The lecture is based on recent two joint papers of Sebastian Lajara and myslef.
Let \((X,\|\cdot\|)\) be a Banach space and \(\mu\) be a probability measure.
Using Luxemburg norm associated to a suitable Orlicz function, we construct
an equivalent norm \(\||\cdot|\|\) on the Lebesgue-Bochner space \(L^1(\mu,X)\) with the property:
If \(\|\cdot\|\) on \(X\) is rotund (or uniformly rotund in every direction, or locally
uniformly rotund, or midpoint locally uniformly rotund, or Gateaux smooth, or uniformly Gateaux smooth), then
the norm \(\||\cdot|\|\) has the respective property (or a combination of them).
Moreover, if \(\|\cdot\|\) on \(X\) is uniformly rotund (or uniformly Fréchet smooth, or has both latter properties), then the restriction
of \(\||\cdot|\|\) to any reflexive subspace of \(L^1(\mu, X)\) is such.
|
