Morrey-Lorentz estimates for Hodge-type systems

We prove up to the boundary regularity estimates in Morrey-Lorentz spaces for weak solutions of the linear system of differential forms with regular anisotropic coefficients {equation*} d^{} ( A dω ) + B^{}d d^{} ( Bω ) = λ Bω + f in Ω, {equation*} with either ν ω and ν d^ ( Bω ) or ν Bω and ν ( A dω ) prescribed on ∂Ω. We derive these estimates from the Lᵖ estimates obtained in {Sil_linearregularity} in the spirit of Campanato's method. Unlike Lorentz spaces, Morrey spaces are neither interpolation spaces nor rearrangement invariant. So Morrey estimates can not be obtained directly from the Lᵖ estimates using interpolation. We instead adapt an idea of Lieberman {Lieberman_morrey_from_Lp} to our setting to derive the estimates. Applications to Hodge decomposition in Morrey-Lorentz spaces, Gaffney type inequalities and estimates for related systems such as Hodge-Maxwell systems and `div-curl' systems are discussed.