The \(L^p\)-to-\(L^q\) compactness of commutators with \(p > q\). (English) Zbl 1539.47075
Summary: Let \(1 < q < p < \infty\), \(1/r:=1/q-1/p\), and \(T\) be a non-degenerate Calderón-Zygmund operator. We show that the commutator \([b,T]\) is compact from \(L^p(\mathbb{R}^n)\) to \(L^q(\mathbb{R}^n)\) if and only if \(b=a+c\) with \(a\in L^r(\mathbb{R}^n)\) and \(c\) a constant. Since neither the corresponding Hardy-Littlewood maximal operator nor the corresponding Calderón-Zygmund maximal operator is bounded from \(L^p(\mathbb{R}^n)\) to \(L^q(\mathbb{R}^n)\), we take the full advantage of the compact support of the approximation element in \(C_{\mathrm{c}}^\infty (\mathbb{R}^n)\), which seems to be redundant for many corresponding estimates when \(p\leq q\) but is crucial when \(p > q\). We also extend the results to the multilinear case.
MSC:
| 47B90 | Operator theory and harmonic analysis |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 42B35 | Function spaces arising in harmonic analysis |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
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