Calderon-type theorems for operators of non-standard endpoint behaviour
Amiran GogatishviliLubos Pick
46B7047B3846E3026D1047G10Calderon theoreminterpolation segmentnon-increasing rearrangementendpoint estimatesHardy operatorssupremum operatorsquasilinear operatorsrearrangement-invariant spaces$K$-functionalfractional maximal operatorSobolev embeddings
The fundamental interpolation theorem of Calder\'on states that a quasilinear operator satisfying, for $1\leq p_0,q_0,p_1,q_1\leq\infty$\[T:L^{p_0,1}\to L^{q_0,\infty}\quad\mbox{and}\quad T:L^{p_1,1}\to L^{q_1,\infty},\] is bounded from a rearrangement-invariant space into another one if and only if an appropriate one-dimensional integral operator is bounded between their respective representation spaces. We establish a Calder\'on-type theorem for operators satisfying \[T:L^{p_0,1}\to L^{q_0,\infty}\quad\textup{and}\quad T:L^{p_1,\infty}\to L^{\infty}, \] and apply this result to the fractional maximal operator. We next prove a Calder\'on-type theorem for operators satisfying \[T:L^1\to L^{q_0,1}\quad\mbox{and}\quad T:L^{p_1,1}\to L^{q_1,\infty}, \] which has applications to sharp Sobolev embeddings and to boundary trace embeddings.
Indiana University Mathematics Journal
2009
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10.1512/iumj.2009.58.3636
10.1512/iumj.2009.58.3636
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Indiana Univ. Math. J. 58 (2009) 1831 - 1852
state-of-the-art mathematics
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