Supermonotone Functions and Uniform Regional Inverse Inequalities on Expanding Planar Squares

This paper deals with an inverse inequality for fractional Sobolev norms on expanding planar squares. We define a partial order by the Euclidean distance from the origin and prove that every continuous function monotone with respect to this order on an open planar square is radial there. We introduce a quantitative supermonotonicity condition and obtain an estimate for the intrinsic fractional Sobolev norm with a constant independent of the size of the square, provided that the quantitative parameters are uniformly bounded. The result holds for every fractional index between zero and one. The proof is based on a uniform estimate of a planar kernel, obtained by splitting the integral into near-field and far-field parts. For positive decreasing functions in L2 of the limiting shifted quadrant, a second near-field/far-field argument yields an inverse inequality on the unbounded domain without a uniform lower bound on the function. For positive increasing functions, we give an explicit uniform bound on each square and show why no nonzero L2 counterpart exists on the unbounded quadrant. For functions that remain positive at the boundary of a square, extension by zero gives an infinite full seminorm when the fractional index is at least one half. Four classes of supermonotone functions illustrate the results.