Estimates of hyperbolic equations in Hardy spaces

The aim of this paper is to study estimates of hyperbolic equations in Hardy classes. Consider the Cauchy problem P ( D t ,D x ) u ( t, x ) = 0 for x ∈ ℝ d and t > 0 with the initial conditions D j t u (0, x ) = g j ( x ), j = 0, 1, …, m – 1. We assume that the symbol ( τ, ξ ) of P ( D t ,D x ) can be factorized as ( τ, ξ ) = $ \prod ^{m}_{j=1} $ ( τ – ϕ j ( ξ )) where ϕ j ( ξ ) = $ \left( \xi ^{2n_{j}}_{1} + \ldots + \xi ^{2n_{j}}_{d} \right)^{{1\over {2n_{j}}}} $ , j = 1, …, m . We assume further that g j ∈ H p k (ℝ d ) for j = 1, …, m – 1. Then the solution u of the problem (3.13) is in H p (ℝ d ) provided k ≥ ( d – 1) $ \left\vert {1 \over p} - {1 \over 2}\right\vert $ and $ {{2n-2} \over {2n-1}} $ < p < ∞. Here n = max{ n 1 , …, n m }. In particular, P ( D t , D x ) u = $ {{\partial ^2 u} \over {\partial t^2}} $ – Δ u = 0 with u (0, x ) = f ( x ) and $ {{\partial u} \over {\partial t}} $ (0, x ) = g ( x ), then the solution u of the wave equation is in H p (ℝ d ) provided k ≥ ( d – 1) $ \left\vert {1 \over p} - {1 \over 2}\right\vert $ and 0 < p < ∞.