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Gehring theory for time-discrete hyperbolic differential equations. (English) Zbl 1060.35527
Let $$h>0$$, $$t_n=nh$$, $$n\in \mathbb Z$$, $$L_h=\sum _{n\in \mathbb Z}(\{t_n\}\times Q_{3/2(0)})$$, where $$Q_r(x)$$ is the cube in $$\mathbb R^m$$, $$| y_i-x_i| <r$$, $$1\leq i\leq m$$. Let the $$h$$-time step function be a mapping $$F$$ defined on $$\mathbb R^1\times Q_{3/2}(0)$$ such that $$F(t,x)=F(nh,x)$$ for $$(t,x)\in ((n-1)h,nh]\times Q_{3/2}(0)$$, $$n\in \mathbb Z$$. Let $$\widetilde Q_{3/2}(0)=(-3/2,3/2)\times Q_{3/2}(0)$$. The following higher integrability estimate in the spirit of Gehring’s lemma is proved for non-negative $$h$$-time step functions $$f\in L^q(\widetilde Q_{3/2}(0))$$ and $$g\in L^r(\widetilde Q_{3/2}(0))$$, $$1<q<r$$: If $\begin{split}| Q_{R/2}(z_0)| ^{-1}\int _{Q_{R/2}(z_0)}g^q\,dz\\ \leq b\bigl \{\bigl (| Q_R(z_0)| ^{-1}\int _{Q_R(z_0)}g\,dz\bigr )^q +| Q_R(z_0)| ^{-1}\int _{Q_R(z_0)}f^q\bigr \} +\theta | Q_R(z_0)| ^{-1}\int _{Q_R(z_0)}g^q\,dz\end{split}$ for some $$\theta \in [0,1)$$, $$b>1$$ and every $$Q_R(z_0)\subset \widetilde Q_{3/2}(0)$$, then $$g\in L^p_{\text{loc}}(\widetilde Q_{3/2}(0))$$ and the appropriate higher integrability estimate holds for $$p\in [q,q+\varepsilon )$$ with some $$\varepsilon >0$$.
Possibility of applications to hyperbolic differential equations is briefly indicated with no explicitly stated claims, recalling the regularity results due to M. Giaquinta and G. Modica [J. Reine Angew. Math. 311/312, 145–169 (1979; Zbl 0409.35015)] and other basic papers in this field.

##### MSC:
 35L90 Abstract hyperbolic equations 26D15 Inequalities for sums, series and integrals 39A10 Additive difference equations 49J40 Variational inequalities
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