In the research of elliptic and parabolic equations, the Schauder estimate is one of the most important issues for them. In this topic, we always bound the norm of higher regularity in the small ball by a bigger one. That is, for the elliptic equation $ \operatorname{div}(A(x)\nabla u)=0 $, we have estimates like $ \left\|u\right\|\_{C^{0,\alpha}(B\_1)}\leq C\left\|u\right\|\_{L^2(B\_2)} $, where $ B\_r=B(0,r) $ is the ball with center $ 0 $ and radius $ r $. I want to ask why we do not study such estimates for hyperbolic equations.



Why don't we study hyperbolic equations as elliptic and parabolic equations?

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> Why we do not study such estimates for hyperbolic equations?
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Because they are false.




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Now: you may ask "why are they false?" This is a fairly deep question, and answers often involve discussion of propagation of singularities and characteristics. Quite a few chapters in Hörmander's *Analysis of Linear Partial Differential Operators* are devoted to this and similar questions.


