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L p – L q Estimates for Parabolic Systems in Non‐Divergence Form with VMO Coefficients
Author(s) -
Haller-Dintelmann Robert,
Heck Horst,
Hieber Matthias
Publication year - 2006
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610706023192
Subject(s) - mathematics , divergence (linguistics) , order (exchange) , realization (probability) , combinatorics , mathematical analysis , statistics , philosophy , linguistics , finance , economics
Consider a parabolic N×N ‐system of order m on ℝ n with top‐order coefficients a α VMO∩ L ∞. Let 1 < p, q < ∞ and let ω be a Muckenhoupt weight. It is proved that systems of this kind possess a unique solution u satisfying∥ u ′ ∥ L q ( J ; L ω p( ℝ n ) N ) + ∥ A u ∥ L q ( J ; L ω p( ℝ n ) N ) ⩽ C ∥ f ∥ L q ( J ; L ω p( ℝ n ) N ) ,where Au = ∑ |α|⩽ m a α D α u and J = [0,∞). In particular, choosing ω = 1, the realization of A in L p (ℝ n ) N has maximal L p – L q regularity.