Lp -Boundedness of a Singular Integral Operator

Al-Hasan, Abdelnaser J. ; Fan, Dashan

Canadian mathematical bulletin, Tome 41 (1998) no. 4, pp. 404-412

Résumé

Let $b(t)$ be an ${{L}^{\infty }}$ function on $\mathbf{R}$ , $\Omega ({y}')$ be an ${{H}^{1}}$ function on the unit sphere satisfying the mean zero property (1) and ${{Q}_{m}}(t)$ be a real polynomial on $\mathbf{R}$ of degree $m$ satisfying ${{Q}_{m}}(0)\,=\,0$ . We prove that the singular integral operator $${{T}_{Qm,}}b\left( f \right)\left( x \right)=p.v.\int\limits_{\mathbf{R}}^{n}{b\left( \left| y \right| \right)}\Omega \left( y \right){{\left| y \right|}^{-n}}f\left( x-{{Q}_{m}}\left( \left| y \right| \right){y}' \right)\,\,dy$$ is bounded in ${{L}^{p}}({{\mathbf{R}}^{n}})$ for $1<p<\infty $ , and the bound is independent of the coefficients of ${{Q}_{m}}(t)$ .

Détail

DOI : 10.4153/CMB-1998-054-5

Mots-clés : 42B20, singular integral, rough kernel, Hardy space