Välkommen till föreläsningen i seminarieserien i matematik.
Young’s inequality ensures that L1 ∗ Lp ⊆ Lp when p ∈ [1, ∞]. Several decades ago, it was proved by Cohen, Hewit and Rudin (in various contributions) that indeed L1 ∗Lp = Lp if we avoid p = ∞, i.e. p ∈ [1,∞). We observe that the identity is non-trivial, because L1 does not contain any identity element in background of convolutions. In the most general setting, the latter equality follows from the fact that L1 possess the property on so-called bounded approximation identity.
In the talk we explain that similar properties hold true when L1 is replaced by any quasi-Banach algebra and the product ∗ is replaced by suitable multiplications with quasi-Banach modules.
As applications we show among others that L1 ∗ Lp = Lp for p ∈ [1, ∞) can be improved into WL1,r ∗ Lp = Lp, p ∈ [1,∞), r ∈ (0,1]. Here WL1,r is a Wiener Lebesgue space which strictly increases with r and we have WL1,1 = L1.
We also give some links on further applications, e.g. convolutions between Wiener Lebesgue spaces on one hand, and spaces of Schatten symbol classes sp in pseudo-differential calculus, or modulation spaces Mp,q on the other hand. In particular we deduce that
WL1,r ∗ sp = sp and WL1,r ∗ Mp,q when r ∈ (0,1] and p,q ∈ [r,∞).
The talk is based on collaborations with Divyand Bhimani.