Endpoint bounds require separate formulations

Assume the Axiom of Countable Choice for the Lebesgue conventions of Riesz potential of order alpha. Recorded orientation, not proved here. Let $0<\alpha <n$, let $I_{\alpha }$ be the unit-normalized Riesz potential on ${\mathbb{R}}^n$ of Riesz potential of order alpha, and let the strict-range theorem of this pair be Hardy–Littlewood–Sobolev fractional integration inequality, whose hypothesis is $1<p<n\mathrm{/}\alpha$. In the notation of Sublinear operators and weak or strong type $(p,q)$ bounds, the following endpoint claims are recorded from the cited source but are not proved, used, or reproduced in this library:

1. Lower endpoint. $I_{\alpha }$ is of weak type $(1,n\mathrm{/}(n-\alpha ))$, and it is not of strong type $(1,n\mathrm{/}(n-\alpha ))$. Consequently the hypothesis $1<p$ of the strong theorem cannot be relaxed to $p=1$: no constant bounds $\mathrm{\parallel }{I}_{\alpha }f{\mathrm{\parallel }}_{n\mathrm{/}(n-\alpha )}$ by $\mathrm{\parallel }f{\mathrm{\parallel }}_1$.
2. Upper endpoint. At $p=n\mathrm{/}\alpha$ the raw potential is not of strong type $(n\mathrm{/}\alpha ,\mathrm{\infty })$: there are $f\in L^{n\mathrm{/}\alpha }({\mathbb{R}}^n)$ for which $I_{\alpha }f$ is not essentially bounded (indeed it may fail to be finite on a set of positive measure).
3. Critical mean oscillation. For $f\in L^{n\mathrm{/}\alpha }({\mathbb{R}}^n)$ with compact support, the potential $I_{\alpha }f$ is finite almost everywhere and its mean-oscillation seminorm modulo additive constants is bounded by $C\mathrm{\parallel }f{\mathrm{\parallel }}_{n\mathrm{/}\alpha }$. For general $f\in L^{n\mathrm{/}\alpha }({\mathbb{R}}^n)$ use the renormalized potential ${\tilde{I}}_{\alpha }f(x):={\int }_{{\mathbb{R}}^n}[K_{\alpha }(x-y)-1_{\{\mathrm{\mid }y\mathrm{\mid }\ge 1\}}{K}_{\alpha }(-y)]f(y)dy,$ with subtraction inside the integral. It is finite almost everywhere and locally integrable, and satisfies the same mean-oscillation bound. If $C_f:={\int }_{\mathrm{\mid }y\mathrm{\mid }\ge 1}{K}_{\alpha }(-y)f(y)dy$ is absolutely convergent, as it is for compactly supported critical data, then ${\tilde{I}}_{\alpha }f=I_{\alpha }f-C_f$ wherever the raw potential is defined. In general the raw integral may diverge everywhere, so no finite additive constant relating it to the renormalized potential is asserted.

These are orientation facts about the boundary of the strict-range theorem, recorded with their exact hypotheses and not established here. The library does not currently define the weak $L^q$ space or the space $BMO$ of functions of bounded mean oscillation, so clauses 1 and 3 are quoted from the source in the source's own vocabulary; clause 1's weak-type inequality is the $p=1$ case of the weak $(p,q)$ estimate stated in the proof of the source's Theorem 1, and clause 3 is the source's Theorem 4 together with the remark that follows it. None of these endpoint claims is a proof supplier for this pair: the strict range $1<p<n\mathrm{/}\alpha$ retains the hypothesis of Hardy–Littlewood–Sobolev fractional integration inequality, and no item of the pair lists this remark among its dependencies. The companion page's two counterexamples exhibit the failures of clause 2 and of the strong part of clause 1 directly, in $L^{p_0}$ and $L^1$ respectively, without proving the weak-type or mean-oscillation bounds recorded above.