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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.
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 Download the full text  
For citation: Tasoev B. B. Contractive Projections in Variable Lebesgue Spaces. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp.72-78. DOI 10.23671/VNC.2017.1.5827
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