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arXiv · 1411.3461

On the upper and lower estimates of norms in variable exponent spaces

Tengiz Kopaliani·Nino Samashvili·Shalva Zviadadze
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Abstract

In the present paper we investigate some geometrical properties of the norms in Banach function spaces. Particularly there is shown that if exponent $1/p(\cdot)$ belongs to $BLO^{1/\log}$ then for the norm of corresponding variable exponent Lebesgue space we have the following lower estimate $$\left\|\sum χ_{Q}\|fχ_{Q}\|_{p(\cdot)}/\|χ_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}\leq C\|f\|_{p(\cdot)}$$ where $\{Q\}$ defines disjoint partition of $[0;1]$. Also we have constructed variable exponent Lebesgue space with above property which does not possess following upper estimation $$\|f\|_{p(\cdot)}\leq C\left\|\sum χ_{Q}\|fχ_{Q}\|_{p(\cdot)}/\|χ_{Q}\|_{p(\cdot)}\right\|_{p(\cdot)}. $$

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Tengiz Kopaliani, Nino Samashvili, Shalva Zviadadze. 2014-11-13. On the upper and lower estimates of norms in variable exponent spaces. https://arxiv.org/abs/1411.3461

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