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

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.