ABSTRACT
Morrey spaces can describe the local properties of functions over Rn. This applies to a general metric measure space. This chapter considers two typical cases. The first is where the Lebesgue measure is replaced by a general Radon measure on Rn. Although fractal theory is not discussed due to its vastness, the ternary Cantor set is recalled. The second is where the underlying space differs from Euclidean space. The chapter establishes the theory of Lebesgue spaces in Euclidean space with general Radon measures and focuses on the theory of Lebesgue spaces in metric measure spaces. One of the strong thrusts of studying general metric measure spaces is to obtain some estimates independent of the underlying measures. Sometimes such estimates are obtained with the constants independent of the dimension. García-Cuerva and Martell obtained the vector-valued estimates for singular integral operators.
