Oscillatory integrals on the unit sphere
The author presents two estimates of the form \[ \bigg(\int_0^\infty|S_{\Omega,P}(t\xi)|^\gamma\frac{dt}{t}\bigg) ^{1/\gamma} \leq C\|\Omega\|\qquad\text{for}\quad 1\leq\gamma<\infty, \] where the phase function \(P(t)\) is a real-valued polynomial in \(t\in \mathbb R\), \(S_{\Omega,P}\) is the oscillatory integral defined by \[ S_{\Omega,P}(\xi)=\int_{S^{n-1}}e^{iP(\xi\cdot y)}\Omega(y)\,d \sigma(y), \] and the norm on the right hand side is either the norm in the space \(L(\log^+L)^{1/\gamma}(S^{n-1})\) or the norm in the Besov space \(B_q^{(0,\frac{1}{\gamma}-1)}(S^{n-1})\). These two estimates generalize the following well known estimate \[ \left(\frac{1}{R}\int_0^R|\widehat{d\mu_{\Omega}}(\xi)|^2\,dt \right)^{1/2} \leq C_\varepsilon(R|\xi|)^{-\varepsilon}\|\Omega\|_{L^q(S^{n-1})} \] \(\text{for all } R>0, \xi\in\mathbb R^n, \text{ and }\;\varepsilon<\frac{1}{4}(1-q^{-1})\) given by [\textit{J. Duoandikoetxea} and \textit{J. L. Rubio de Francia}, Invent. Math. 84, 541--561 (1986; Zbl 0568.42012)]. As applications, the author establishes \(L^2\) boundedness of the Marcinkiewicz integral and the singular integral operators along the surfaces of revolution, whenever \(\Omega \in L(\log^+ L)^{1/2}(S^{n-1})\) or \(\Omega \in B_q^{(0, -1/2)}(S^{n-1})\).
scientific article; zbMATH DE number 5141235