Coarse-graining support:

Support.CoarseGraining.MagicIdentities.MuOrdering.EllipticConsequences.BCoarseAveraged

Imported from the Apache-2.0 CoarseGraining development at commit c7ddd76c08ade64fed1b8d2ca51be14dfee8deb4.

MuOrdering -- bCoarse averaged-blockmatrix bounds

bCoarse_le_average_blockMatrixOfCoeff_upperLeft_… plus the three bCoarse_le_averaged_symmPart_plus_correction_… wrappers. These reach back to the Mu/blockEnergy variational lower bound and show the canonical bCoarse matrix is dominated by the volume-average of (blockMatrixOfCoeff (a x)).upperLeft, i.e., the symmetric part plus the (skew⊤ · symm⁻¹ · skew) Schur correction.

theorem HCPolySupport.bCoarse_le_average_blockMatrixOfCoeff_upperLeft_of_isEllipticFieldOn_of_isSobolevRegularDomain {d : ℕ} {U : Set (Vec d)} [MeasureTheory.IsFiniteMeasure (volumeMeasureOn U)] (hU : IsSobolevRegularDomain U) {a : CoeffField d} {lam Lam : ℝ} (hEll : IsEllipticFieldOn lam Lam U a) (hvol : (MeasureTheory.volume U).toReal ≠ 0) {sigma sigmaStar kappa : Mat d} (hA : IsCoarseBlockMatrix U a (deterministicCoarseBlockMatrix U a)) (hS : IsSigmaStarCoarse U a sigmaStar) (hK : IsKappaCoarse U a sigmaStar kappa) (hSigma : IsSigmaCoarse U a sigma sigmaStar kappa) (hdet : IsUnit (Matrix.det sigmaStar)) (p : Vec d) : vecDot p (matVecMul (bCoarse (sigmaCoarse U a) (sigmaStarCoarse U a) (kappaCoarse U a)) p) ≤ volumeAverage U fun (x : Vec d) => vecDot p (matVecMul (blockMatrixOfCoeff (a x)).upperLeft p)

theorem HCPolySupport.bCoarse_le_averaged_symmPart_plus_correction_of_isEllipticFieldOn_of_isSobolevRegularDomain {d : ℕ} {U : Set (Vec d)} [MeasureTheory.IsFiniteMeasure (volumeMeasureOn U)] (hU : IsSobolevRegularDomain U) {a : CoeffField d} {lam Lam : ℝ} (hEll : IsEllipticFieldOn lam Lam U a) (hvol : (MeasureTheory.volume U).toReal ≠ 0) {sigma sigmaStar kappa : Mat d} (hA : IsCoarseBlockMatrix U a (deterministicCoarseBlockMatrix U a)) (hS : IsSigmaStarCoarse U a sigmaStar) (hK : IsKappaCoarse U a sigmaStar kappa) (hSigma : IsSigmaCoarse U a sigma sigmaStar kappa) (hdet : IsUnit (Matrix.det sigmaStar)) (p : Vec d) : vecDot p (matVecMul (bCoarse (sigmaCoarse U a) (sigmaStarCoarse U a) (kappaCoarse U a)) p) ≤ volumeAverage U fun (x : Vec d) => vecDot p (matVecMul (symmPart (a x) + matTranspose (skewPart (a x)) * (symmPart (a x))⁻¹ * skewPart (a x)) p)

theorem HCPolySupport.bCoarse_le_averaged_symmPart_plus_correction_of_isEllipticFieldOn_of_hodgeConverseCriterion {d : ℕ} {U : Set (Vec d)} [MeasureTheory.IsFiniteMeasure (volumeMeasureOn U)] (R : PotentialSolenoidalL2RecoveryData U) (hU : IsSobolevRegularDomain U) {a : CoeffField d} {lam Lam : ℝ} (hEll : IsEllipticFieldOn lam Lam U a) (hHodge : HodgeConverseCriterion U) (hvol : 0 < (MeasureTheory.volume U).toReal) (compat : R.MuRecoveryCompatibilityData (R.toMuOperatorSystemDataOfIsEllipticFieldOn hEll hvol)) {sigma sigmaStar kappa : Mat d} (hA : IsCoarseBlockMatrix U a (deterministicCoarseBlockMatrix U a)) (hS : IsSigmaStarCoarse U a sigmaStar) (hK : IsKappaCoarse U a sigmaStar kappa) (hSigma : IsSigmaCoarse U a sigma sigmaStar kappa) (p : Vec d) : vecDot p (matVecMul (bCoarse (sigmaCoarse U a) (sigmaStarCoarse U a) (kappaCoarse U a)) p) ≤ volumeAverage U fun (x : Vec d) => vecDot p (matVecMul (symmPart (a x) + matTranspose (skewPart (a x)) * (symmPart (a x))⁻¹ * skewPart (a x)) p)

theorem HCPolySupport.bCoarse_le_averaged_symmPart_plus_correction_of_isEllipticFieldOn_of_isOpenBoundedConvexDomain {d : ℕ} {U : Set (Vec d)} [MeasureTheory.IsFiniteMeasure (volumeMeasureOn U)] (R : PotentialSolenoidalL2RecoveryData U) (hConv : IsOpenBoundedConvexDomain U) {a : CoeffField d} {lam Lam : ℝ} (hEll : IsEllipticFieldOn lam Lam U a) (hvol : 0 < (MeasureTheory.volume U).toReal) (compat : R.MuRecoveryCompatibilityData (R.toMuOperatorSystemDataOfIsEllipticFieldOn hEll hvol)) {sigma sigmaStar kappa : Mat d} (hA : IsCoarseBlockMatrix U a (deterministicCoarseBlockMatrix U a)) (hS : IsSigmaStarCoarse U a sigmaStar) (hK : IsKappaCoarse U a sigmaStar kappa) (hSigma : IsSigmaCoarse U a sigma sigmaStar kappa) (p : Vec d) : vecDot p (matVecMul (bCoarse (sigmaCoarse U a) (sigmaStarCoarse U a) (kappaCoarse U a)) p) ≤ volumeAverage U fun (x : Vec d) => vecDot p (matVecMul (symmPart (a x) + matTranspose (skewPart (a x)) * (symmPart (a x))⁻¹ * skewPart (a x)) p)