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. |
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| + | Many operator from Harmonic Analysis are bounded on Morrey spaces. We recall the Maximal operator and the Singular Integral Operators. | ||
Latest revision as of 15:41, 28 August 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.
Many operator from Harmonic Analysis are bounded on Morrey spaces. We recall the Maximal operator and the Singular Integral Operators.
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