Difference between revisions of "Morrey spaces"
From Encyclopedia of Mathematics
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is called the Morrey space $L^{p,\lambda}(\R^n)$. | is called the Morrey space $L^{p,\lambda}(\R^n)$. | ||
Different values of $\lambda$ give rise to trivial situations. | Different values of $\lambda$ give rise to trivial situations. | ||
| − | + | It is a Banach space (Hilbert if $p=2$) under the norm | |
$$ | $$ | ||
\|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} | \|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} | ||
$$ | $$ | ||
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| − | Morrey spaces were introduced by C.B.Morrey in 1938 in connection with the regularity for systems of partial differential equations. | + | Morrey spaces were introduced by C.B.Morrey in 1938 in connection with the study of regularity for systems of partial differential equations. |
Revision as of 16:36, 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. It is a Banach space (Hilbert if $p=2$) under the norm $$ \|f\|_{p,\lambda} \equiv \sup_{r>0, x \in \R^n} \left(\dfrac{1}{r^\lambda} \int_{B_r(x)} |f(y)|^p dy\right)^{1/p} $$
Morrey spaces were introduced by C.B.Morrey in 1938 in connection with the study of 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