Difference between revisions of "Morrey spaces"
From Encyclopedia of Mathematics
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\sup_{r>0, x \in \R^n} \dfrac{1}{r^\lambda} \int_{B_r(x)} |f(y)|^p dy <+\infty | \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. | ||
Revision as of 16:33, 8 June 2024
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.
How to Cite This Entry:
Morrey spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morrey_spaces&oldid=55815
Morrey spaces. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morrey_spaces&oldid=55815