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I'm reading 'Navier-Stokes flow in $\mathbb{R}^3$ with measures as initial vorticity and Morrey spaces' (a paper of Giga and Miyakawa) but there is something that I don't understand about the definition of Morrey spaces they give.

They introduce them (for each $1\leq p\leq \infty$) as the spaces of Radon signed measures $\mu$ in $\mathbb{R}^N$ verifying:

$$\Vert \mu\Vert_p:=\sup\limits_{x\in \mathbb{R}^N,r>0} r^{-\frac{N}{p'}}\cdot \vert\mu\vert(B(x,r))<\infty$$

They proof that $\Vert \cdot \Vert_p$ is a Banach norm, but I don't understand how could one sum two Radon signed measure without having problems at computing $+\infty-\infty$. Notice that we are NOT asuming $\mu$ to have finite total variation.

Could anyone help me? Thanks.

PD: Sorry for my English.

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  • $\begingroup$ The addition operation on the space of signed measures is really only a partial binary operation, exactly to avoid the issue of $\infty - \infty$. I am not familiar with the work of Giga and Miyakawa, but they probably just avoid the issue entirely. Namely, by considering the triangle inequality $\|x +y\| \le \|x\|+\|y\|$ only when the sum $x+y$ is well-defined. $\endgroup$
    – Tom LaGatta
    Commented Aug 27, 2013 at 20:02
  • $\begingroup$ Thak you very much. I thought they were using an implicit construction of addition by means of inner regularity but your answer makes sense. $\endgroup$
    – JDPoincare'S
    Commented Aug 27, 2013 at 20:13

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It depends on what one means by a signed Radon measure. In the context of non-compact spaces it does not really make sense to require that it is well-defined on the whole space, so that I am pretty sure that what they mean by a signed Radon measure $\mu$ is just $\mu=\mu_+-\mu_-$, where $\mu_{\pm}$ are two positive (not necessarily finite) Radon measures. Then $\mu$ makes sense on any compact subspace, whereas under additional growth assumptions on $\mu$ (like the one made in the quoted paper) one can make integrals against $\mu$ well-defined for certain functions with non-compact support as well.

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  • $\begingroup$ I agree with you. I'm reading about related topic in other references just know and it seems that they only consider meausures locally (defined for bounded Borel sets). So, what you mean is that the space of signed Radon measures they are problably considerying is kind of the vector space of formal sums $\alpha\cdot \mu+\beta\cdot \mu$ with $\mu,\nu$ positive Radon measures, isn't it? $\endgroup$
    – JDPoincare'S
    Commented Aug 28, 2013 at 9:54
  • $\begingroup$ Yes, precisely. $\endgroup$
    – R W
    Commented Aug 28, 2013 at 9:55
  • $\begingroup$ I was afraid of changing the standard definition because I wanted to go on seeing this measures as distributions in the same way we see locally finite borel measures, but there is no problem. The only problem I have, if we use $\mathcal{M}^p$ for noting this subspace is to describe in an easy way the intersection $L^1_{loc}\cap \mathcal{M}^p$ (in the sense of distributions). Such functions $f$ verify: $$\int f\cdot\varphi \,dx=\int \phi \,d\mu^+ +\int \phi \,d\mu^-$$ for every $\varphi\in C^{\infty}_c$ and certain $\mu^+,\mu^-$ positive Radon measures $\endgroup$
    – JDPoincare'S
    Commented Aug 28, 2013 at 10:03
  • $\begingroup$ I say "problems" because $\mu^+,\mu^-$ seem to be too arbitrary. But perhaps there are not better descriptions. I'll go on reading. Thank you very much! $\endgroup$
    – JDPoincare'S
    Commented Aug 28, 2013 at 10:06
  • $\begingroup$ Sorry for my typing mistakes. I wanted to write: $$\int f\cdot \varphi\,dx=\int \varphi\,d\mu^+-\int \varphi \,d\mu^-$$ $\endgroup$
    – JDPoincare'S
    Commented Aug 28, 2013 at 10:12

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