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Specificaly:

  • Does convergence in $L^{\frac{1}{2}}$ imply weak $L^2$ convergence?

  • Having a limit in $L^{\frac{1}{2}}$ topology and a limit in weak $L^2$ topology whether these are always equal? If not, what additional assumption would enable the conclusion?

Generally:

  • Whether strong $L^q$ topology is finer (stronger) than a weak $L^p$ topology for some $q<p$?
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