Advanced Search

We prove the $L^2{(}^2)$ boundedness of the oscillatory singular integrals $P_{0}f(x,y)={\int }_{D_x}{e}^{i(M_2\left(x\right)y^{\text{'}}+M_1\left(x\right)x^{\text{'}})}over{x}^{\text{'}}{y}^{\text{'}}f(x-x^{\text{'}},y-y^{\text{'}})d{x}^{\text{'}}d{y}^{\text{'}}$ for arbitrary real-valued $L^{\infty }$ functions $M_1\left(x\right),M_2\left(x\right)$ and for rather general domains $D_x{\subseteq }^2$ whose dependence upon x satisfies no regularity assumptions.

We prove some extrapolation results for operators bounded on radial $L^p$ functions with p ∈ (p₀,p₁) and deduce some endpoint estimates. We apply our results to prove the almost everywhere convergence of the spherical partial Fourier integrals and to obtain estimates on maximal Bochner-Riesz type operators acting on radial functions in several weighted spaces.

Let $ℒ$ be a right-invariant sub-Laplacian on a connected Lie group $G,$ and let $S_{R}f:={\int }_0^{R}d{E}_{\lambda }f,R\ge 0,$ denote the associated “spherical partial sums,” where $ℒ={\int }_0^{\infty }\lambda d{E}_{\lambda }$ is the spectral resolution of $ℒ.$ We prove that $S_{R}f\left(x\right)$ converges a.e. to $f\left(x\right)$ as $R\to \infty$ under the assumption $log(2+ℒ)f\in L^2\left(G\right).$