Spaces of functions similar to $L^p$ but with different local and global sizes

Question

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.

Answer 1

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:

• Wiener Amalgams Over Euclidean Spaces And Some of Their Applications by Feichtinger
• An Introduction to Weighted Wiener Amalgams by Heil (it deals with the weighted space, but gives references describing non-weighted spaces in the introduction)