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In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) $L^{\lambda ,p}(\Omega )$ are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of $\lambda$, elements of the space $L^{\lambda ,p}(\Omega )$ are Hölder continuous functions over the domain $\Omega$.

The seminorm of the Morrey spaces is given by

$\left[f\right]_{\lambda ,p}^{p}=\sup _{0<r<\operatorname {diam} (\Omega ),x_{0}\in \Omega }{\frac {1}{r^{\lambda }}}\int _{B_{r}(x_{0})\cap \Omega }|u(y)|^{p}dy.$

When $\lambda =0$, the Morrey space is the same as the usual $L^{p}$ space. When $\lambda =n$, the spatial dimension, the Morrey space is equivalent to $L^{\infty }$, due to the Lebesgue differentiation theorem. When $\lambda >n$, the space contains only the 0 function.

The seminorm of the Campanato space is given by

$\left[f\right]_{\lambda ,p}^{p}=\sup _{0<r<\operatorname {diam} (\Omega ),x_{0}\in \Omega }{\frac {1}{r^{\lambda }}}\int _{B_{r}(x_{0})\cap \Omega }|u(y)-u_{r,x_{0}}|^{p}dy$

where

$u_{r,x_{0}}={\frac {1}{|B_{r}(x_{0})\cap \Omega |}}\int _{B_{r}(x_{0})\cap \Omega }u(y)dy.$

It is known that the Morrey spaces with $0\leq \lambda <n$ are equivalent to the Campanato spaces with the same value of $\lambda$ when $\Omega$ is a sufficiently regular domain, that is to say, when there is a constant A such that $|\Omega \cap B_{r}(x_{0})|>Ar^{n}$ for every $x_{0}\in \Omega$ and $r<\operatorname {diam} (\Omega )$.

When $n=\lambda$, the Campanato space is the space of functions of bounded mean oscillation. When $n<\lambda \leq n+p$, the Campanato space is the space of Hölder continuous functions $C^{\alpha }(\Omega )$ with $\alpha ={\frac {\lambda -n}{p}}$. For $\lambda >n+p$, the space contains only constant functions.

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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