Endpoint Campanato estimate (α = 1) for Dirichlet-energy minimizing constraint maps

Establish whether the Campanato-type decay estimate ∫_{B_r(y)} |Dv − (Dv)_{y,r}|^2 dx ≤ c (r/s)^{n+2} ∫_{B_s(y)} |Dv − (Dv)_{y,s}|^2 dx + c (r/R)^{n+2} ∫_{B_{2R}(x_0)} |Dv|^2 dx holds with the exponent α = 1 for every minimizing constraint map v ∈ W^{1,2}(B_{2R}(x_0); \overline M) of the Dirichlet energy E_0[v] = ∫_{B_{2R}(x_0)} (1/2)|Dv|^2 dx under the pointwise constraint v(x) ∈ \overline M a.e., assuming the small-energy condition (2R)^{2−n} ∫_{B_{2R}(x_0)} |Dv|^2 dx < ε_0. Determine whether the constant c can be chosen independent of r, s, and R, depending only on n and the geometry of ∂M.

Background

In Section 2, the authors establish a Campanato-type estimate (equation (eq:camp)) for Dirichlet-energy minimizing constraint maps v with small energy, valid for every Hölder exponent α ∈ (0,1). This yields a C^{1,α} estimate for the gradient, complementing a separate optimal C^{1,1} estimate obtained under the same small-energy hypothesis.

However, it remains unclear whether the same Campanato decay holds at the endpoint α = 1, which would correspond to the decay rate with exponent n+2. Proving the endpoint estimate would sharpen the quantitative regularity theory and align the decay inequality with the already-available C^{1,1} bound.