Mixed Lebesgue spaces are a generalization of Lp spaces that occur naturally when considering functions that depend on quantities with different properties, such as space and time. We ﬁrst present mixed Lebesgue versions of several classical results, including the boundedness of Calderon-Zygmund operators, a Littlewood-Paley theorem, and some other vector-valued inequalities. As applications we present a Leibniz's rule for fractional derivatives in the context of mixed-Lebesgue spaces, some sampling theorems and a characterization of mixed Lebesgue spaces in terms of wavelet coefﬁcients.
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