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Difference between revisions of "Morrey spaces"

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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.
The quantity
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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}  
 
$$
 
$$
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.
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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