Let $ \sigma \geq \frac{n}{2} $. And consider the inhomogeneous Besov space $B^{\sigma}_{2,1}$ with the norm $$ \Vert f \Vert_{B^{\sigma}_{2,1}}= \sum_{k=0}^{\infty} 2^{\sigma k} \Vert \Delta_{k} f \Vert_{L^{2}} $$ with the understanding that $ \Delta_{0} f = \mathcal{F}^{-1} ( \eta \hat{f}) $ where $\eta$ is supported on the unit ball. The question is as follows: is it true that $$ \Vert f g \Vert_{B^{\sigma}_{2,1}} \leq C(n) ( \Vert f \Vert_{B^{\sigma}_{2,1}} \Vert g \Vert_{B^{n/2}_{2,1}} + \Vert g \Vert_{B^{\sigma}_{2,1}} \Vert f \Vert_{B^{n/2}_{2,1}}) $$ ? I know this inequality is true but I am not sure if the constant depend on $\sigma$ or not ? Any help is appreciated.
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$\begingroup$ $2^{k\sigma}$ missing in the definition of the norm. Since this factor increases with $\sigma$, obvioulsy the constant does not depend on $\sigma$ $\endgroup$Piero D'Ancona– Piero D'Ancona2025-11-26 14:54:25 +00:00Commented Nov 26 at 14:54
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$\begingroup$ But the left hand side increases with \sigma too. $\endgroup$User091099– User0910992025-11-29 20:44:55 +00:00Commented yesterday
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