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We establish sharp (H1,L1,q) and local (L logrL,L1,q) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H1 to L1,∞ and L log1/2L to L1,∞, and that these estimates are sharp.
Generalizing the classical BMO spaces defined on the unit circle with vector or scalar values, we define the spaces and , where for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that and by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of . Pisier conjectured that the supports of idempotent multipliers of form a Boolean...
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