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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.18035 |
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Table of Contents:
- In this article, we show the necessary and sufficient conditions for the inequality $\|u\|_{L_t^qL_x^r}\lesssim \|u\|_{X^{s,b}}$, where $$\|u\|_{X^{s,b}}:=\|\hat{u}(τ,ξ)\langle ξ\rangle^s\langle τ-ξ^3\rangle^b \|_{L_{τ,ξ}^2}. $$ Here, we provide a complete classification of the indices relationships for which this inequality holds true. Such estimates will be very useful in solving the well-posedness for low regularity well-posedness of the Korteweg--de Vries equations and stochastic Korteweg--de Vries equations.