Variable Lebesgue Spaces by David V. Cruz-Uribe & Alberto Fiorenza

Author:David V. Cruz-Uribe & Alberto Fiorenza
Language: eng
Format: epub
Publisher: Springer Basel, Basel

(4.37)

Finally, we will show that p( ⋅) satisfies the K 0 condition for intervals I = (x, y), where . In this case it will suffice to show this for intervals of the form J = (0, y). For if it were true in this special case, then

Moreover, given and J = (0, y), it will suffice to show that there exists a constant C (independent of k) such that

(4.38)

For then, since p′( ⋅) is an increasing function, by Corollary 2.23,

To prove (4.38), fix such that . Define

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