Heli Tuominen (University of Jyväskylä)
Orlicz-Sobolev spaces on metric measure spaces
Abstract.
In the talk, I will define Orlicz-Sobolev spaces on metric measure
spaces using weak upper gradients and discuss the basic properties of the
spaces. The definition is a generalization of Newtonian spaces, Sobolev
spaces on metric measure spaces; instead of being p-integrable, functions
and upper gradients are supposed to be in an Orlicz space. New
tools, such as modulus, capacity, and Poincaré-type inequalities,
connected to Young function are needed for these spaces.
The Orlicz-Sobolev space is always a Banach space and its functions are
absolutely continuous on almost all curves. If the Young function that
defines the Orlicz space is doubling, then the Orlicz-Sobolev space has
more important properties, for example density of bounded functions and
existence of a minimal weak upper gradient. If the underlying metric space
supports a Poincaré-type inequality, then Lipschitz functions are
dense both in norm and "in the Lusin sense". The talk is based on my PhD
thesis.
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