Off-diagonal estimates for the first order commutators in higher dimensions
Motivated by the first Calderón commutator the authors of this paper study the operator \[ T_{m}(f,g)(x) := \iint_{\mathbb{R}^d}\left[ \int_{0}^{1}m(\xi+t\eta) dt \right] \widehat{f}(\xi)\widehat{g}(\eta)e^{2\pi i x\cdot (\xi+\eta)} d\xi d\eta \] where \(m\) is a multiplier that either arises as the Fourier transform of a standard Calderón-Zygmund kernel or is a bounded multiplier satisfying a standard Hörmander derivative condition. The main result of this paper is that \(T_m : L^{p}(\mathbb{R}^d)\times L^{q}(\mathbb{R}^d) \rightarrow L^r(\mathbb{R}^d)\) for every \(1<p\), \(q\leq \infty\), \(\frac{1}{r} = \frac{1}{p} + \frac{1}{q}\) with \(\frac{d}{d+1}<r<\infty\). These results are sharp up to the endpoint. In the case \(d=1\) these results are classic and but when \(d\geq 2\) they are only well understood when \(r \geq 1\). The main contribution of this paper is to establish these bounds in the higher dimensional case \(d\geq 2\) when \(r<1\). Prior to this work this had only been done in the model case where \(m\) arises from the Riesz kernels [\textit{P. W. Fong}, Smoothness properties of symbols, Calderón commutators and generalizations. Cornell University (PhD Thesis) (2016)]. The methods build on the new proofs of the boundedness of the first Calderón commutator by \textit{C. Muscalu} [Rev. Math. Iberoam. 30, No. 2, 727--750 (2014; Zbl 1306.42026)].
scientific article; zbMATH DE number 7233272
scientific article