On local oscillatory integrals with variable Calderón-Zygmund kernels
Let \(p\in (1,\infty)\). In the paper, the authors study the uniform \(L^p(\mathbb{R}^n)\) boundedness in \(\lambda\) of the oscillatory singular integral operator \(T_\lambda\) defined by \[ T_\lambda f(x)= \text{p.v. }\int_{\mathbb{R}^n} e^{i\lambda\Phi(x, y)} K(x,x- y) \psi(x, y) f(y) dy, \] where \(\lambda\in \mathbb{R}\), \(\Phi\) is a real analytic function or a real \(C^\infty\) function on \(\mathbb{R}^n\times \mathbb{R}^n\), \(\psi\in C^\infty_0(\mathbb{R}^n\times \mathbb{R}^n)\) and \(K\) is a variable Calderón-Zygmund kernel. Moreover, after obtaining the weighted \(L^p\) boundedness, the authors also study the uniform boundedness in \(\lambda\) of the commutators generated by \(T_\lambda\) and \(\text{BMO}(\mathbb{R}^n)\) functions. In the proof, the authors assume that the kernel \(K\) satisfies the condition \[ \max_{|j|\leq 2n} \Biggl\|{\partial^{|j|}\over\partial y^j} K(x,y)\Biggl\|_{L^\infty(\mathbb{R}^n\times S^{n- 1})}= M< \infty. \] So the proof can follow a standard argument by Calderón and Zygmund and an integration by parts yields the necessary estimate. Clearly, the paper would be more interesting if the above condition on the kernel \(K\) can be weakened.
scientific article; zbMATH DE number 1310385