Strong endpoint continuity of planar centered maximal gradients

We prove strong L¹ continuity of planar centered maximal gradients at the Sobolev endpoint from a uniform signed finite-band gradient estimate for disk averages. For the centered Hardy–Littlewood maximal operator M, convergence of inputs in W1,1(ℝ²) gives convergence of their maximal vector gradients in L¹(ℝ²; ℝ²). The result extends to the canonical homogeneous space of L² functions with integrable weak gradient. Our joint approximation theorem permits input smoothing and removal of radius cutoffs in a single limit that preserves maximal values and spatial variation. If inputs converge in the homogeneous gradient norm and aj → 0, bj → ∞, the radius-restricted outputs Maj,bjfj converge to Mf in L², and their vector gradients converge in L¹. The input error and the two cutoffs may approach their limits independently. Affine localization yields uniform integrability along the input sequence; contact gradients and global tails give strong convergence. Zero total integral of the signed vector gradient controls the moving upper cutoff.