L p estimates for wave equations with specific C 0 , 1 coefficients

Peral–Miyachi’s celebrated theorem states that the operator ( I - Δ ) - α 2 exp ( i - Δ ) is bounded on L p ( ℝ d ) if and only if α ≥ s p : = ( d - 1 ) 1 p - 1 2 . We extend this result to operators of the form ℒ = - ∑ j = 1 d a j + d ∂ j a j ∂ j , such that, for j = 1 , ⋯ , d , the functions a j and a j + d only depend on x j , are bounded above and below, but are merely Lipschitz continuous. This is below the C 1 , 1 regularity that is required in general situations. We construct spaces on which exp ( i ℒ ) is bounded by lifting L p functions to tent spaces, using wave packets adapted to the coefficients. The result then follows from Sobolev embedding properties of these spaces.