On the Boundedness of Hypersingular Integrals Along Certain Radial Hypersurfaces

TL;DR

The paper studies L^p and Sobolev boundedness for oscillatory hypersingular operators along radial hypersurfaces $\\Gamma(t)=(t,\\varphi(t))$ in $\\mathbb{R}^{n+1}$, with a kernel incorporating $e^{-2\\pi i|t|^{-\\beta}}$ and a homogeneous angular factor. By a dyadic phase analysis, multiplier estimates, and carefully designed oscillatory integral bounds, it proves $L^2$-boundedness under $\\beta>2\\alpha>0$ and $L^p$-boundedness for $\\frac{\\beta}{\\beta-\\alpha}<p<\\frac{\\beta}{\\alpha}$, along with Sobolev estimates up to $s_0=\\frac{\\beta/2-\\alpha}{\\beta+k_3+2}$. The methodology combines a detailed phase-space decomposition, Van der Corput-type control, and complex interpolation via an analytic family of operators to obtain $L^p$-$L^p_s$ bounds for $s\,\le\,s_0$. Collectively, the results extend prior works on hypersingular oscillatory integrals to a broader class of radial surfaces and higher dimensions, with implications for harmonic analysis along curved hypersurfaces and potential applications to PDEs and geometric measure theory.

Abstract

We study a class of oscillatory hypersingular integral operators associated to a radial hypersurface of the form $Γ(t)=(t,\varphi(t)), t\in\R{n}$. When $\varphi$ satisfies suitable curvature and monotonicity conditions, we prove $L^p(\R{n+1})$ boundedness of the operator, where the range of $p$ depends on the hypersingularity of the operator. We also establish certain Sobolev estimates of the operator under consideration.

Theorems & Definitions (17)

• Remark 1.1
• Remark 1.2
• Remark 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1: Van der corput's lemma MR1232192
• Theorem 2.2: Mean Value Theorem MR344384
• Theorem 2.3: Polar decomposition in $\mathbb{R}^n$ MR1970295
• Theorem 2.4: MR3887684
• Theorem 2.5: MR2827930