Original Paper
Fractional integration operators of variable order: continuity and compactness properties
Mikhail Lifshits,
Werner Linde,
Mikhail Lifshits
Department of Mathematics and Mechanics, St. Petersburg State University, 198504 St. Petersburg, Russia
MAI, Linköping University, 58183 Linköping, Sweden
e-mail: [email protected]
Search for more papers by this authorCorresponding Author
Werner Linde
Friedrich-Schiller-Universität Jena, Institut für Stochastik, Ernst-Abbe-Platz 2, 07743 Jena, Germany
Corresponding author: e-mail: [email protected]Search for more papers by this authorMikhail Lifshits,
Werner Linde,
Mikhail Lifshits
Department of Mathematics and Mechanics, St. Petersburg State University, 198504 St. Petersburg, Russia
MAI, Linköping University, 58183 Linköping, Sweden
e-mail: [email protected]
Search for more papers by this authorCorresponding Author
Werner Linde
Friedrich-Schiller-Universität Jena, Institut für Stochastik, Ernst-Abbe-Platz 2, 07743 Jena, Germany
Corresponding author: e-mail: [email protected]Search for more papers by this authorAbstract
Let 
be a Lebesgue-almost everywhere positive function. We consider the Riemann-Liouville operator of variable order defined by
to 
. Our first aim is to study its continuity properties. For example, we show that 
is always bounded (continuous) in 
provided that 
. Surprisingly, this becomes false for 
. In order 
to be bounded in L1[0, 1], the function 
has to satisfy some additional assumptions.
In the second, central part of this paper we investigate compactness properties of 
. We characterize functions 
for which 
is a compact operator and for certain classes of functions 
we provide order-optimal bounds for the dyadic entropy numbers 
.
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