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DOI: 10.23671/VNC.2017.1.5827 Contractive Projections in Variable Lebesgue Spaces
Tasoev B. B.
Vladikavkaz 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.
Language: Russian
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For citation: Tasoev B. B. Contractive Projections in Variable Lebesgue Spaces. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.7278.
DOI 10.23671/VNC.2017.1.5827 ← Contents of issue 
 

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