[proofplan] The lower bound $\|f\|_{\mathrm{BMO}} \le \|f\|_{\mathrm{BMO}_p}$ is a single application of [Hölder's inequality](/theorems/???) on each cube. The upper bound $\|f\|_{\mathrm{BMO}_p} \le C_{p,n}\|f\|_{\mathrm{BMO}}$ requires the [John–Nirenberg Inequality](/theorems/3180): the exponential bound on level sets of $|f - f_Q|$ converts into a $p$-th moment bound by the layer-cake formula and a single change of variables in the resulting integral. The seminorm property and the value of the constant $C_{p,n}$ both fall out of these two estimates. [/proofplan]

[step:Verify that $\|\cdot\|_{\mathrm{BMO}_p}$ is a seminorm] Fix $p \in [1, \infty)$. We check the three seminorm axioms — non-negativity, absolute homogeneity, and the triangle inequality — for the map $\|\cdot\|_{\mathrm{BMO}_p}: \mathrm{BMO}(\mathbb{R}^n) \to [0, \infty]$ on the space $\mathrm{BMO}(\mathbb{R}^n) \subseteq L^1_{\mathrm{loc}}(\mathbb{R}^n)$. Finiteness on $\mathrm{BMO}(\mathbb{R}^n)$ follows from the upper bound $\|f\|_{\mathrm{BMO}_p} \le C_{p,n}\|f\|_{\mathrm{BMO}}$ proved in step 3. (a) **Non-negativity.** Each cube average $\frac{1}{|Q|}\int_Q |f - f_Q|^p\, d\mathcal{L}^n \ge 0$, so its supremum is in $[0, \infty]$. (b) **Absolute homogeneity.** For $c \in \mathbb{R}$ and $f \in \mathrm{BMO}(\mathbb{R}^n)$, $(cf)_Q = c\cdot f_Q$ by linearity of the integral. Hence $|cf(x) - (cf)_Q|^p = |c|^p|f(x) - f_Q|^p$, so \begin{align*} \frac{1}{|Q|}\int_Q |cf - (cf)_Q|^p\, d\mathcal{L}^n = |c|^p \cdot \frac{1}{|Q|}\int_Q |f - f_Q|^p\, d\mathcal{L}^n. \end{align*} Taking the $p$-th root and the supremum over $Q$ gives $\|cf\|_{\mathrm{BMO}_p} = |c|\,\|f\|_{\mathrm{BMO}_p}$. (c) **Triangle inequality.** For $f, g \in \mathrm{BMO}(\mathbb{R}^n)$ and any cube $Q$, the map $h \mapsto h - h_Q$ is linear, so $(f + g)(x) - (f+g)_Q = (f(x) - f_Q) + (g(x) - g_Q)$. By Minkowski's inequality on the probability space $(Q, \mathcal{B}(Q), |Q|^{-1}\mathcal{L}^n|_Q)$, \begin{align*} \left(\frac{1}{|Q|}\int_Q |(f+g) - (f+g)_Q|^p\, d\mathcal{L}^n\right)^{1/p} &\le \left(\frac{1}{|Q|}\int_Q |f - f_Q|^p\, d\mathcal{L}^n\right)^{1/p} + \left(\frac{1}{|Q|}\int_Q |g - g_Q|^p\, d\mathcal{L}^n\right)^{1/p} \\ &\le \|f\|_{\mathrm{BMO}_p} + \|g\|_{\mathrm{BMO}_p}. \end{align*} Taking the supremum over $Q$ gives $\|f + g\|_{\mathrm{BMO}_p} \le \|f\|_{\mathrm{BMO}_p} + \|g\|_{\mathrm{BMO}_p}$. This completes the proof that $\|\cdot\|_{\mathrm{BMO}_p}$ is a seminorm. [/step]

[step:Lower bound $\|f\|_{\mathrm{BMO}} \le \|f\|_{\mathrm{BMO}_p}$ via Hölder's inequality] Fix $p \in [1, \infty)$ and $f \in \mathrm{BMO}(\mathbb{R}^n)$. For any cube $Q$, write \begin{align*} \frac{1}{|Q|}\int_Q |f(x) - f_Q|\, d\mathcal{L}^n(x) = \frac{1}{|Q|}\int_Q 1\cdot |f(x) - f_Q|\, d\mathcal{L}^n(x). \end{align*} We apply [Hölder's inequality](/theorems/???) on the finite measure space $(Q, \mathcal{B}(Q), \mathcal{L}^n|_Q)$ with conjugate exponents $p$ and $q = p/(p-1)$ if $p > 1$ (with the convention $q = \infty$ if $p = 1$). For $p > 1$, \begin{align*} \frac{1}{|Q|}\int_Q |f - f_Q|\, d\mathcal{L}^n &\le \frac{1}{|Q|}\left(\int_Q |f - f_Q|^p\, d\mathcal{L}^n\right)^{1/p}\left(\int_Q 1^q\, d\mathcal{L}^n\right)^{1/q} \\ &= \frac{1}{|Q|}\cdot |Q|^{1/q}\left(\int_Q |f - f_Q|^p\, d\mathcal{L}^n\right)^{1/p} \\ &= |Q|^{-1/p}\left(\int_Q |f - f_Q|^p\, d\mathcal{L}^n\right)^{1/p} \\ &= \left(\frac{1}{|Q|}\int_Q |f - f_Q|^p\, d\mathcal{L}^n\right)^{1/p}, \end{align*} using $1 - 1/q = 1/p$ and combining $|Q|^{-1}|Q|^{1/q} = |Q|^{-1/p}$. For $p = 1$, both sides equal $\frac{1}{|Q|}\int_Q |f - f_Q|\, d\mathcal{L}^n$ and the inequality is an equality. Taking the supremum over $Q$, \begin{align*} \|f\|_{\mathrm{BMO}} = \sup_Q \frac{1}{|Q|}\int_Q |f - f_Q|\, d\mathcal{L}^n \le \sup_Q \left(\frac{1}{|Q|}\int_Q |f - f_Q|^p\, d\mathcal{L}^n\right)^{1/p} = \|f\|_{\mathrm{BMO}_p}. \end{align*} [/step]

[step:Upper bound $\|f\|_{\mathrm{BMO}_p} \le C_{p,n}\|f\|_{\mathrm{BMO}}$ via the John–Nirenberg layer-cake] We assume $f \in \mathrm{BMO}(\mathbb{R}^n)$ with $\|f\|_{\mathrm{BMO}} > 0$ (the case $\|f\|_{\mathrm{BMO}} = 0$ has $f$ a.e. constant, both sides vanish, and the inequality is automatic). Fix a cube $Q \subset \mathbb{R}^n$. The [layer-cake representation](/theorems/???) of an $L^p$ integral applied to the non-negative measurable function $|f - f_Q|^p$ on the finite measure space $(Q, \mathcal{B}(Q), \mathcal{L}^n|_Q)$ states \begin{align*} \int_Q |f(x) - f_Q|^p\, d\mathcal{L}^n(x) = \int_0^\infty p\lambda^{p-1}\,\mathcal{L}^n(\{x \in Q : |f(x) - f_Q| > \lambda\})\, d\mathcal{L}^1(\lambda). \end{align*} Apply the [John–Nirenberg Inequality](/theorems/3180) to the inner level set: there exist $c_1, c_2 > 0$ depending only on $n$ such that \begin{align*} \mathcal{L}^n(\{x \in Q : |f(x) - f_Q| > \lambda\}) \le c_1|Q|\exp\!\left(-\frac{c_2\lambda}{\|f\|_{\mathrm{BMO}}}\right). \end{align*} Substituting and pulling out $|Q|$, \begin{align*} \int_Q |f - f_Q|^p\, d\mathcal{L}^n \le c_1 p|Q| \int_0^\infty \lambda^{p-1}\exp\!\left(-\frac{c_2\lambda}{\|f\|_{\mathrm{BMO}}}\right)\, d\mathcal{L}^1(\lambda). \end{align*} Substitute $t := c_2\lambda/\|f\|_{\mathrm{BMO}}$, so that $\lambda = (\|f\|_{\mathrm{BMO}}/c_2)\,t$ and $d\mathcal{L}^1(\lambda) = (\|f\|_{\mathrm{BMO}}/c_2)\,d\mathcal{L}^1(t)$, and the integration interval $(0, \infty)$ is preserved (substitution by an increasing diffeomorphism of $(0, \infty)$): \begin{align*} \int_0^\infty \lambda^{p-1}\exp(-c_2\lambda/\|f\|_{\mathrm{BMO}})\, d\mathcal{L}^1(\lambda) &= \left(\frac{\|f\|_{\mathrm{BMO}}}{c_2}\right)^p \int_0^\infty t^{p-1}e^{-t}\, d\mathcal{L}^1(t) \\ &= \left(\frac{\|f\|_{\mathrm{BMO}}}{c_2}\right)^p \Gamma(p), \end{align*} where $\Gamma(p) := \int_0^\infty t^{p-1}e^{-t}\, d\mathcal{L}^1(t)$ is the [Gamma function](/theorems/???) value at $p$, finite because $p > 0$. Therefore \begin{align*} \frac{1}{|Q|}\int_Q |f - f_Q|^p\, d\mathcal{L}^n \le c_1 p\,c_2^{-p}\,\Gamma(p)\,\|f\|_{\mathrm{BMO}}^p. \end{align*} Taking the $p$-th root and the supremum over $Q$ gives \begin{align*} \|f\|_{\mathrm{BMO}_p} \le \left(c_1 p\,\Gamma(p)\right)^{1/p}c_2^{-1}\|f\|_{\mathrm{BMO}} =: C_{p,n}\|f\|_{\mathrm{BMO}}, \end{align*} where $C_{p, n} := \left(c_1 p\,\Gamma(p)\right)^{1/p}c_2^{-1}$ depends only on $n$ (through $c_1, c_2$) and $p$. [/step]

[step:Combine the bounds to conclude equivalence] Steps 2 and 3 together state that for every $1 \le p < \infty$ and every $f \in \mathrm{BMO}(\mathbb{R}^n)$, \begin{align*} \|f\|_{\mathrm{BMO}} \le \|f\|_{\mathrm{BMO}_p} \le C_{p, n}\|f\|_{\mathrm{BMO}}, \end{align*} with $C_{p, n} := \left(c_1 p\,\Gamma(p)\right)^{1/p}c_2^{-1}$ depending only on $n$ and $p$. The two-sided estimate shows that the seminorm $\|\cdot\|_{\mathrm{BMO}_p}$ is equivalent to $\|\cdot\|_{\mathrm{BMO}_1} = \|\cdot\|_{\mathrm{BMO}}$ on $\mathrm{BMO}(\mathbb{R}^n)$, finite for each $f \in \mathrm{BMO}(\mathbb{R}^n)$, and zero exactly when $f$ is a.e. constant — confirming that $\mathrm{BMO}(\mathbb{R}^n)$ does not depend on the exponent $p$ used to measure mean oscillation. [/step]