Some new Regularity properties of Elliptic Equations in generalized Zygmund Spaces with VMO Coefficients
Keywords:
Zygmund spaces; Orlicz–Zygmund spaces; divergence-form elliptic operators; VMO; Calder´on– Zygmund theory; H¨older gradient estimatesAbstract
Let ℒu = −Σᵢⱼ(aᵢⱼuₓᵢ)ₓⱼ = ∇·h on D ⊂ ℝⁿ, n ≥ 3, (aᵢⱼ) ∈ V ∩ L∞. For 1 < q < n, 0 < λ < n, β ≥ 0, set Φ(t) = tᵠ logᵝ(e + t) and let Zᵠ,λᵝ(D) be the Φ-modulated ball-oscillation (Zygmund) class. For h ∈ [Zᵠ,λᵝ(D)]ⁿ we prove ∂ₓᵢu ∈ Zᵠ,λᵝ(K), K ⊂ D, with a quantitative bound, via a parametrix decomposition, a Calderón–Zygmund operator estimate on Zᵠ,λᵝ obtained from the Orlicz-space singular-integral theory together with the norm comparison (6), a V-smallness estimate, and a Banach contraction. A Hölder corollary follows at λ + q > n.
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