Morrey spaces

Let $1\leq p <\infty$ and $0<\lambda<n$. The set of locally integrable functions $f$ such that $$ \sup_{r>0, x \in \R^n} \dfrac{1}{r^\lambda} \int_{B_r(x)} |f(y)|^p dy <+\infty $$ is called the Morrey space $L^{p,\lambda}(\R^n)$. Different values of $\lambda$ give rise to trivial situations. The quantity $$ \|f\|_{p,\lambda} \equiv \sup_{r>0, x \in \R^n} \left(\dfrac{1}{r^\lambda} \int_{B_r(x)} |f(y)|^p dy\right)^{1/p} $$ is a norm on $L^{p,\lambda}(\R^n)$ that is Banach. If $p=2$ it is a Hilbert space.

Morrey spaces were introduced by C.B.Morrey in 1938 in connection with the regularity for systems of partial differential equations.