Singular integrals and fractional integrals in Triebel-Lizorkin spaces and in weighted LP spaces

We study the hypersingular integral T(h,alpha)f(x) = lim(epsilon -> 0)integral(vertical bar y vertical bar>epsilon)b(vertical bar y vertical bar)e(ih(vertical bar y vertical bar))Omega(y')/vertical bar y vertical bar(n+alpha)f(x-y)dy and the fractional integral I(h,alpha)f(x) = integral(Rn)b(vertical bar y vertical bar)e(ih(vertical bar y vertical bar))Omega(y')/vertical bar y vertical bar(n-alpha)f(x-y)dy in Triebel-Lizorkin spaces and weighted LP spaces. Here Omega epsilon H-r(Sn-1), and b(vertical bar y vertical bar) and h(vertical bar y vertical bar) are measurable radial functions which satisfy some suitable conditions. We also consider the above integrals along some surfaces of revolution. The results in this paper extend some known results about hypersingular integrals and fractional integrals.