Abstract
We find the best possible constant C in the inequality‖φ‖Lrpr⩽C‖φ‖Lppr‖φ‖BMO1−pr for all values of parameters p and r such that 1⩽p<r<+∞. To solve the problem, we define and explicitly compute the corresponding Bellman function. This function depends on three variables and has a rather complicated structure. We first solve the problem on an interval and then transfer our results to the circle and the line. We also obtain explicit estimates in several higher-dimensional settings.
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