Morrey-Lorentz estimates for Hodge-type systems

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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$ \begin{equation*} d^{\ast} \left( A d\omega \right) + B^{\intercal}d d^{\ast} \left( B\omega \right) = \lambda B\omega + f \text{ in } \Omega, \end{equation*} $with either $ \nu\wedge \omega $ and $ \nu\wedge d^{\ast} \left( B\omega \right) $ or $ \nu\lrcorner B\omega $ and $ \nu\lrcorner \left( A d\omega \right) $ prescribed on $ \partial\Omega. $ We derive these estimates from the $ L^{p} $ estimates obtained in [32] 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 $ L^{p} $ estimates using interpolation. We instead adapt an idea of Lieberman [19] 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.