Sharp bounds for multiplier operators of negative indices associated with degenerate curves
Studied are the Bochner-Riesz type means associated with the plane curve \(C=\{(t,\gamma(t)):t\in[-1,1]\}\), where \(\gamma:[-1,1] \to\mathbb R\) is a smooth function. They are multiplier operators and are defined as \[ \widehat{S^\alpha f}(x)=\frac{(x_2-\gamma(x_1))^\alpha_+}{\Gamma(\alpha+1)} \chi(x) \widehat{f}(x), \] where \(x=(x_1,x_2)\in\mathbb R^2\) and \(\chi\) is a smooth function compactly supported in a small neighborhood of \((0,\gamma(0))\). When \(\text{Re}(\alpha)\leq-1\), \(S^\alpha\) is defined by analytic continuation. Also considered are related multiplier operators. When \(C\) is a part of the circle, one gets usual Bochner-Riesz means for which classical results on \(L^p\)-\(L^q\) boundedness are known. In the paper under review these results are generalized to multiplier operators of negative indices associated with degenerate curves. The proofs are quite sophisticated and the obtained results are sharp.
scientific article; zbMATH DE number 5859316
scientific article