For every ball Banach function space X on which the Hardy-Littlewood maximal operator is bounded on the associate space, the vanishing Campanato spaces VLα, XLα, and CLα coincide with their X-based analogues.
Garc ´ıa-Cuerva and J
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Characterization of Vanishing Campanato Spaces via Ball Banach Function Spaces and Its Applications
For every ball Banach function space X on which the Hardy-Littlewood maximal operator is bounded on the associate space, the vanishing Campanato spaces VLα, XLα, and CLα coincide with their X-based analogues.