Convolution singular integrals on Lipschitz surfaces

Abstract

We prove the L p {L_p} -boundedness of convolution singular integral operators on a Lipschitz surface \[ Σ = { g ( x ) e 0 + x ∈ R n + 1 : x ∈ R n } \Sigma = \{ g({\mathbf {x}}){e_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:{\mathbf {x}} \in {\mathbb {R}^n}\} \] where g g is a Lipschitz function which satisfies ‖ ∇ g ‖ ∞ ≤ tan ω > ∞ {\left \| {\nabla g} \right \|_\infty } \leq {\text {tan}}\omega > \infty . Here we have embedded R n + 1 {\mathbb {R}^{n + 1}} in the Clifford algebra R ( n ) {\mathbb {R}_{(n)}} with identity e 0 {e_0} , and are considering convolution with right-monogenic functions ϕ \phi which satisfy | ϕ ( x ) | ≤ C | x | − n \left | {\phi (x)} \right | \leq C{\left | x \right |^{ - n}} on a sector \[ S μ o = { x = x 0 + x ∈ R n + 1 : | x 0 | > | x | tan μ } S_\mu ^o = \{ x = {x_0} + {\mathbf {x}} \in {\mathbb {R}^{n + 1}}:\left | {{x_0}} \right | > \left | {\mathbf {x}} \right |{\text {tan}} \mu \} \] where μ > ω \mu > \omega . Provided there exists an L ∞ {L_\infty } function ϕ _ \underline \phi satisfying \[ ϕ _ ( R ) − ϕ _ ( r ) = ∫ r > | x | > R x ∈ R n ϕ ( x ) d x \underline \phi (R) - \underline \phi (r) = \int _{\substack {r > |x| > R\\x \in {\mathbb {R}^{n}}}} \phi (x)\;dx \] , then the related convolution singular integral operator \[ ( T ( ϕ , ϕ ) _ u ) ( x ) = lim ε → 0 + { ∫ y ∈ Σ | x − y | ≥ ε ϕ ( x − y ) n ( y ) u ( y ) d S y + ϕ _ ( ε n ( x ) ) u ( x ) } ({T_{(\phi ,\underline {\phi )} }}u)(x) = \lim _{\varepsilon \to 0+}\left \{\int _{\substack {y \in \Sigma \\|x - y| \geq \varepsilon }} \phi (x - y)n(y)u(y)\;d{S_y} + \underline \phi (\varepsilon n(x))u(x) \right \} \] is bounded on L p ( Σ ) {L_p}(\Sigma ) for 1 > p > ∞ 1 > p > \infty .