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Obviously much of analysis on $\mathbb R^n$ considers $L^p$ spaces or other Banach spaces derived from them. The definition of $L^p(\mathbb R^n)$ looks very natural, but I've been bothered for some time that what it measures might not so natural. My issue specifically is that it measures the "local" size of $f$ (say $f\cdot\mathbf1_{|f|>1}$) with the same $p$ that it measures the "global" size (say $f\cdot\mathbf1_{|f|\leq1}$). I intuitively feel like the local and global sizes have nothing to do with each other and therefore it is arbitrary to say they should be measured with the same exponent.

It seems to me then that analysis would benefit from some Banach space $X^{p,q}$ that lets you measure the function locally in $p$ and globally in $q$. To illustrate my concern, consider $|x|^{-a}$ ($a>0$), which are not in any $L^p$ spaces! But we have $|x|^{-a}\in X^{p,q}$ as long as $n/q<a<n/p$. It's then clear how $L^p=X^{p,p}$ is just a special case, and there's no $a$ to satisfy $n/p<a<n/p$.

An explicit norm I thought of that might do the job is $$\|\varphi\|_{X^{p,q}}=\inf_{K\subset\subset\mathbb R^n}\left(\|\varphi\cdot\mathbf1_K\|_{L^p}+\|\varphi\cdot\mathbf1_{K^c}\|_{L^q}\right).$$

The closest I've seen to this is what Lemarie-Rieusset (Recent Developments in the Navier-Stokes Problem, 2002) calls $WL^\infty$ with norm $$\|\varphi\|_{WL^\infty}=\sum_{k\in\mathbb Z^n}\sup_{x-k\in[0,1]^n}|\varphi(x)|,$$ which resembles what my above notation would call $X^{\infty,1}$.

So please tell me, am I wrong that this is something natural to consider? It seems that analysts have gotten pretty far without it. Perhaps there's something very special about measuring the local and global parts of a function in the same way. Or maybe such a space is known and I haven't come across it.

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  • $\begingroup$ you won't have anymore $\|C \varphi\|_{X^{p,q}} = C\| \varphi\|_{X^{p,q}}$, so good luck for using it $\endgroup$ – reuns May 31 '16 at 6:46
  • $\begingroup$ You're right, I deleted it. $\endgroup$ – Funktorality May 31 '16 at 6:48
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analysis would benefit from some Banach space $X^{p,q}$ that lets you measure the function locally in $p$ and globally in $q$.

Such spaces do exist, in several flavors.

Interpolation spaces

The spaces $L^p\cap L^q$ and $L^p+L^q$ sometimes appear in PDE and frequently in interpolation theory. In $L^p\cap L^q$, the smaller exponent controls the global behavior, the larger controls the global "tail". The norms can be simply $\max(\|f\|_p,\|f\|_q)$. The sum $L^p+L^q$ consists of the sums $ h= f+g$ with $f\in L^p$, $g\in L^q$; the norm is the infimum $\|h\| = \inf_{f+g=h}(\|f\|_p+\|g\|_q)$.

Amalgam spaces

Wiener amalgam spaces are defined differently depending on the domain of functions under consideration (sometimes involving a continuous partition of unity, or convolution with a window function), but the simplest form, on $\mathbb{R}^n$, is

$$\|f\|_{W(L^p,L^q)} = \left(\sum_{k\in\mathbb{Z}^n}\left(\int_{[0,1]^n+k}|f|^p \right)^{q/p}\right)^{1/q}$$

The functions in $W(L^p,L^q)$ are locally in $L^p$ and their global behavior is like $L^q$.

References:

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  • $\begingroup$ what about the $\inf$ ? this one is clearly a norm, but with a $\inf$ for choosing your lattice or the size of the boxes, can we get $\| f+g\| \le \|f\|+\|g\|$ ? ($\|Cf\| = C\|f\|$ is obviously fulfilled). as you wrote it, without a $\inf$ it is the same as $l^q(L^p([0,1]^n))$ $\endgroup$ – reuns May 31 '16 at 23:44
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    $\begingroup$ The space with $\inf_K$ looks like $L^p+L^q$ (which I just added to my answer). One needs to take the infimum over arbitrary decompositions to have the triangle inequality, If the only allowed decomposition is $f\chi_K + f\chi_{K^c}$, I don't expect the triangle inequality to hold. $\endgroup$ – user147263 May 31 '16 at 23:50
  • $\begingroup$ hence the answer to his first question is wiki/Interpolation_space, sort of. I didn't know those things $\endgroup$ – reuns May 31 '16 at 23:53
  • $\begingroup$ it seems very unobvious to me that for any $2$ norms on $\mathbb{R} \to \mathbb{R}$, $\|h\| = \inf_{h = f+g} \|f\|_A+\|g\|_B$ is also a norm (this would prove that $\|h\| = \inf_{h = f_1+f_2+f_3+\ldots } \|f_1\|_A+\|f_2\|_B+\|f_3\|_C+\ldots$ works too) $\endgroup$ – reuns May 31 '16 at 23:57
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    $\begingroup$ @StanCoreyCarter You may want to ask a separate question on the motivation of $L^p$ spaces. Most of what you had in this question was about the $X^{p,q}$ idea, so I addressed that (and edited the title to align it with the question). $\endgroup$ – user147263 Jun 1 '16 at 0:23

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