Let \(n\geq 2\) and \(S^{n-1}\) be the unit sphere in \({\mathbb R}^n\). Let \(H^r(S^{n-1})\) be the Hardy spaces on \(S^{n-1}\). For a smooth function \(f\) on \({\mathbb R}^n\), we write \(f_{(x,s)}(y')=f(x-|y|y')\), where \(y'=y/|y|\) and \(s=|y|\). Further, let \(b\in L^{\infty}(\mathbb R_+\times\mathbb R)\) and define \[ F_{\Omega,t}(f)(x)=2^{-t}\int_0^{2^t}b(s,2^t)\langle\Omega,f_{x,s}\rangle \,ds. \] In this paper the authors study certain hypersingular integral operators \[ {\mathcal J}_{\Omega,\alpha}(f)(x)=\int_{\mathbb R}2^{-t\alpha} e^{i2^{-t\beta}}F_{\Omega,t}(f)(x)\,dt, \] and \[ {\bar \mu}_{\Omega,\alpha}(f)(x)=\left|\int_{\mathbb R}2^{-2t\alpha} e^{i2^{-t\beta+1}}|F_{\Omega,t}(f)(x)|^2\,dt\,\right|^{1/2}. \] They prove that if \(\Omega\) is in the Hardy space \(H^r(S^{n-1})\) with \(00)\), and satisfies certain cancellation condition, then the operators \({\mathcal J}_{\Omega,\alpha}\) and \( {\bar \mu}_{\Omega,\alpha}\) are bounded from Sobolev space \(L^p_{\gamma}\) to Lebesgue space \(L^p\) for some \(p.\)