A singular integral operator related to block spaces - Hokkaido ...

Hokkaido Mathematical Journal Vol. 28 (1999) p. 285-299

A singular integral operator related to block spaces AbdelNaser J. AL-HASAN* and Dashan FAN (Received November 10, 1997; Revised July 3, 1998) Abstract. Let h(t) be an function on (0, \infty) , \Omega(y’) be a function on the unit sphere satisfying the mean zero property (1.1) and P_{N}(t) be a real polynomial on R of degree N satisfying P_{N}(0)=0 . We prove that the singular integral operator L^{\infty}

B_{q}^{0,0}

( \tau_{P_{N\prime}h}f)(x)=p.v.\int_{R^{n}}h(|y|)\Omega(y’)|y|^{-n}f(x-P_{N}(|y|)y’)dy

is bounded in of P_{N}(t) .

L^{p}(R^{n})

for 11

.

r>1 , we have

.

such that

\lambda\leq\min\{q, 2\}

. Following the

|\hat{\sigma}_{b_{m},k,r}(\xi)|\leq C|2^{rk}\beta_{r}\xi|^{\frac{-\delta}{2r}}||b_{m}||_{L^{\lambda}(S^{n-1})}

\cross\{\int_{S^{n-1}}\int_{\cross S^{n-1}}(|\xi’.(y’-x’)|^{\frac{1}{r}})^{-\delta\lambda’}d\sigma(y’)d\sigma(x’)\}^{2\lambda}\neg 1

for any \delta\in(0,1] . Since

||b_{m}||_{L^{\lambda}(S^{n-1})}\leq C

, letting

, we obtain

\delta2 . Let \varphi(t)=\phi(t^{2}) . In order to use an induction argument, we define the \phi\in C_{0}^{\infty}(R)

294

A.J. Al-Hasan and D. Fan

measures

\{\tau_{\Omega,k,N-\lambda}\}

and

\{\tau_{b_{m},k,N-\lambda}\}

by

\hat{\tau}_{\Omega,k,N-\lambda}(\xi)=\hat{\sigma}_{\Omega,k,N-\lambda}(\xi)\prod_{N-\lambda