A Quantitative Hardy Scale for Mixed Local–Fractional Energies and Applications to Singular Schrödinger Forms

We develop a quantitative Hardy scale for mixed quadratic energies combining the classical Dirichlet form and a fractional Dirichlet form, Eλ,s(u)=∫Rn|∇u(x)|2dx+λ∫Rn|(−Δ)s/2u(x)|2dx,0<s<1,λ>0. Here, the word scale denotes a parameterized family with a fixed interpolation variable, explicit constants, and the scaling exponent forced by the coexistence of the orders 2 and 2s. For n≥3, we prove weighted L2 inequalities indexed by γ∈[s,1], which control |x|−2γ by Eλ,s with the factor λ−θ, where θ=(1−γ)/(1−s). In dimension n=2, the local endpoint is replaced by the logarithmic Hardy weight and gives a mixed log–power family governed by the same parameter. The novelty lies in organizing the endpoint Hardy estimates into a λ-adapted form suitable for mixed-order operators, with explicit constants, scaling-level optimality of the λ exponent, a planar endpoint formulation, and directly usable singular-potential thresholds. The operator consequences are stated at the level of form boundedness, coercivity, spectral lower bounds on bounded domains, semigroup generation, and variational well-posedness; they are presented as consequences of the Hardy scale rather than as a separate spectral theory.