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\title[Fractional Gagliardo--Nirenberg and BMO]{Fractional Gagliardo--Nirenberg interpolation inequality and bounded mean oscillation}

\author{\firstname{Jean} \lastname{Van Schaftingen}\CDRorcid{0000-0002-5797-9358}}
\address{Universit\'e catholique de Louvain, Institut de Recherche en Math\'ematique et Physique, Chemin du Cyclotron 2 bte L7.01.01, 1348 Louvain-la-Neuve, Belgium}
\email{Jean.VanSchaftingen@UCLouvain.be}
\thanks{The author was supported by the Projet de Recherche T.0229.21 ``Singular Harmonic Maps and Asymptotics of Ginzburg--Landau Relaxations'' of the Fonds de la Recherche Scientifique--FNRS}

%~\keywords{Gagliardo--Nirenberg interpolation inequality, bounded mean oscillation, homogeneous fractional Sobolev--Slobodeckiĭ space, homogeneous first-order Sobolev space, maximal function, Gagliardo semi-norm}

\subjclass[2020]{26D10, 35A23, 42B35, 46B70, 46E35}

\begin{abstract}
We prove Gagliardo--Nirenberg interpolation inequalities estimating the Sobolev semi-norm in terms of the bounded mean oscillation semi-norm and of a Sobolev semi-norm, with some of the Sobolev semi-norms having fractional order.
\end{abstract}


\begin{document}
\maketitle

\section{Introduction}

The homogeneous Gagliardo--Nirenberg interpolation inequality for Sobolev space states that if \(d \in \Nset \setminus \set{0}\) and if \(0 \le s_0 < s < s_1\), \(1 \le p, p_0, p_1 \le \infty\) and \(0 < \theta < 1\) fulfil the condition
\begin{equation}
\bigl(s, \tfrac{1}{p}\bigr) = (1 - \theta) \brk[\big]{s_0, \tfrac{1}{p_0}} + \theta\brk[\big]{s_1, \tfrac{1}{p_1}},
\end{equation}
then, for every function \(f \in \dot{W}^{s_0, p_0} (\Rset^d) \cap \dot{W}^{s_1, p_1} (\Rset^d)\), one has \(f \in \dot{W}^{s, p} (\Rset^d)\), and
\begin{equation}\label{eq_eev7phaipouxooGuoYooGhah}
\norm{f}_{\dot{W}^{s, p}(\Rset^d)} \le C \norm{f}_{\dot{W}^{s_0, p_0}(\Rset^d)}^{1 - \theta} \norm{f}_{\dot{W}^{s_1, p_1}(\Rset^d)}^{\theta},
\end{equation}
unless \(s_1\) is an integer, \(p_1 = 1\) and \(s_1 - s_0 \le 1 - \frac{1}{p_0}\).

When \(s = 0\), we use the convention that \(\dot{W}^{0, p} (\Rset^d) = L^p (\Rset^d)\), and when \(s \in\Nset \setminus \set{0}\) is a positive integer, \(\dot{W}^{s, p} (\Rset^d)\) is the classical integer-order \emph{homogeneous Sobolev space} of \(s\) times weakly differentiable functions \(f : \Rset^d \to \Rset\) such that \(\Deriv^s f \in L^{p} (\Rset^d)\) and
\begin{equation}
\norm{f}_{\dot{W}^{s, p} (\Rset^d)}
\defeq \brk*{\int_{\Rset^d} \abs{\Deriv^s f}^p}^\frac{1}{p}.
\end{equation}
For \(s_0, s_1, s \in \Nset\) the inequality~\eqref{eq_eev7phaipouxooGuoYooGhah} was proved by Gagliardo~\cite{Gagliardo_1959} and Nirenberg~\cite{Nirenberg_1959} (see also~\cite{Fiorenza_Formica_Roskovec_Soudsky_2021}).

When \(s \not \in \Nset\), the \emph{homogeneous fractional Sobolev--Slobodeckiĭ space} \(\dot{W}^{s, p} (\Rset^d)\) can be defined as the set of measurable functions \(f : \Rset^d \to \Rset\) which are \(k\) times weakly differentiable with a finite Gagliardo semi-norm:
\begin{equation}
\norm{f}_{\dot{W}^{s, p} (\Rset^d)}
\defeq \brk*{\int_{\Rset^d} \int_{\Rset^d} \frac{\abs{\Deriv^{k} f (y) - \Deriv^{k} f (x)}^p}{\abs{y - x}^{d + \sigma p}} \dif y \dif x}^\frac{1}{p} < \infty,
\end{equation}
with \(k \in \Nset\), \(\sigma \in \intvo{0}{1}\) and \(s = k + \sigma\); the characterisation of the range in which the Gagliardo--Nirenberg interpolation inequality~\eqref{eq_eev7phaipouxooGuoYooGhah} holds was performed in a series of works \cite{Cohen_2000, Brezis_Mironescu_2001, Cohen_Dahmen_Daubechies_DeVore_2003, Cohen_Meyer_Oru_1998} up to the final complete settlement by Brezis and Mironescu~\cite{Brezis_Mironescu_2018}.

We focus on the endpoint case where \(s_0 = 0\) and \(p_0 = \infty\). In this case, the inequality~\eqref{eq_eev7phaipouxooGuoYooGhah} becomes
\begin{equation}\label{eq_ouzieFa8NeeSoonei0pae8ku}
\norm{f}_{\dot{W}^{s, p}(\Rset^d)}^p \le C \norm{f}_{L^\infty (\Rset^d)}^{p - p_1} \norm{f}_{\dot{W}^{s_1, p_1}(\Rset^d)}^{p_1},
\end{equation}
and holds under the assumption that \(s p = s_1 p_1\) and either \(s_1 \ne 1\) or \(p_1 > 1\). It is natural to ask whether the inequality~\eqref{eq_ouzieFa8NeeSoonei0pae8ku} can be strengthened by replacing the uniform norm \(\norm{{\,\cdot\,}}_{L^\infty (\Rset^d)}\) by John and Nirenberg’s \emph{bounded mean oscillation} (BMO) semi-norm \(\norm{{\,\cdot\,}}_{\BMO(\Rset^d)}\), which plays an important role in harmonic analysis, calculus of variations and partial differential equations~\cite{John_Nirenberg_1961}, that is, whether we have the inequality
\begin{equation}\label{eq_iethieNie4Zou8ohch0cee2n}
\norm{f}_{\dot{W}^{s, p}(\Rset^d)}^p \le C \norm{f}_{\BMO(\Rset^d)}^{p - p_1} \norm{f}_{\dot{W}^{s_1, p_1}(\Rset^d)}^{p_1},
\end{equation}
where the bounded mean oscillation semi-norm \(\norm{{\,\cdot\,}}_{\BMO(\Rset^d)}\) is defined for any measurable function \(f : \Rset^d \to \Rset\) as
\begin{equation}\label{eq_oothaKahp8pe7OGheix4phoo}
\norm{f}_{\BMO(\Rset^d)}
\defeq
\sup_{\substack {x \in \Rset^d\\ r > 0}}
\fint_{B_r (x)} \fint_{B_r (x)} \abs{f (y) - f (z)}\dif y \dif z.
\end{equation}
The estimate~\eqref{eq_iethieNie4Zou8ohch0cee2n} was proved indeed when \(s = 1\), \(p = 4\), \(s_1 = 2\) and \(p_1 = 2\) via a Littlewood--Paley decomposition by Meyer and Rivière \cite[Theorem~1.4]{Meyer_Riviere_2003}, and for \(s, s_1 \in \Nset\) via the duality between \(\BMO (\Rset^d)\) and the real Hardy space \(\mathcal{H}^1 (\Rset^d)\) by Strezelecki~\cite{Strzelecki_2006}; a direct proof was been given recently by Miyazaki~\cite{Miyazaki_2020} (in the limiting case \(s_0 = s_1 = 0\), see \cite[Theorem~2.2]{Kozono_Wadade_2008}, \cite{Chen_Zhu_2005}); when \(s_1 < 1\), the estimate~\eqref{eq_iethieNie4Zou8ohch0cee2n} has been proved by Brezis and Mironescu through a Littlewood--Paley decomposition \cite[Lemma~15.7]{Brezis_Mironescu_2021} (see also \cite{Adams_Frazier_1992, Kozono_Wadade_2008} for similar estimates in Riesz potential spaces).

The main result (\cref{proposition_GN_Wsp_W1p_BMO}) of the present work is the estimate~\eqref{eq_iethieNie4Zou8ohch0cee2n} when \(s_1 = 1\) and \(0 < s < 1\), with a proof which is quite elementary: the main analytical tool is the classical maximal function theorem. We also show how the same ideas can be used to give a direct proof of~\eqref{eq_iethieNie4Zou8ohch0cee2n} when \(s_1 < 1\), depending only on the definitions of the Gagliardo and bounded mean oscillation semi-norms (\cref{proposition_GN_Wsp_Wsp_BMO}). Finally, we show how a last interpolation result (\cref{proposition_higher_order}) allows one to obtain the full range of interpolation between \(\BMO (\Rset^d)\) and higher-order fractional Sobolev--Slobodeckiĭ spaces \(\dot{W}^{s, p} (\Rset^d)\) with \(s \in \intvo{1}{\infty}\).

Our proofs can be considered as fractional counterparts of Miyazaki's direct proof in the integer-order case~\cite{Miyazaki_2020}. We also refer to Dao's recent work~\cite{Dao} for an alternative approach via negative-order Besov spaces to the results in the present paper.

\section{Interpolation between first-order Sobolev semi-norm and mean oscillation}

We prove the following interpolation inequality between the fist-order Sobolev semi-norm and the mean oscillation seminorm into fractional Sobolev spaces.

\begin{theo}\label{proposition_GN_Wsp_W1p_BMO}
For every \(d \in \Nset \setminus \set{0}\) and every \(p\in \intvo{1}{\infty}\), there exists a constant \(C(p) > 0\) such that for every \(s \in \intvo{1/p}{1}\), every open convex set \(\Omega \subseteq \Rset^d\) satisfying \(\varkappa (\Omega) < \infty\) and every function \(f \in \dot{W}^{1, sp} (\Omega) \cap \BMO (\Omega)\), one has
\(f \in \dot{W}^{s, p} (\Omega)\) and
\begin{equation}\label{eq_aijei2Xaighei4obeichiiKi}
\iint_{\Omega \times \Omega}
\frac{\abs{f (y) - f (x)}^p}{\abs{y - x}^{d + sp}}
\dif y
\dif x \le
\frac{C(p) \varkappa(\Omega)^{sp}}{(sp - 1)(1 - s)}
\norm{f}_{\BMO(\Omega)}^{(1 -s) p}
\int_{\Omega} \abs{\Deriv f}^{sp}.
\end{equation}
\end{theo}

We define here for a domain \(\Omega \subseteq \Rset^d\), the \emph{bounded mean oscillation semi-norm} of a measurable function \(f: \Omega \to \Rset\) as
\begin{equation}\label{eq_HakeeGhae0ku0yohy1jied9E}
\norm{f}_{\BMO(\Omega)}
\defeq
\sup_{\substack {x \in \Omega\\ r > 0}}\,
\fint_{\Omega \cap B_r (x)} \fint_{\Omega \cap B_r (x)} \abs{f (y) - f (z)}\dif y \dif z,
\end{equation}
and the geometric quantity
\begin{equation}\label{eq_waiC3theiwooDaithee1aeLa}
\varkappa (\Omega)
\defeq \sup \set*{\frac{\mathcal{L}^d (B_r (x))}{\mathcal{L}^d (\Omega \cap B_r (x))}
\st x \in \Omega \text{ and } r \in \intvo{0}{\diam (\Omega)} }.
\end{equation}
For the latter quantity, one has for example
\begin{equation}
\varkappa (\Rset^d) = 1
\end{equation}
and
\begin{equation}
\varkappa (\Rset^d_+) = 2.
\end{equation}
If the set \(\Omega\) is convex and bounded, we have \(\Omega \subseteq B_{\diam (\Omega)} (x)\) and
\(t \Omega + (1 - t)x \subseteq \Omega \cap B_r (x)\), with \(t
\defeq r/\diam (\Omega)\), so that
\begin{equation*}
\mathcal{L}^d (\Omega \cap B_{r} (x))
\ge t^d \mathcal{L}^d (\Omega) =
\frac{\mathcal{L}^d (\Omega)}{\diam (\Omega)^d}r^d,
\end{equation*}
and thus
\begin{equation}
\varkappa (\Omega) \le \frac{\mathcal{L}^d (B_1)} {\mathcal{L}^d (\Omega)} \diam (\Omega)^d.
\end{equation}
The quantity \(\varkappa (\Omega)\) can be infinite for some unbounded convex sets such as \(\Omega = \intvo{0}{1}\times \Rset^{d - 1}\) and \(\Omega = \set{(x', x_d) \in \Rset^d \st x_d \ge \abs{x'}^2}\).


Our first tool to prove \cref{proposition_GN_Wsp_W1p_BMO} is an estimate by the maximal function of the derivative of the average distance of values on a ball to a fixed value; this formula is related to the \emph{Lusin--Lipschitz inequality} \cite[Lemma~2]{Liu_1977}, \cite[Lemma~II.1]{Acerbi_Fusco_1984}, \cite{Bojarski_1990}, \cite[p.~404]{Hajlasz_1996}, \cite[(3.3)]{Jabin_2010}.

\begin{lemm}\label{lemma_osc_MDu}
If the set \(\Omega \subseteq \Rset^d\) is open and convex and if \(f \in \dot{W}^{1, 1}_{\loc} (\Omega)\), then for every \(r \in \intvo{0}{\diam(\Omega)}\) and almost every \(x \in \Omega\),
\begin{equation}\label{eq_eutai3kaeviuGhaisiechoqu}
\fint_{\Omega \cap B_{r} (x)} \abs{f (z) - f (x)} \dif z
\le \varkappa (\Omega)\, r\,
\maxfun \abs{\Deriv f} (x).
\end{equation}
\end{lemm}

Here \(\maxfun g : \Rset^d \to \intvc{0}{+\infty}\) denotes the classical \emph{Hardy--Littlewood maximal function} of the function \(g : \Omega \to \Rset\), defined for each \(x \in \Rset^d\) by
\begin{equation}\label{eq_JahvaePieG5cahth4AepheCi}
\maxfun g (x)
\defeq
\sup_{r > 0} \frac{1}{\mathcal{L}^d (B_r (x))} \int_{\Omega \cap B_r (x)} \abs{g}.
\end{equation}

\begin{proof}[Proof of \cref{lemma_osc_MDu}]
Since \(\Omega\) is convex and \(f \in \dot{W}^{1, 1} (\Omega)\), for almost every \(x \in \Omega\) and every \(r \in \intvo{0}{\infty}\), we have
\begin{equation}\label{eq_wa1lohNgiechei5oos0ae0Oh}
\int_{\Omega \cap B_{r} (x)} \abs{f (z) - f (x)} \dif z
\le {\int_{\Omega \cap B_{r}(x)}} \int_{0}^{1} \abs{\Deriv f ((1-t) x + t z)[z - x]} \dif t \dif z.
\end{equation}
By convexity of the set \(\Omega\), for every \(z \in \Omega \cap B_{r}(x)\) and \(t \in \intvc{0}{1}\) we have \((1 -t)x + t z \in \Omega \cap B_{t r}(x)\). We deduce from~\eqref{eq_JahvaePieG5cahth4AepheCi} and~\eqref{eq_wa1lohNgiechei5oos0ae0Oh} through the change of variable \(y = (1-t)x + tz\) that
\begin{equation}
\begin{split}
\int_{\Omega \cap B_{r} (x)} \abs{f (z) - f (x)} \dif z &\le \int_0^1 \int_{\Omega \cap B_{tr} (x)} \frac{\abs{\Deriv f (y)[y - x]}}{t^{d + 1}} \dif y \dif t\\
&\le r \maxfun{\abs{\Deriv f}}(x) \int_0^1 \frac{\mathcal{L}^d (B_{tr} (x))}{t^{d}} \dif t
\le r \mathcal{L}^d (B_r (0))
\maxfun \abs{\Deriv f} (x),
\end{split}
\end{equation}
in view of the definition~\eqref{eq_HakeeGhae0ku0yohy1jied9E} of the maximal function, and the conclusion~\eqref{eq_eutai3kaeviuGhaisiechoqu} then follows from the definition of the geometric quantity \(\varkappa (\Omega)\) in~\eqref{eq_waiC3theiwooDaithee1aeLa}.
\end{proof}

Our second tool to prove \cref{proposition_GN_Wsp_W1p_BMO} is the following property of averages of functions of bounded mean oscillation (see \cite[\S 3]{Carleson_1976}).

\begin{lemm}\label{lemma_BMO_log}
If the set \(\Omega \subseteq \Rset^d\) is open and convex, if \(f \in \BMO (\Omega) \) and if \(r_0 < r_1\), then
\begin{equation}\label{eq_eiboh7im4ZahghuLaiv9Eene}
\fint_{\Omega \cap B_{r_0}(x)} \fint_{\Omega \cap B_{r_1} (x)}
\abs{f (y) - f (z)}\dif y \dif z
\le e \brk[\big]{1 + d \ln \brk*{r_1/r_0}} \norm{f}_{\BMO(\Omega)}.
\end{equation}
\end{lemm}

In~\eqref{eq_eiboh7im4ZahghuLaiv9Eene}, \(e\) denotes Euler's number.

The proof of \cref{lemma_BMO_log} will use the following triangle inequality for averages
\begin{lemm}\label{lemma_triangle}
Let \(\Omega \subseteq \Rset^d\). If the function \(f : \Omega \to \Rset\) is measurable, and the sets \(A, B, C \subseteq \Rset^d\) are measurable and have positive measure, then
\begin{equation*}
\fint_{A} \fint_{B} \abs{f (y) - f (x)} \dif y \dif x
\le \fint_{A} \fint_{C} \abs{f (z) - f (x)} \dif z \dif x +
\fint_{C} \fint_{B} \abs{f (y) - f (z)} \dif y \dif z.
\end{equation*}
\end{lemm}
\begin{proof}
We have successively, in view of the triangle inequality,
\begin{equation}
\begin{split}
\fint_{A} \fint_{B} \abs{f (y) - f (x)} \dif y \dif x &= \fint_{A} \fint_{B} \fint_{C} \abs{f (y) - f (x)} \dif z \dif y \dif x\\
& \le \fint_{A} \fint_{B} \abs{f (z) - f (x)} + \abs{f (y) - f (z)} \dif z \dif y \dif x\\
&= \fint_{A} \fint_{C} \abs{f (z) - f (x)} \dif z \dif x +
\fint_{C} \fint_{B} \abs{f (y) - f (z)} \dif y \dif z. \qedhere
\end{split}
\end{equation}
\end{proof}


\begin{proof}[Proof of \cref{lemma_BMO_log}]
We first note that since \(r_1 > r_0\), we have in view of~\eqref{eq_HakeeGhae0ku0yohy1jied9E}
\begin{equation}\label{eq_voh1xobai3Eshai7aeh8Na1u}
\begin{split}
\fint_{\Omega \cap B_{r_0}(x)} &\fint_{\Omega \cap B_{r_1}(x)} \abs{f (y) - f (z)}\dif y \dif z\\
&\le \frac{\mathcal{L}^d (\Omega \cap B_{r_1} (x))}{\mathcal{L}^d (\Omega \cap B_{r_0} (x))} \fint_{\Omega \cap B_{r_1}(x)} \fint_{\Omega \cap B_{r_1} (x)} \abs{f (y) - f (z)}\dif y \dif z \le \brk*{\frac{r_1}{r_0}}^d \norm{f}_{\BMO(\Omega)},
\end{split}
\end{equation}
since by convexity \(r_0/r_1(\Omega \cap B_{r_1} (x)) \subseteq\Omega \cap B_{r_0} (x)\) and thus
\(\mathcal{L}^d (\Omega \cap B_{r_1} (x))/r_1^d \le \mathcal{L}^d (\Omega \cap B_{r_0} (x))/r_0^d\). Applying \(k \in \Nset\setminus \set{0}\) times the inequality~\eqref{eq_voh1xobai3Eshai7aeh8Na1u}, we get thanks to the triangle inequality for mean oscillation of \cref{lemma_triangle},
\begin{equation}
\begin{split}
\fint_{\Omega \cap B_{r_0}(x)}& \fint_{\Omega \cap B_{r_1} (x)} \abs{f (y) - f (z)}\dif y \dif z\\
&\le \sum_{j = 0}^{k - 1}\fint_{\Omega \cap B_{r_0(r_1/r_0)^{j/k}}(x)} \fint_{\Omega \cap B_{r_0(r_1/r_0)^{(j + 1)/k}} (x)} \abs{f (y) - f (z)}\dif y \dif z
\le k \brk*{\frac{r_1}{r_0}}^{d/k} \norm{f}_{\BMO(\Omega)}.
\end{split}
\end{equation}
Taking \(k \in \Nset\setminus \set{0}\) such that \(k - 1 < d \ln (r_1/r_0) \le k\), we obtain the conclusion~\eqref{eq_eiboh7im4ZahghuLaiv9Eene}.
\end{proof}

Our last tool to prove \cref{proposition_GN_Wsp_W1p_BMO} is the following integral identity.


\begin{lemm}\label{lemma_Gamma}
For every \(p \in \intvo{1}{\infty}\) and \(\alpha \in \intvo{0}{\infty}\), one has
\begin{equation*}
\int_1^\infty \frac{(\ln r)^{p}}{r^{1 + \alpha}} \dif r = \frac{\Gamma (p + 1)}{\alpha^{p + 1}}.
\end{equation*}
\end{lemm}
\begin{proof}
One performs the change of variable \(r = \exp (t/\alpha)\) in the left-hand side integral and uses the classical integral definition of the Gamma function.
\end{proof}

We now proceed to the proof of \cref{proposition_GN_Wsp_W1p_BMO}.

\begin{proof}
[Proof of \cref{proposition_GN_Wsp_W1p_BMO}]
\resetconstant For every \(x, y \in \Omega\), we have by the triangle inequality and the domain monotonicity of the integral
\begin{equation}\label{eq_ieph0eiwoose5IeJooxee1wj}
\begin{split}
\abs{f (y) - f (x)} &\le {\fint_{\Omega \cap B_{\abs{y - x}/2}(\frac{x + y}{2})}} \abs{f (y) - f} +{
\fint_{\Omega \cap B_{\abs{y - x}/2}(\frac{x + y}{2})}} \abs{f - f (x)}\\
& \le {2^d \fint_{\Omega \cap B_{\abs{y - x}}(y)}} \abs{f (y) - f } +{2^d
\fint_{\Omega \cap B_{\abs{y - x}}(x)}} \abs{f - f (x)},
\end{split}
\end{equation}
since by convexity \(\Omega \cap B_{\abs{y - x}/2} \bigl(\frac{x + y}{2}\bigr) \subseteq \frac{1}{2} (B_{\abs{y - x}} (x) \cap \Omega) + \frac{y}{2}\). It follows thus from~\eqref{eq_ieph0eiwoose5IeJooxee1wj} by integration and by symmetry that
\begin{equation}
\begin{split}
\iint_{\Omega \times \Omega}
\frac{ \abs{f (y) - f (x)}^p}{\abs{y - x}^{d + sp}} \dif y \dif x & \le \C \iint_{\Omega\times \Omega}
\brk*{\fint_{\Omega \cap B_{\abs{y - x}}(x)} \abs{f - f (x)}}^p \frac{\dif y \dif x}{\abs{y - x}^{d + sp}}\\
& \le \C \int_{\Omega} \int_0^{\diam \Omega}
\brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)} }^p \frac{\dif r}{r^{1 + sp}} \dif x.
\end{split}
\end{equation}
If \(\varrho \in \intvo{0}{\diam (\Omega)}\), we first have by \cref{lemma_osc_MDu}, for almost every \(x \in \Omega\),
\begin{equation}\label{eq_pheiph4dei2uc0iefo5KaR0j}
\begin{split}
\int_0^\varrho \brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)} }^p \frac{\dif r}{r^{1 + sp}} & \le \brk[\big]{\varkappa (\Omega) \, \maxfun \abs{\Deriv f} (x)}^p \int_0^{\varrho} r^{(1 - s)p - 1} \dif r \\
&= \frac{\varrho^{(1 - s)p} \brk[\big]{\varkappa (\Omega)\, \maxfun\abs{\Deriv f}(x)}^p}{(1 - s)p}.
\end{split}
\end{equation}
Next we have by the triangle inequality, by \cref{lemma_osc_MDu} again and by \cref{lemma_BMO_log}, for every \(r \in \intvo{\varrho}{\diam (\Omega)}\),
\begin{equation}\label{eq_AeJu8Uveed8teiqu1soR5que}
\begin{split}
\fint_{\Omega \cap B_r(x)} \abs{f - f (x)} & \le \fint_{\Omega \cap B_\varrho (x)} \abs{f - f (x)} + \fint_{\Omega \cap B_r (x)} \fint_{\Omega \cap B_\varrho (x)} \abs{f (y) - f (z)} \dif y \dif z\\
&\le \brk*{\varrho\, \varkappa (\Omega)\,\maxfun{\abs{\Deriv f}}(x) + e (1 + d \ln (r/\varrho)) \norm{f}_{\BMO(\Omega)} },
\end{split}
\end{equation}
and hence, integrating~\eqref{eq_AeJu8Uveed8teiqu1soR5que}, we get
\begin{equation}\label{eq_Tah3Loosiela3aQuoothio2p}
\begin{split}
\int_\varrho^{\diam (\Omega)} &\brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}}^p \frac{\dif r}{r^{1 + sp}}\\
& \le \C \brk*{\int_{\varrho}^{\infty} \frac{\varrho^p \maxfun{\abs{\Deriv f}}(x)^p}{r^{1 + sp}} \dif r + \int_{\varrho}^\infty \varkappa (\Omega)^p \norm{f}_{\BMO(\Omega)}^p \frac{(1 + d \ln(r/\varrho))^p}{r^{1 + sp}} \dif r}\\
&\le \C \brk*{\frac{\varrho^{(1 - s)p} \varkappa(\Omega)^p \maxfun\abs{\Deriv f}(x)^p}{s} + \frac{\Gamma (p + 1) \norm{f}_{\BMO(\Omega)}^p}{(sp)^{p + 1}\varrho^{sp}}},
\end{split}
\end{equation}
in view of \cref{lemma_Gamma}. Putting~\eqref{eq_pheiph4dei2uc0iefo5KaR0j} and~\eqref{eq_Tah3Loosiela3aQuoothio2p} together, we get, since \(sp > 1\),
\begin{equation}\label{eq_quohjeicaedaiY4aewoh5Ooc}
\int_0^{\diam (\Omega)} \brk*{\fint_{\Omega \cap B_r(x)} \abs[\big]{f - f (x)} }^p \frac{\dif r}{r^{1 + sp}}
\le \C\brk*{\frac{\varrho^{(1 - s) p} \varkappa (\Omega)^p \maxfun\abs{\Deriv f}(x)^p}{1 - s} + \frac{\norm{f}_{\BMO(\Omega)}^p}{\varrho^{sp}}}.
\end{equation}
If \(\norm{f}_{\BMO(\Omega)} \le \diam (\Omega) \varkappa (\Omega) \maxfun \abs{\Deriv f}(x)\), taking \(\varrho
\defeq \norm{f}_{\BMO(\Omega)}/(\varkappa (\Omega) \maxfun \abs{\Deriv f}(x))\) in~\eqref{eq_quohjeicaedaiY4aewoh5Ooc}, we obtain
\begin{equation}\label{eq_oophae8ocee6Ohphoh4foox8}
\int_0^{\diam (\Omega)} \brk*{\fint_{\Omega \cap B_r(x)} \abs[\big]{f - f (x)}}^p \frac{\dif r}{r^{1 + sp}} \le \frac{\C}{1 - s} \brk*{\varkappa (\Omega)\maxfun\abs{\Deriv f}(x)}^{sp} \,\norm{f}_{\BMO(\Omega)}^{(1-s)p};
\end{equation}
otherwise we take \(\varrho
\defeq \diam (\Omega) \le \norm{f}_{\BMO(\Omega)}/(\varkappa (\Omega) \maxfun \abs{\Deriv f}(x))\) in~\eqref{eq_pheiph4dei2uc0iefo5KaR0j} and also obtain~\eqref{eq_oophae8ocee6Ohphoh4foox8}. Integrating the inequality~\eqref{eq_oophae8ocee6Ohphoh4foox8}, we reach the conclusion~\eqref{eq_aijei2Xaighei4obeichiiKi} by the quantitative version of the classical maximal function theorem in \(L^{sp}(\Rset^d)\) since \(sp > 1\) (see for example \cite[\linebreak Theorem~I.1]{Stein_1970}).
\end{proof}

We conclude this section by pointing out that \cref{proposition_GN_Wsp_W1p_BMO} admits a \emph{localised version} in terms of Fefferman and Stein's \emph{sharp maximal function} \(f^\sharp:\Omega \to \intvc{0}{\infty}\) which is defined for every \(x \in \Omega\) (see \cite[(4.1)]{Fefferman_Stein_1972}) as
\begin{equation}
f^\sharp (x)
\defeq \sup_{r > 0} \fint_{\Omega \cap B_r (x)} \fint_{\Omega \cap B_r (x)} \abs{f (y) - f (z)}\dif y \dif z;
\end{equation}
noting that the proof of \cref{lemma_BMO_log} yields in fact the estimate
\begin{equation}
\fint_{\Omega \cap B_{r_0}(x)} \fint_{\Omega \cap B_{r_1} (x)}
\abs{f (y) - f (z)}\dif y \dif z
\le e \brk[\big]{1 + d \ln \brk*{r_1/r_0}} f^\sharp (x)
\end{equation}
and following then the proof of \cref{proposition_GN_Wsp_W1p_BMO}, we reach the following local counterpart of~\eqref{eq_oophae8ocee6Ohphoh4foox8}.

\begin{prop}\label{proposition_local_GN_Wsp_W1p_BMO}
For every \(d \in \Nset \setminus \set{0}\) and for every \(p\in \intvo{1}{\infty}\), there exists a constant \(C > 0\) such that for every \(s \in \intvo{1/p}{1}\), for every open convex set \(\Omega \subseteq \Rset^d\) satisfying \(\varkappa (\Omega) < \infty\), for every function \(f \in \dot{W}^{1, 1}_{\loc} (\Omega)\) and for almost every \(x \in \Omega\), we have
\begin{equation}\label{eq_thaelohloh9UcahsieBei7Ah}
\int_0^{\diam (\Omega)} \brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}}^p \frac{\dif r}{r^{1 + sp}}
\le \frac{C}{1 - s} \brk[\big]{f^\sharp(x)}^{(1-s)p} \brk[\big]{ \varkappa (\Omega)
\maxfun{\abs{\Deriv f}} (x)}^{sp}.
\end{equation}
\end{prop}

\Cref{proposition_local_GN_Wsp_W1p_BMO} is stronger than \cref{proposition_GN_Wsp_W1p_BMO} in the sense that the integration of the estimate~\eqref{eq_thaelohloh9UcahsieBei7Ah} yields~\eqref{eq_aijei2Xaighei4obeichiiKi}.


\Cref{proposition_local_GN_Wsp_W1p_BMO} is a counterpart of the interpolation involving maximal and sharp maximal function of derivatives \cite[(4)]{Lokharu_2011}, which generalised a priori estimates in terms of maximal functions \cite[Theorem~1]{Mazya_Shaposhnikova_1999}, \cite{Kalamajska_1994}; \cref{proposition_local_GN_Wsp_W1p_BMO} generalises the corresponding result for integer-order Sobolev spaces \cite[Remark~2.2]{Miyazaki_2020}.

\section{Interpolation between first-order Sobolev semi-norm and mean oscillation}

We explain how the tools of the previous section can be used to prove the fractional BMO Gagliardo--Nirenberg interpolation inequality as persented by Brezis and Mironescu's \cite[Lem\-ma~15.7]{Brezis_Mironescu_2021}.

\begin{theo}\label{proposition_GN_Wsp_Wsp_BMO}
For every \(d \in \Nset \setminus \set{0}\), every \(s, s_1 \in \intvo{0}{1}\) and every \(p, p_1 \in (1, +\infty)\) satisfying \(s< s_1\) and \(s_1 p_1 = sp\), there exists a constant \(C > 0\) such that for every open convex set \(\Omega \subseteq \Rset^d\) satisfying \(\varkappa (\Omega)<\infty\) and for every function \(f \in \dot{W}^{s_1, p_1} (\Omega) \cap\BMO(\Omega)\), one has
\(f \in \dot{W}^{s, p} (\Omega)\) and
\begin{equation}\label{eq_aeshohRe7chuFohyiizieyo4}
\iint_{\Omega \times \Omega}
\frac{\abs{f (y) - f (x)}^p}{\abs{y - x}^{d + sp}}
\dif y
\dif x
\le C
\norm{f}_{\BMO(\Omega)}^{p - p_1}\varkappa (\Omega)^{p_1}
\iint_{\Omega \times \Omega}
\frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1 p_1}}
\dif y
\dif x.
\end{equation}
\end{theo}

The proof of \cref{proposition_GN_Wsp_Wsp_BMO} will follow essentially the proof of \cref{proposition_GN_Wsp_W1p_BMO}, the main difference being the replacement of \cref{lemma_osc_MDu} by its easier fractional counterpart.

\begin{lemm}\label{lemma_osc_Wsp}
For every \(p \in \intvo{1}{\infty}\), there exists a constant \(C > 0\) such that if the set \(\Omega \subseteq \Rset^d\) is open and convex, if \(s \in \intvo{0}{1}\) and if \(f : \Omega \to \Rset\) is measurable, then for every \(r \in \intvo{0}{\diam(\Omega)}\) and every \(x \in \Omega\),
\begin{equation}\label{eq_jaC2Aedee8Pheefashiphuiw}
\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}
\le C \varkappa (\Omega) \, r^s \brk*{\int_{\Omega} \frac{\abs{f (y) - f (x)}^p}{\abs{y - x}^{d + sp}} \dif y}^\frac{1}{p}.
\end{equation}
\end{lemm}

\begin{proof}
By Hölder's inequality we have for every \(r \in \intvo{0}{\diam(\Omega)}\) and for every \(x \in \Omega\),
\begin{equation}
\int_{\Omega \cap B_r(x)} \abs{f - f (x)}
\le \brk*{\int_{\Omega} \frac{\abs{f (y) - f (x)}^p}{\abs{y - x}^{d + sp}} \dif y}^\frac{1}{p} \brk*{\int_{B_r (x)} \abs{y - x}^{\frac{d + sp}{p - 1}} \dif y}^{1 - \frac{1}{p}}.
\end{equation}
Noting that
\begin{equation}
\int_{B_r (x)} \abs{y - x}^\frac{d + sp}{p - 1} \dif y = \C \frac{p - 1}{d + s p} \brk[\big]{r^s \mathcal{L}^d (B_r (x)) }^\frac{p}{p - 1}
\le \C \brk[\big]{r^s \mathcal{L}^d (B_r (x)) }^\frac{p}{p - 1},
\end{equation}
we reach the conclusion~\eqref{eq_jaC2Aedee8Pheefashiphuiw} thanks to the definition of the geometric quantity \(\varkappa (\Omega)\) in~\eqref{eq_waiC3theiwooDaithee1aeLa}.
\end{proof}

\begin{proof}[Proof of \cref{proposition_GN_Wsp_Wsp_BMO}]
We begin as in the proof of \cref{proposition_GN_Wsp_W1p_BMO}. Instead of~\eqref{eq_pheiph4dei2uc0iefo5KaR0j}, we have by \cref{lemma_osc_Wsp},
\begin{equation}\label{eq_ohThaequi9eiThoh0aidoh4a}
\begin{split}
\int_0^\varrho \brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}}^p \frac{\dif r}{r^{1 + sp}} &\le \C \varkappa(\Omega)^p
\brk*{\int_{\Omega} \frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1p_1}} \dif y}^\frac{p}{p_1}
\int_0^\varrho r^{(s_1 -s) p - 1} \dif r\\
& \le \frac{\C \varkappa(\Omega)^p \varrho^{(s_1 - s)p}}{(s_1 -s) p} \brk*{\int_{\Omega} \frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1p_1}} \dif y}^\frac{p}{p_1}.
\end{split}
\end{equation}
Next instead of~\eqref{eq_Tah3Loosiela3aQuoothio2p}, we have
\begin{equation}\label{eq_ahYeipe1quaiZ3lei8voh4as}
\begin{split}
\int_\varrho^{\diam (\Omega)} &\brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}}^p \frac{\dif r}{r^{1 + sp}}\\
&\le \C \brk*{\frac{\varkappa(\Omega)^p \varrho^{(s_1 - s)p} }{sp} \brk*{\int_{\Omega} \frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1p_1}} \dif y}^\frac{p}{p_1} + \frac{\norm{f}_{\BMO(\Omega)}^p}{(sp)^{p + 1} \varrho^{sp}}}.
\end{split}
\end{equation}
Taking \(\varrho \in \intvo{0}{\diam (\Omega)}\) such that
\begin{equation}\label{eq_Iel5aigheeng3geeghiqu8ge}
\norm{f}_{\BMO(\Omega)}^p = \varrho^{s_1 p}\varkappa(\Omega)^p \brk*{\int_{\Omega} \frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1p_1}} \dif y}^\frac{p}{p_1}
\end{equation}
if possible, and otherwise taking \(\varrho
\defeq \diam (\Omega)\), we obtain, since \(s_1 p_1 = sp\), by~\eqref{eq_ohThaequi9eiThoh0aidoh4a}, \eqref{eq_ahYeipe1quaiZ3lei8voh4as} and~\eqref{eq_Iel5aigheeng3geeghiqu8ge}
\begin{equation}\label{eq_ohng5eeRas3Ohcithoociit4}
\begin{split}
\int_0^{\diam \Omega} \brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}}^p &\frac{\dif r}{r^{1 + sp}}
\le \C \norm{f}_{\BMO(\Omega)}^{p - p_1} \varkappa(\Omega)^{p_1}
\int_{\Omega} \frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1p_1}} \dif y.
\end{split}
\end{equation}
We conclude by integration of~\eqref{eq_ohng5eeRas3Ohcithoociit4}.
\end{proof}

As previously, we point out that the estimate~\eqref{eq_ohng5eeRas3Ohcithoociit4} admits a localised version, which is the fractional counterpart of \cref{proposition_local_GN_Wsp_W1p_BMO}.

\begin{prop}
For every \(d \in \Nset \setminus \set{0}\), every \(s, s_1 \in \intvo{0}{1}\) and every \(p, p_1 \in (1, +\infty)\) satisfying \(s< s_1\) and \(s_1 p_1 = sp\), there exists a constant \(C > 0\) such that for every open convex set \(\Omega \subseteq \Rset^d\) satisfying \(\varkappa (\Omega)<\infty\), for every measurable function \(f : \Omega \to \Rset\) and for every \(x \in \Omega\),
\begin{equation}\label{eq_ael6ohf6rooK4shaiMe0thai}
\int_0^{\diam (\Omega)} \brk*{\fint_{\Omega \cap B_r(x)} \abs{f - f (x)}}^p \frac{\dif r}{r^{1 + sp}}\le C \brk[\big]{f^\sharp(x)}^{p - p_1}\varkappa (\Omega)^{p_1}
\int_{\Omega} \frac{\abs{f (y) - f (x)}^{p_1}}{\abs{y - x}^{d + s_1p_1}} \dif y.
\end{equation}
\end{prop}

The estimate~\eqref{eq_aeshohRe7chuFohyiizieyo4} can be seen as a consequence of the integration of~\eqref{eq_ael6ohf6rooK4shaiMe0thai}.

\section{Higher-order fractional spaces estimates}

The last ingredient to obtain the full scale of Gagliardo--Nirenberg interpolation inequalities between fractional Sobolev--Slobodeckiĭ spaces and the bounded mean oscillation space is the following estimate.

\begin{theo}\label{proposition_higher_order}
For every \(d \in \Nset \setminus \set{0}\), every \(k_1 \in \Nset \setminus\set{0}\), every \(\sigma_1 \in \intvo{0}{1}\) and every \(p, p_1 \in \intvo{1}{\infty}\) satisfying
\begin{equation}\label{eq_Yoxaikeengah4deil3Aingae}
k_1p = (k_1 + \sigma_1)p_1,
\end{equation}
there exists a constant \(C > 0\) such that for every function \(f \in \dot{W}^{k_1 + \sigma_1, p_1} (\Rset^d)\cap \BMO (\Rset^d)\), one has \(f \in \dot{W}^{k_1, p} (\Rset^d)\) and
\begin{equation}\label{eq_ohjooYai3xeijo1aiphiivie}
\int_{\Rset^d} \abs{D^{k_1} f}^p
\le C \norm{f}_{\BMO (\Rset^d)}^{p - p_1}
\smashoperator{\iint_{\Rset^d \times \Rset^d}} \frac{\abs{D^{k_1} f(x) - D^{k_1} f(y)}^{p_1}}{\abs{x - y}^{d + \sigma_1p_1}} \dif x \dif y.
\end{equation}
\end{theo}

As a consequence of \cref{proposition_higher_order}, we have that \(f \in \dot{W}^{k + \sigma, p}(\Rset^d)\) whenever \(k \in \Nset\), \(\sigma \in \intvr{0}{1}\) and \(p \in \intvo{1}{\infty}\) satisfy \(k + \sigma < k_1 + \sigma_1\) and
\((k + \sigma)p = (k_1 + \sigma_1)p_1\). Indeed for \(\sigma = 0\) and \(k = k_1\), this follows from \cref{proposition_higher_order} and then for \(k \in \set{1, \dotsc, k_1 - 1}\) by the Gagliardo--Nirenberg interpolation inequality for integer-order Sobolev space \cite{Strzelecki_2006, Miyazaki_2020}; for \(0 < \sigma < 1\) and \(k = 0\) one then uses \cref{proposition_GN_Wsp_W1p_BMO} whereas for \(0 < \sigma < 1\) and \(k \in \Nset \setminus \set{0}\) one uses the classical fractional Gagliardo--Nirenberg interpolation inequality~\cite{Brezis_Mironescu_2018}.


\begin{proof}[Proof of \cref{proposition_higher_order}]
Fixing a function \(\eta \in C^\infty_c (\Rset^d)\) such that \(\int_{\Rset^d} \eta = 1\) and \(\supp \eta \subseteq B_1\), we have for every \(x \in \Rset^d\) and every \(\varrho \in \intvo{0}{\infty}\),
\begin{equation}\label{eq_iemai1quuyeiSh2eebo9aiw8}
D^{k_1} f (x) = \frac{1}{\varrho^d} \int_{\Rset^d} \eta\brk*{\tfrac{x - y}{\varrho}} \brk[\big]{D^{k_1} f (x) - D^{k_1} f (y)} \dif y + \frac{1}{\varrho^d} \int_{\Rset^d} \eta\brk*{\tfrac{x - y}{\varrho}} D^{k_1} f (y) \dif y.
\end{equation}
We estimate the first term in the right-hand side of~\eqref{eq_iemai1quuyeiSh2eebo9aiw8} by Hölder's inequality
\begin{equation}\label{eq_iumooxoh6ohhePijahw6leev}
\begin{split}
\abs[\bigg]{\frac{1}{\varrho^d} \int_{\Rset^d}& \eta\brk*{\tfrac{x - y}{\varrho}} \brk[\big]{D^{k_1} f (x) - D^{k_1} f (y)} \dif y}\\
&\le \frac{\C}{\varrho^d} \brk*{\int_{\Rset^d} \frac{\abs{D^{k_1} f (x) - D^{k_1} f (y)}^{p_1}}{\abs{x - y}^{d+\sigma_1 p_1}}\dif x}^\frac{1}{p_1}
\brk*{\int_{B_\varrho(x)} \abs{x - y}^{\frac{d+\sigma_1 p_1}{p_1 - 1}}\dif x}^{1 - \frac{1}{p_1}}\\
&\le \C \varrho^{\sigma_1} \brk*{\int_{\Rset^d} \frac{\abs{D^{k_1} f (x) - D^{k_1} f (y)}^{p_1}}{\abs{x - y}^{d+\sigma_1 p_1}}\dif x}^\frac{1}{p_1}.
\end{split}
\end{equation}
For the second-term in the right-hand side of~\eqref{eq_iemai1quuyeiSh2eebo9aiw8}, for every \(x \in \Rset^d\), we have by weak differentiability,
\begin{equation}\label{eq_Shai9yauPh6Ohph3phei2Uic}
\begin{split}
\frac{1}{\varrho^d} \int_{\Rset^d} \eta\brk*{\tfrac{x - y}{\varrho}} D^{k_1} f (y) \dif y &= \frac{1}{\varrho^{d + k_1}} \int_{\Rset^d} D^{k_1} \eta\brk*{\tfrac{x - y}{\varrho}} f (y) \dif y\\
&= \frac{1}{\varrho^{2 d + k_1}} \int_{\Rset^d} \int_{\Rset^d} D^{k_1} \eta\brk*{\tfrac{x - y}{\varrho}} \eta\brk*{\tfrac{x - z}{\varrho}} \brk[\big]{f (y) - f (z)} \dif y \dif z,
\end{split}
\end{equation}
and thus by~\eqref{eq_Shai9yauPh6Ohph3phei2Uic} and by definition of bounded mean oscillation~\eqref{eq_oothaKahp8pe7OGheix4phoo}, we have
\begin{equation}\label{eq_egai1Isae4vie5opeida5le8}
\abs*{\frac{1}{\varrho^d} \int_{\Rset^d} \eta\brk*{\tfrac{x - y}{\varrho}} D^{k_1} f (y) \dif y}
\le \frac{\C}{\varrho^{k_1}} \norm{f}_{\BMO(\Rset^d)}.
\end{equation}
Choosing \(\varrho \in \intvo{0}{\infty}\) such that
\begin{equation}
\varrho^{k_1 + \sigma_1} \brk*{\int_{\Rset^d} \frac{\abs{D^{k_1} f (x) - D^{k_1} f (y)}^{p_1}}{\abs{x - y}^{d+\sigma_1 p_1}}\dif y}^\frac{1}{p_1} = \norm{f}_{\BMO(\Rset^d)},
\end{equation}
we get from~\eqref{eq_iemai1quuyeiSh2eebo9aiw8}, \eqref{eq_iumooxoh6ohhePijahw6leev} and~\eqref{eq_egai1Isae4vie5opeida5le8}, for every \(x \in \Rset^d\),
\begin{equation}
\abs{D^{k_1} f (x)}
\le \C \norm{f}_{\BMO(\Rset^d)}^{1 - \frac{k_1}{k_1 + \sigma_1}} \brk*{\int_{\Rset^d} \frac{\abs{D^{k_1} f (x) - D^{k_1} f (y)}^{p_1}}{\abs{x - y}^{d+\sigma_1 p_1}}\dif y}^\frac{k_1}{(k_1 + \sigma_1) p_1},
\end{equation}
and thus in view of the condition~\eqref{eq_Yoxaikeengah4deil3Aingae}, the estimate~\eqref{eq_ohjooYai3xeijo1aiphiivie} follows by integration.
\end{proof}

\Cref{proposition_higher_order} also admits a localised version involving the sharp maximal function which follows from the replacement of \(\norm{f}_{\BMO(\Rset^d)}\) by \(f^\sharp (x)\) in~\eqref{eq_egai1Isae4vie5opeida5le8}.

\begin{prop}\label{proposition_higher_order_local}
For every \(d \in \Nset \setminus \set{0}\), every \(k_1 \in \Nset \setminus\set{0}\), every \(\sigma_1 \in \intvo{0}{1}\) and every \(p_1 \in \intvo{1}{\infty}\), there exists a constant \(C>0\) such that for every function \(f \in \dot{W}^{k_1, 1}_{\loc} (\Rset^d)\) and every \(x \in \Rset^d\),
\begin{equation}\label{eq_jaeliu1aeph2tooTh8Wul5ox}
\abs{D^{k_1} f (x)}
\le C \brk[\big]{f^\sharp (x)}^{1 - \frac{k_1}{k_1 + \sigma_1}} \brk*{\int_{\Rset^d} \frac{\abs{D^{k_1} f (x) - D^{k_1} f (y)}^{p_1}}{\abs{x - y}^{d+\sigma_1 p_1}}\dif y}^\frac{k_1}{(k_1 + \sigma_1) p_1}.
\end{equation}
\end{prop}

As previously, the integration of~\eqref{eq_jaeliu1aeph2tooTh8Wul5ox} yields~\eqref{eq_ohjooYai3xeijo1aiphiivie}.

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