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Let $B$ be the unit ball in $\mathbb{R}^n$. A local version of the Sobolev inequality on $\mathbb{R}^n$ says that for any $p\in[1,\infty]$ there exist constants $C >0$ and $k \in \mathbb{N}$ such that

$$f(0)\leq C\ \sum_{|\alpha|\leq k} \left(\int_B |D^\alpha f|^p\right)^{1/p}$$

for all sufficiently smooth $f$.

Question: Is there a generalization of this inequality to nonarchimedean local fields, and if so, what is the proper interpretation of the derivative in this context.

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For various imbedding theorems for spaces of complex-valued functions on non-Archimedean local fields, see

M. Taibleson, Harmonic Analysis on n-Dimensional Vector Spaces over Local Fields. I. Basic Results on Fractional Integration. Math. Ann. 176 (1968), 191-207;

S. Haran, Quantizations and symbolic calculus over p-adic numbers, Ann. Inst. Fourier 43 (1993), 997-1053.

The role of derivatives is played by pseudo-differential operators, analogs of fractional derivatives.

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