Regularity estimates for critical d-Laplace systems involving gauge transformations and Muckenhoupt weights

We establish local gradient regularity for solutions 𝑢 ∈ 𝑊 1 , 𝑑 ⁡ ( Ω , ℝ 𝑚 ) to a class of nonlinear elliptic conformal-type systems with critical growth of the form − d i v ⁢ ( | ∇ 𝑢 | 𝑑 − 2 ⁢ 𝔸 ∇ 𝑢 ) = d i v ⁢ 𝔾 , i n Ω ⊂ ℝ 𝑑 . This equation exhibits interesting features due to the criticality of the exponent, which corresponds to the limiting case of the Sobolev embedding, and the matrix-valued coefficient 𝔸 arises as a gauge transformation of Rivière or Uhlenbeck-Coulomb-type. The main motivation for this study stems from the recent contributions of Martino and Schikorra in [18] , [19] , who investigated a class of conformal systems with right-hand sides involving antisymmetric potentials. The analysis of such d -Laplace systems with an antisymmetric structure crucially allows for the application of compensated compactness, which leads to gradient estimates in the optimal scale of Lorentz spaces. Under suitable assumptions on the Muckenhoupt 𝔸 ∞ weights and structural conditions on the given data, we control the gradient of solutions through the fractional maximal operator 𝐌 𝛼 in the weighted Lorentz framework. Furthermore, as an immediate consequence of our main result, we also derive the sharp gradient bounds in Morrey spaces and a pointwise gradient estimate via the Riesz potential 𝐈 𝛽 .