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DOI: 10.23671/VNC.2017.1.5827

# Contractive Projections in Variable Lebesgue Spaces

Tasoev B. B.
Vladikakazian Mathematical Journal 2017. Vol. 19. Issue 1.
Abstract:
In this article we describe the structure of positive contractive projections in variable Lebesgue spaces $$L_{p(\cdot)}$$ with $$\sigma$$-finite measure and essentially bounded exponent function $$p(\cdot)$$. It is shown that every positive contractive projection $$P:L_{p(\cdot)}\rightarrow L_{p(\cdot)}$$ admits a matrix representation, and the restriction of $$P$$ on the band, generated by a weak order unite of its image, is weighted conditional expectation operator. Simultaneously we get a description of the image $$\mathcal{R}(P)$$ of the positive contractive projection $$P$$. Note that if measure is finite and exponent function $$p(\cdot)$$ is constant, then the existence of a weak order unit in $$\mathcal{R}(P)$$ is obvious. In our case, the existence of the weak order unit in $$\mathcal{R}(P)$$ is not evident and we build it in a constructive manner. The weak order unit in the image of positive contractive projection plays a key role in its representation.
Keywords: conditional expectation operator, contractive projection, variable Lebesgue space, Nakano space, $$\sigma$$-finite measure.