High-contrast homogenization: #
Provider.PolynomialHomogenization.ConvexFractionalChainPhysicalKernel
Imported from the Apache-2.0 HighContrastHomogenization development at commit
7a13dbcd8d6609264a713373f5c69ceeac870472.
Physical kernel bound for convex fractional chains #
theorem
HCPolySupport.HighContrast.tsum_convexFractionalChainBall_weight_le_rieszKernel
{d : ℕ}
{U : Set (Vec d)}
{c x y : Vec d}
{rho Rad a : ℝ}
(hrho : 0 < rho)
(hRad : 0 < Rad)
(houter : U ⊆ euclideanBallAt c Rad)
(hx : x ∈ U)
(hy : y ∈ U)
(hxy : x ≠ y)
(ha : 0 < a)
:
∑' (n : ℕ), (convexFractionalChainBall c x rho n).indicator (fun (x : Vec d) => ENNReal.ofReal ((2 ^ a) ^ n)) y ≤ ENNReal.ofReal ((2 * Rad) ^ a) * ENNReal.ofReal (euclideanDist x y ^ (-a)) + ENNReal.ofReal ((Rad + rho) ^ a) * ENNReal.ofReal (euclideanDist x y ^ (-a)) * ENNReal.ofReal (2 ^ a / (2 ^ a - 1))
The growing dyadic weights of all chain balls containing y are
controlled by the order-a Riesz kernel at x-y.