Function spaces arising from kernel operators
Tóm tắt
Given a probability space (Ω, μ) and a rearrangement invariant space X on [0,1], in certain situations inequalities for spaces of
$${\mathbb {R}}$$
-valued functions on Ω are equivalent to the boundedness of an associated operator T
K
: L
∞ ([0, 1]) → X generated by a kernel K ≥ 0 on the unit square (e.g. Sobolev type inequalities or Riesz potentials on subsets
$${\Omega \subset \mathbb {R}^n}$$
). A natural class of spaces for treating such inequalities is given by
$${[T_{K}, X](\Omega) := \{u : \Omega\to \mathbb {R} : T_{K} u^* \in X\}}$$
together with the functional
$${u \mapsto ||T_{K} u^*||_X}$$
, where u* is the decreasing rearrangement of u. The investigation of these spaces is our main aim; the nature of the base space X and of K (via its monotonicity/growth properties) play a crucial role.
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