THE BOUNDEDNESS OF MARCINKIEWICZ INTEGRAL WITH VARIABLE KERNEL

In this article, we study the fractional Marcinkiewicz integral with variable kernel defined by mu Omega,alpha(f)(x) = (integral(infinity)(0)vertical bar integral(vertical bar x-y vertical bar <= t) Omega(x, x - y)/vertical bar x - y vertical bar(n - 1)f(y) dy vertical bar(2)dt/t3-alpha)(1/2) where 0 < alpha <= 2. We first prove that mu Omega,alpha is bounded from L2n/n+alpha(R-n) to L-2(R-n) without any smoothness assumption on the kernel Omega. Then we show that, if the kernel Omega satisfies a class of Dini condition, mu Omega,alpha is bounded from H-p (R-n) (p <= 1) to H-q (R-n), where = 1/q = 1/p - alpha/2n. As corollary of the above results, we obtain the L-p - L-q (1 < p < 2) botmdedness of this fractional Marcinkiewicz integral.