Quasi-Banach Valued Inequalities via the Helicoidal method
By C. Benea, Camil Muscalu. 2016
48 citations recorded by the source corpus.
We extend the helicoidal method that we previously developed to the quasi-Banach context, proving in this way multiple Banach and quasi-Banach vector-valued inequalities for paraproducts $\Pi$ and for the bilinear Hilbert transform $BHT$. As an immediate application, we obtain mixed norm estimates for $\Pi \otimes \Pi$ in the whole range of Lebesgue exponents. One of the novelties in the quasi-Banach framework (that is, when $0<r<1$), which we expect to be useful in other contexts as well, is the "linearization" of the operator $ \left( \sum_k | T(f_k, g_k) |^r \right)^{1/r}$ by dualizing its weak-$L^p$ quasinorms through $L^r$. Another important role is played by the sharp evaluation of the operatorial norm $\| T_{I_0}(f \cdot \mathbf{1}_F, g \cdot \mathbf{1}_G) \cdot \mathbf{1}_{H'}\|_r$, which is obtained by dualizing the weak-$L^p$ quasinorms through $L^\tau$, with $\tau \leq r$. In the Banach case, the linearization of the operator and the sharp estimates for the localized operatorial norm can be both achieved through the classical (generalized restricted type) $L^1$ dualization.
Subfields: Applied Mathematics, Mathematical Physics, Mechanics of Materials
View Paper: https://www.semanticscholar.org/paper/356f268d0d509624482076ed28a8cae7f8def7c7
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In Applied Mathematics
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In Mechanics of Materials
Citations in the corpus
- On Function Spaces with Mixed Norms — A Survey (2019, background): “Another recent interest in developing the theory of function spaces with mixed norms comes from bilinear estimates and their vector-valued extensions which have proved useful in partial differential equations involving functions in n dimension space variable x and one dimension time variable t; see,”
- Extrapolation for multilinear Muckenhoupt classes and applications (2018, methodology): “Here it is important to emphasize that [11] did not recover the full range of vector-valued estimates from [3, 4] in the unweighted situation.”
- Smoothing properties of bilinear operators and Leibniz-type rules in Lebesgue and mixed Lebesgue spaces (2017, background): “Moreover, in an even more recent version [5] the same authors reduced the condition on β to β > max(0, 1q − 1).”