical linear potential estimates recovered as limiting cases.

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