Equivalent Conditions for the Consistency of the Helmholtz Decomposition of Muckenhoupt Ap-Weighted Lp-Spaces

Abstract

We are concerned with basic properties of the Helmholtz decomposition of the Muckenhoupt Ap-weighted Lp-space (\n $$L_w^p(\\Omega )$$\n )n for any domain Ω in ℝn (n ∈ ℤ, n ≥ 2), 1 < p < ∞ and Muckenhoupt Ap-weight w ∈ Ap. Set $${p^\\prime}: = {p \\over {p - 1}}$$\n and $${w^\\prime}: = {w^{ - {1 \\over {p - 1}}}}$$\n . Then the consistency of the Helmholtz decomposition of $${(L_w^p(\\Omega ))^n}$$\n and $${(L_{{w^\\prime}}^{{p^\\prime}}(\\Omega ))^n}$$\n are equivalent to the validity of the variational estimate of $$L_{w,\\pi}^p(\\Omega ) + L_{{w^\\prime},\\pi}^{{p^\\prime}}(\\Omega )$$\n . The variational estimate of $$L_{w,\\pi}^p(\\Omega )$$\n and $$L_{{w^\\prime},\\pi}^{{p^\\prime}}(\\Omega )$$\n plays an important role in the proof. Furthermore, we can replace $$L_{w,\\pi}^p(\\Omega )$$\n , $$L_{{w^\\prime},\\pi}^{{p^\\prime}}(\\Omega )$$\n and $$L_{w,\\pi}^p(\\Omega ) + L_{{w^\\prime},\\pi}^{{p^\\prime}}(\\Omega )$$\n by $$L_{w,\\sigma}^p(\\Omega ),\\,\\,L_{{w^\\prime},\\sigma}^{{p^\\prime}}(\\Omega )$$\n and $$L_{w,\\sigma}^p(\\Omega ) + L_{{w^\\prime},\\sigma}^{{p^\\prime}}(\\Omega )$$\n respectively.