Singular Integrals on Morrey Spaces

Abstract

Here we want to consider the action of the Calderon-Zygmund singular integrals on the Morrey Spaces. This has a brief history, namely due to Stampacchia and later Peetre. What was known up to 1969 is summarized in the paper [Pe1]. But their approach was really one via interpolation: knowing that the singular operator T maps L ^p → L ^p for different p (and λ = n), allows that T then maps an L ^p for an intermediate p into $L^{p,\lambda },\;\lambda = \frac{p} {p_{0}} n(1-\theta ) + \frac{p} {p_{1}} n\theta$. But one thing he could not do was have L ^p, λ in the domain space. Later in Chapter 11, we will remove this restriction, but there are some relevant counterexamples that need to be addressed.