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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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