Exact inequalities for derivatives of functions of low smoothness defined on an axis and a semiaxis
We obtain new exact inequalities of the form $$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$ for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where $$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$ for functions defined on the axis $R$ in the case where $$r = 2,\...
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| Date: | 2006 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2006
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3454 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | We obtain new exact inequalities of the form
$$∥x(k)∥_q ⩽ K∥x∥^{α}_p ∥x(r)∥^{1−α}_s$$
for functions defined on the axis $R$ or the semiaxis $R_{+}$ in the case where
$$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s=1,$$
for functions defined on the axis $R$ in the case where
$$r = 2,\; k = 1,\; q ∈ [2,∞),\; p = ∞,\; s= 1,$$
and for functions of constant sign on $R$ or $R_{+}$ in the case where
$$r = 2,\; k = 0,\; p ∈ (0,∞),\; q ∈ (0,∞],\; q > p,\; s = ∞$$
and in the case where
$$r = 2,\; k = 1,\; p ∈ (0,∞),\; q = s = ∞.$$ |
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