Riemann-Liouville Operators of Varying Order

Werner Linde (Newark, DE) Joint work with Mikhail Lifshits (St. Petersburg, Russia)

Abstract: We present continuity and compactness properties of the integration operator

(R α(·) f)(t) := 1 Γ(α(t)) Z t 0 (t − s) α(t)−1 f(s) ds , 0 ≤ t ≤ 1 .

Here α(·) is a given measurable function on [0, 1] possessing a.e. positive values. Operators Rα(·) are generalizations of classical Riemann-Liouville operators Rα of order α > 0 which correspond to α(t) ≡ α. Thus Rα(·) may be viewed as fractional integration operator of varying order.

Our interest to investigate operators Rα(·) stems from the theory of multi-fractional random processes. These are fractional Brownian motions {BH(t) : t ≥ 0} with time depending Hurst index H = H(t).

In the talk we will treat the following problems:

- Under which conditions on α(·) is Rα(·) bounded from Lp[0, 1] into Lq[0, 1] ?
- In which cases is Rα(·) not only bounded but even a compact operator?
- How does the degree of compactness (measured by the behavior of its entropy numbers) depend on properties of the function α(·) ?

References: M. Lifshits, Linde, W.: Fractional integration operators of variable order: continuity and compactness properties, Math. Nachr. 287, 980 – 1000 (2014).

Extended version in arXiv: 1211.3826