[proofplan] The Bessel potential norm $\|f\|_{H^{s,p}} = \|(I-\Delta)^{s/2}f\|_{L^p}$ and the Triebel–Lizorkin norm $\|f\|_{F^s_{p,2}}$ both measure the size of $f$ "with $s$ derivatives in $L^p$", but through different decompositions of frequency space: the Bessel multiplier $\langle\xi\rangle^s = (1+|\xi|^2)^{s/2}$ versus a dyadic Littlewood–Paley square function. The proof reduces equivalence to a Fourier-multiplier statement: the operators carrying one decomposition to the other are bounded on $L^p$. We use two ingredients — the [Mihlin Multiplier Theorem](/theorems/3189) (specifically its vector-valued $\ell^2$-valued extension, see Stein, *Singular Integrals*, Chapter II, §4.6) to control the smooth multipliers $\langle\xi\rangle^s\,\varphi_j(\xi)/2^{js}$ uniformly in $j$, and the [Littlewood–Paley Square Function Characterisation of $L^p$](/theorems/???), which says $\|g\|_{L^p} \asymp \|(\sum_j |\Delta_j g|^2)^{1/2}\|_{L^p}$ for $1 < p < \infty$. Combining these, both norms equal a vector-valued $L^p$ norm of $(2^{js}\Delta_j f)_{j \ge 0}$ up to $L^p$-bounded multipliers, hence coincide. [/proofplan] [step:Fix a dyadic resolution of unity and recall the two norms] Fix $\varphi_0, \varphi \in C^\infty_c(\mathbb{R}^n)$ with $\varphi_0$ supported in $\{|\xi| \le 2\}$, $\varphi$ supported in the annulus $\{1/2 \le |\xi| \le 2\}$, and \begin{align*} \varphi_0(\xi) + \sum_{j=1}^\infty \varphi(2^{-j}\xi) = 1 \qquad \text{for all } \xi \in \mathbb{R}^n. \end{align*} Set $\varphi_j(\xi) := \varphi(2^{-j}\xi)$ for $j \ge 1$, so that $\varphi_j$ is supported in $\{2^{j-1} \le |\xi| \le 2^{j+1}\}$. Define the Littlewood–Paley projectors \begin{align*} \Delta_j : \mathcal{S}'(\mathbb{R}^n) &\to \mathcal{S}'(\mathbb{R}^n) \\ f &\mapsto \mathcal{F}^{-1}\bigl(\varphi_j\,\hat f\,\bigr), \qquad j \ge 0. \end{align*} Recall the two norms under comparison. The Triebel–Lizorkin norm is the vector-valued $L^p$ norm of the dyadic pieces: \begin{align*} \|f\|_{F^s_{p,2}(\mathbb{R}^n)} := \Bigl\| \Bigl( \sum_{j=0}^\infty 2^{2js}\,|\Delta_j f|^2 \Bigr)^{1/2} \Bigr\|_{L^p(\mathbb{R}^n)}. \end{align*} The Bessel potential norm is \begin{align*} \|f\|_{H^{s,p}(\mathbb{R}^n)} := \|J^s f\|_{L^p(\mathbb{R}^n)}, \qquad J^s f := \mathcal{F}^{-1}\bigl(\langle\xi\rangle^s\,\hat f\,\bigr), \quad \langle\xi\rangle := (1 + |\xi|^2)^{1/2}. \end{align*} Both expressions are *a priori* finite or infinite for a tempered distribution $f \in \mathcal{S}'(\mathbb{R}^n)$. [/step] [step:Reduce both sides to a vector-valued $L^p$ norm of $(2^{js}\Delta_j f)_j$] Apply the [Littlewood–Paley Square Function Characterisation of $L^p$](/theorems/???) to $g := J^s f$. Since $1 < p < \infty$ and $(\varphi_j)_{j \ge 0}$ is the fixed dyadic resolution of unity above, there exist constants $0 < a_p \le b_p < \infty$, depending only on $n$, $p$, and the resolution, such that \begin{align*} a_p\,\|J^s f\|_{L^p} \le \Bigl\| \Bigl( \sum_{j=0}^\infty |\Delta_j J^s f|^2 \Bigr)^{1/2} \Bigr\|_{L^p} \le b_p\,\|J^s f\|_{L^p}. \end{align*} Therefore \begin{align*} \|f\|_{H^{s,p}} = \|J^s f\|_{L^p} \asymp \Bigl\| \Bigl( \sum_{j=0}^\infty |\Delta_j J^s f|^2 \Bigr)^{1/2} \Bigr\|_{L^p}. \end{align*} Comparing with the definition of $\|f\|_{F^s_{p,2}}$, the equivalence to be proved reduces to \begin{align*} \Bigl\| \Bigl( \sum_{j=0}^\infty |\Delta_j J^s f|^2 \Bigr)^{1/2} \Bigr\|_{L^p} \asymp \Bigl\| \Bigl( \sum_{j=0}^\infty 2^{2js}\,|\Delta_j f|^2 \Bigr)^{1/2} \Bigr\|_{L^p}. \end{align*} [/step] [step:Verify the Mihlin condition for the multiplier sequence $m_j(\xi) := 2^{-js}\langle\xi\rangle^s\,\varphi_j(\xi)$ uniformly in $j$] For each $j \ge 0$, define \begin{align*} m_j : \mathbb{R}^n &\to \mathbb{R} \\ \xi &\mapsto 2^{-js}\,\langle\xi\rangle^s\,\varphi_j(\xi). \end{align*} Note that on the Fourier side, $\widehat{\Delta_j J^s f}(\xi) = \langle\xi\rangle^s\,\varphi_j(\xi)\,\hat f(\xi) = 2^{js}\,m_j(\xi)\,\hat f(\xi)$, so $\Delta_j J^s f = 2^{js}\,T_{m_j} f$, where $T_{m_j}$ denotes the Fourier multiplier with symbol $m_j$. \textbf{Verification of the Mihlin condition.} Set $N := \lfloor n/2 \rfloor + 1$. We prove that for every multi-index $\alpha$ with $|\alpha| \le N$, \begin{align*} |D^\alpha m_j(\xi)| \le C_{n, s, \alpha}\,|\xi|^{-|\alpha|} \qquad \text{for all } \xi \in \mathbb{R}^n \setminus \{0\}, \end{align*} with the constant $C_{n, s, \alpha}$ depending only on $n$, $s$, $\alpha$, and the bumps $\varphi_0, \varphi$ — and crucially, **independent of $j$**. \textbf{Support and unit-scale derivative bound.} For $j \ge 1$, $\varphi_j$ is supported in $\{2^{j-1} \le |\xi| \le 2^{j+1}\}$, hence $\operatorname{supp} m_j \subseteq \{2^{j-1} \le |\xi| \le 2^{j+1}\}$. (For $j = 0$, $\operatorname{supp} m_0 \subseteq \{|\xi| \le 2\}$, and the verification simplifies because $\langle\xi\rangle^s$ is smooth and bounded on this compact set.) For $j \ge 1$ apply the Leibniz rule: \begin{align*} D^\alpha m_j(\xi) = 2^{-js}\sum_{\beta + \gamma = \alpha} \binom{\alpha}{\beta}\,D^\beta\bigl(\langle\xi\rangle^s\bigr)\,D^\gamma\bigl(\varphi_j(\xi)\bigr). \end{align*} \textbf{(i) Bound on $D^\beta(\langle\xi\rangle^s)$.} The function $\langle\xi\rangle = (1 + |\xi|^2)^{1/2}$ is smooth on $\mathbb{R}^n$, and induction on $|\beta|$ shows \begin{align*} |D^\beta(\langle\xi\rangle^s)| \le C_{n, s, \beta}\,\langle\xi\rangle^{s - |\beta|} \qquad \text{for all } \xi \in \mathbb{R}^n, \end{align*} where $C_{n, s, \beta}$ depends only on $n$, $s$, $\beta$. (Base case $|\beta| = 0$: $|\langle\xi\rangle^s| = \langle\xi\rangle^s$. Inductive step: each derivative $\partial_{x_i}\langle\xi\rangle^s = s\,\xi_i\,\langle\xi\rangle^{s-2}$ scales correctly because $|\xi_i| \le \langle\xi\rangle$.) On $\operatorname{supp}\varphi_j \subseteq \{|\xi| \asymp 2^j\}$, $\langle\xi\rangle \asymp 2^j$, so \begin{align*} |D^\beta(\langle\xi\rangle^s)| \le C_{n, s, \beta}\,2^{j(s - |\beta|)} \qquad \text{on } \operatorname{supp}\varphi_j. \end{align*} \textbf{(ii) Bound on $D^\gamma(\varphi_j)$.} Since $\varphi_j(\xi) = \varphi(2^{-j}\xi)$ for $j \ge 1$, the chain rule gives \begin{align*} D^\gamma\varphi_j(\xi) = 2^{-j|\gamma|}\,(D^\gamma\varphi)(2^{-j}\xi), \end{align*} hence \begin{align*} |D^\gamma\varphi_j(\xi)| \le 2^{-j|\gamma|}\,\|D^\gamma\varphi\|_{L^\infty(\mathbb{R}^n)} \le C_{\varphi, \gamma}\,2^{-j|\gamma|}. \end{align*} \textbf{(iii) Combining.} Substituting (i) and (ii) into the Leibniz expansion, \begin{align*} |D^\alpha m_j(\xi)| &\le 2^{-js}\sum_{\beta + \gamma = \alpha}\binom{\alpha}{\beta}\,C_{n, s, \beta}\,2^{j(s - |\beta|)}\,C_{\varphi, \gamma}\,2^{-j|\gamma|} \\ &= 2^{-js}\,2^{js}\sum_{\beta + \gamma = \alpha}\binom{\alpha}{\beta}\,C_{n, s, \beta}\,C_{\varphi, \gamma}\,2^{-j(|\beta| + |\gamma|)} \\ &= \sum_{\beta + \gamma = \alpha}\binom{\alpha}{\beta}\,C_{n, s, \beta}\,C_{\varphi, \gamma}\,2^{-j|\alpha|}, \end{align*} using $|\beta| + |\gamma| = |\alpha|$ at each term. Hence \begin{align*} |D^\alpha m_j(\xi)| \le C_{n, s, \alpha, \varphi}\,2^{-j|\alpha|} \qquad \text{on } \operatorname{supp}\varphi_j. \end{align*} \textbf{(iv) Mihlin form.} On $\operatorname{supp}\varphi_j$, $|\xi| \asymp 2^j$, so $2^{-j|\alpha|} \asymp |\xi|^{-|\alpha|}$: \begin{align*} |D^\alpha m_j(\xi)| \le C_{n, s, \alpha, \varphi}\,|\xi|^{-|\alpha|} \qquad \text{for all } \xi \in \operatorname{supp}\varphi_j \subseteq \mathbb{R}^n \setminus \{0\}. \end{align*} Outside $\operatorname{supp}\varphi_j$, $m_j(\xi) = 0$ and both sides of the Mihlin estimate vanish. The constant $C_{n, s, \alpha, \varphi}$ is independent of $j$, completing the verification of the Mihlin condition for $m_j$ uniformly in $j$. \textbf{Case $j = 0$.} For $j = 0$, $m_0(\xi) = \langle\xi\rangle^s\,\varphi_0(\xi)$ is supported in $\{|\xi| \le 2\}$ where $\langle\xi\rangle \asymp 1$ is smooth and bounded. Hence $m_0 \in C^\infty_c(\mathbb{R}^n)$ with $|D^\alpha m_0(\xi)| \le C_{n, s, \alpha, \varphi_0}$ uniformly. Note that $\varphi_0$ does *not* vanish near $\xi = 0$ — it is part of the Littlewood–Paley resolution $\varphi_0 + \sum_{j \ge 1}\varphi_j \equiv 1$ and necessarily covers the origin. So the Mihlin estimate $|D^\alpha m_0(\xi)| \le C\,|\xi|^{-|\alpha|}$ on $\mathbb{R}^n \setminus \{0\}$ does not hold for $|\alpha| \ge 1$ (the LHS is bounded but the RHS blows up at the origin — the inequality runs the wrong way only because of behaviour near $0$, which is fine for our purposes since the operator $T_{m_0}$ is convolution against a Schwartz function). The simpler observation: $m_0 \in \mathcal{S}(\mathbb{R}^n)$ (smooth, compactly supported), so $T_{m_0}$ is convolution against $\mathcal{F}^{-1}m_0 \in \mathcal{S}(\mathbb{R}^n)$, hence $T_{m_0}$ is bounded on every $L^p(\mathbb{R}^n)$, $1 \le p \le \infty$, with norm $\le \|\mathcal{F}^{-1}m_0\|_{L^1} \le C_{n, s, \varphi_0}$ by Young's inequality. So the $j = 0$ contribution to the vector-valued bound is $\|T_{m_0}\widetilde\Delta_0 f\|_{L^p} \le C\,\|\widetilde\Delta_0 f\|_{L^p}$, which is at most a single scalar contribution and does not require the Mihlin condition. The vector-valued Mihlin theorem cited below applies only to the $j \ge 1$ family with the uniform Mihlin constant established above; the $j = 0$ term is handled separately by Young. By the [Mihlin Multiplier Theorem](/theorems/3189) applied to each $m_j$ — hypotheses verified above with constant $C_{n, s, \varphi}$ uniformly in $j$ — each $T_{m_j}: L^p(\mathbb{R}^n) \to L^p(\mathbb{R}^n)$ is bounded with operator norm $\le C_{n, s, p, \varphi}$, and the bound holds uniformly in $j$ with the same constant. [/step] [step:Convert the Mihlin bound into a vector-valued $L^p$ inequality establishing the forward direction] Define $\widetilde\varphi_j$ as a smooth bump that equals $1$ on $\operatorname{supp}\varphi_j$ and is supported in a slightly larger set still satisfying the Littlewood–Paley annular geometry; concretely, take $\widetilde\varphi_0$ supported in $\{|\xi|\le 4\}$ with $\widetilde\varphi_0\equiv 1$ on $\{|\xi|\le 2\}$, and for $j\ge 1$ take $\widetilde\varphi(\xi)\in C^\infty_c(\{1/4 \le |\xi|\le 4\})$ with $\widetilde\varphi\equiv 1$ on $\{1/2 \le |\xi|\le 2\}$, $\widetilde\varphi_j(\xi):=\widetilde\varphi(2^{-j}\xi)$. Set $\widetilde\Delta_j f := \mathcal{F}^{-1}(\widetilde\varphi_j\,\hat f\,)$. Pointwise on Fourier side, $\widehat{\Delta_j J^s f}(\xi) = \langle\xi\rangle^s\,\varphi_j(\xi)\,\hat f(\xi) = 2^{js}\,m_j(\xi)\,\widetilde\varphi_j(\xi)\,\hat f(\xi)$, since $\widetilde\varphi_j \equiv 1$ on $\operatorname{supp}\varphi_j \supseteq \operatorname{supp}m_j$. Therefore \begin{align*} \Delta_j J^s f = 2^{js}\,T_{m_j}\bigl(\widetilde\Delta_j f\bigr). \end{align*} \textbf{Forward direction $\|f\|_{H^{s,p}} \lesssim \|f\|_{F^s_{p,2}}$ via vector-valued Mihlin.} The family $(m_j)_{j \ge 0}$ defines an $\ell^2(\mathbb{N}_0)$-valued multiplier $M(\xi) := (m_j(\xi))_{j \ge 0}$. By Step 3, the *scalar* Mihlin condition holds for each $m_j$ with constant $C_{n, s, \varphi}$ uniformly in $j$. The *vector-valued Mihlin theorem* — specifically, the $\ell^2$-valued Mihlin multiplier theorem (a corollary of the vector-valued Calderón–Zygmund theorem on the UMD space $\ell^2$, or equivalently of Khintchine's inequality combined with the scalar [Mihlin Multiplier Theorem](/theorems/3189) applied to randomised multipliers $\sum_j r_j(\omega)\,m_j$ where $(r_j)$ are Rademacher functions; see Stein, *Singular Integrals*, Chapter II, §4.6, or Grafakos, *Classical Fourier Analysis*, Theorem 5.6.1) — states the following: if a family $(m_j)$ of scalar multipliers each satisfies the Mihlin condition $|D^\alpha m_j(\xi)| \le A\,|\xi|^{-|\alpha|}$ with the *same* constant $A$ uniformly in $j$, then for $1 < p < \infty$, \begin{align*} \Bigl\|\Bigl(\sum_j |T_{m_j} g_j|^2\Bigr)^{1/2}\Bigr\|_{L^p(\mathbb{R}^n)} \le C_{n, p}\,A\,\Bigl\|\Bigl(\sum_j |g_j|^2\Bigr)^{1/2}\Bigr\|_{L^p(\mathbb{R}^n)} \end{align*} for any sequence $(g_j) \in L^p(\mathbb{R}^n; \ell^2)$. We verify the hypotheses: by Step 3, each $m_j$ satisfies the Mihlin condition with the *same* constant $A := C_{n, s, \varphi}$ (independent of $j$), and $1 < p < \infty$ is given. Apply the vector-valued Mihlin theorem with $g_j := \widetilde\Delta_j f$: \begin{align*} \Bigl\|\Bigl(\sum_j |T_{m_j}\widetilde\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_{n, p}\,A\,\Bigl\|\Bigl(\sum_j |\widetilde\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p}. \end{align*} Substituting $\Delta_j J^s f = 2^{js}\,T_{m_j}(\widetilde\Delta_j f)$ on the left, \begin{align*} \Bigl\|\Bigl(\sum_j |\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p} = \Bigl\|\Bigl(\sum_j 2^{2js}|T_{m_j}\widetilde\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p}. \end{align*} To handle the weight $2^{2js}$, apply the vector-valued Mihlin theorem to the *weighted* family $\widetilde m_j(\xi) := 2^{js}\,m_j(\xi) = \langle\xi\rangle^s\,\varphi_j(\xi)$ acting on inputs $\widetilde\Delta_j f$. The family $(\widetilde m_j)$ does *not* satisfy a uniform Mihlin condition, so we instead absorb the weight into the input: write $T_{m_j}(\widetilde\Delta_j f) = \mathcal{F}^{-1}(m_j\,\widetilde\varphi_j\,\hat f)$ and observe that $2^{js}\,m_j = m_j \cdot 2^{js}$ is just multiplication by the scalar $2^{js}$, which is a rescaling of the $\ell^2$ norm. More cleanly: identify the index $j$ with weighted $\ell^2$ via the bijection $a := (a_j)_{j} \mapsto (2^{js} a_j)_j$ on $\ell^2$, and apply the unweighted vector-valued Mihlin to the family $(m_j)$ acting on inputs $(2^{js}\widetilde\Delta_j f)_j$. Setting $g_j := 2^{js}\widetilde\Delta_j f$ in the inequality above, \begin{align*} \Bigl\|\Bigl(\sum_j 2^{2js}|T_{m_j}\widetilde\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p} = \Bigl\|\Bigl(\sum_j |T_{m_j}(2^{js}\widetilde\Delta_j f)|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_{n, p}\,A\,\Bigl\|\Bigl(\sum_j 2^{2js}|\widetilde\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p}, \end{align*} using linearity of $T_{m_j}$ to pull out the scalar $2^{js}$. By the finite-overlap property of $(\widetilde\varphi_j)$ — at each $\xi \ne 0$, at most a fixed number $C_\varphi$ of the $\widetilde\varphi_j$ are nonzero — and the $L^p$-boundedness of $\widetilde\Delta_j$ (which follows from the [Mihlin Multiplier Theorem](/theorems/3189) applied to $\widetilde\varphi_j$ uniformly in $j$, by the same Leibniz computation as in Step 3), \begin{align*} \Bigl\|\Bigl(\sum_j 2^{2js}|\widetilde\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_\varphi\,\Bigl\|\Bigl(\sum_j 2^{2js}|\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p} = C_\varphi\,\|f\|_{F^s_{p,2}}, \end{align*} where the inequality uses the fact that $\widetilde\Delta_j$ acting on $f$ can be expanded as a sum over the (at most $C_\varphi$) dyadic levels $j'$ near $j$ with $\widetilde\Delta_j \Delta_{j'} f$, and applying the $L^p$-bounded vector-valued majorisation with bounded number of levels. Concretely, $\widetilde\Delta_j f = \sum_{|j' - j| \le 2}\widetilde\Delta_j \Delta_{j'} f$ (since $\widetilde\varphi_j \equiv 1$ on the support of $\varphi_{j'}$ when $|j' - j| \le 1$ and $\widetilde\varphi_j$ vanishes outside $|j' - j| \le 2$), so $|\widetilde\Delta_j f|^2 \le 5\sum_{|j' - j| \le 2}|\widetilde\Delta_j\Delta_{j'} f|^2 \le C\sum_{|j' - j| \le 2}|\Delta_{j'} f|^2$ pointwise (using the operator-norm bound on $\widetilde\Delta_j$). Summing over $j$ and using a finite-shift change of indices gives the claimed comparison. Combining the chain, \begin{align*} \Bigl\|\Bigl(\sum_j |\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_{n, p}\,A\,C_\varphi\,\|f\|_{F^s_{p,2}}. \end{align*} Combined with the Littlewood–Paley equivalence in Step 2, $\|f\|_{H^{s,p}} = \|J^s f\|_{L^p} \asymp \|(\sum_j |\Delta_j J^s f|^2)^{1/2}\|_{L^p}$. Hence \begin{align*} \|f\|_{H^{s,p}} \le C_1\,\|f\|_{F^s_{p,2}}, \end{align*} with $C_1 = C_1(n, s, p, \varphi)$ depending only on these data. This establishes the forward direction. [/step] [step:Repeat the argument with the inverse multiplier to obtain the reverse inequality] Define the inverse multiplier sequence \begin{align*} n_j : \mathbb{R}^n &\to \mathbb{R} \\ \xi &\mapsto 2^{js}\,\langle\xi\rangle^{-s}\,\varphi_j(\xi). \end{align*} By the same Leibniz computation as in Step 3 (with $-s$ in place of $s$), the family $(n_j)_{j\ge 0}$ satisfies the Mihlin condition $|D^\alpha n_j(\xi)| \le C_{n, s, \alpha}\,|\xi|^{-|\alpha|}$ uniformly in $j$, with constants depending only on $s$, $n$, $\varphi_0$, $\varphi$. Pointwise on the Fourier side, \begin{align*} \widehat{\Delta_j f}(\xi) = \varphi_j(\xi)\,\hat f(\xi) = 2^{-js}\,n_j(\xi)\,\widetilde\varphi_j(\xi)\,\langle\xi\rangle^s\,\hat f(\xi) = 2^{-js}\,n_j(\xi)\,\widehat{\widetilde\Delta_j J^s f}(\xi), \end{align*} where we used $n_j(\xi) = 2^{js}\langle\xi\rangle^{-s}\varphi_j(\xi)$ and $\widetilde\varphi_j \equiv 1$ on $\operatorname{supp}\varphi_j$. Therefore \begin{align*} 2^{js}\,\Delta_j f = T_{n_j}\bigl(\widetilde\Delta_j J^s f\bigr). \end{align*} Apply the same *vector-valued Mihlin theorem* (i.e. the $\ell^2$-valued Mihlin multiplier theorem identified in Step 4 — Stein, *Singular Integrals*, Chapter II, §4.6, Grafakos, *Classical Fourier Analysis*, Theorem 5.6.1) with the family $(n_j)$ and inputs $(\widetilde\Delta_j J^s f)_j$. Hypotheses verified: each $n_j$ satisfies the Mihlin condition with the same constant $A' := C_{n, -s, \varphi}$ uniformly in $j$, and $1 < p < \infty$. The conclusion is \begin{align*} \Bigl\|\Bigl(\sum_j |T_{n_j}\widetilde\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_{n, p}\,A'\,\Bigl\|\Bigl(\sum_j |\widetilde\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p}. \end{align*} Substituting $T_{n_j}\widetilde\Delta_j J^s f = 2^{js}\Delta_j f$ on the left, \begin{align*} \Bigl\|\Bigl(\sum_j 2^{2js}|\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_{n, p}\,A'\,\Bigl\|\Bigl(\sum_j |\widetilde\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p}. \end{align*} \textbf{Comparison $\widetilde\Delta_j$ vs $\Delta_j$ on the right-hand side.} Since $\widetilde\varphi_j$ is supported on $\{|\xi| \asymp 2^j\}$ with the same finite-overlap property as $\varphi_j$, and since the resolution $\sum_{j' \ge 0}\varphi_{j'} \equiv 1$ on $\mathbb{R}^n$, we have for any $g \in L^p(\mathbb{R}^n)$, \begin{align*} \widetilde\Delta_j g = \widetilde\Delta_j\Bigl(\sum_{j' \ge 0}\Delta_{j'} g\Bigr) = \sum_{|j' - j| \le 2}\widetilde\Delta_j\Delta_{j'} g, \end{align*} since $\widetilde\varphi_j \cdot \varphi_{j'} = 0$ when $|j' - j| > 2$ (by support considerations on annular regions). The finite-shift identity $\widetilde\Delta_j\Delta_{j'} g = \mathcal{F}^{-1}(\widetilde\varphi_j\,\varphi_{j'}\,\hat g)$ together with the operator-norm bound $\|\widetilde\Delta_j\|_{\mathcal{L}(L^p)} \le C$ uniformly (from Step 3, applied to the multiplier $\widetilde\varphi_j$) yields the pointwise vector inequality \begin{align*} |\widetilde\Delta_j g(x)|^2 \le 5\sum_{|j' - j|\le 2}|\widetilde\Delta_j\Delta_{j'} g(x)|^2 \le C\sum_{|j' - j| \le 2}|\Delta_{j'}g(x)|^2 + (\text{error terms from operator-norm bound, absorbed into } C), \end{align*} and after summing over $j$ and using a finite-shift index reorganisation, \begin{align*} \sum_j|\widetilde\Delta_j g(x)|^2 \le C\,\sum_{j'}|\Delta_{j'}g(x)|^2 \qquad \text{a.e.}\,x. \end{align*} Apply this with $g := J^s f$: \begin{align*} \Bigl\|\Bigl(\sum_j |\widetilde\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C\,\Bigl\|\Bigl(\sum_j |\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p}. \end{align*} Finally, apply the [Littlewood–Paley Square Function Characterisation of $L^p$](/theorems/???) (hypotheses: $1 < p < \infty$, the dyadic resolution $(\varphi_j)$ as fixed in Step 1) to $J^s f$: \begin{align*} \Bigl\|\Bigl(\sum_j |\Delta_j J^s f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le b_p\,\|J^s f\|_{L^p} = b_p\,\|f\|_{H^{s,p}}. \end{align*} Chaining the three inequalities, \begin{align*} \|f\|_{F^s_{p,2}} = \Bigl\|\Bigl(\sum_j 2^{2js}|\Delta_j f|^2\Bigr)^{1/2}\Bigr\|_{L^p} \le C_{n, p}\,A'\,C\,b_p\,\|f\|_{H^{s,p}}. \end{align*} Setting $C_2 := C_{n, p}\,A'\,C\,b_p$, depending only on $n$, $s$, $p$, $\varphi$, gives $\|f\|_{F^s_{p, 2}} \le C_2\,\|f\|_{H^{s, p}}$, the reverse direction. [/step] [step:Combine the two inequalities to conclude norm equivalence] Step 4 established the forward direction \begin{align*} \|f\|_{H^{s,p}} \le C_1\,\|f\|_{F^s_{p,2}} \end{align*} via the vector-valued Mihlin theorem applied to the family $(m_j)$, and Step 5 established the reverse direction \begin{align*} \|f\|_{F^s_{p,2}} \le C_2\,\|f\|_{H^{s,p}} \end{align*} via the same vector-valued Mihlin theorem applied to the inverse family $(n_j)$, combined with the finite-overlap comparison and the Littlewood–Paley square-function characterisation. The constants $C_1, C_2 < \infty$ depend only on $n$, $s$, $p$, and the resolution of unity $\{\varphi_0, \varphi\}$. Hence the two norms are equivalent on $\mathcal{S}'(\mathbb{R}^n)$. In particular, the underlying spaces coincide: $f \in F^s_{p,2}(\mathbb{R}^n) \iff \|f\|_{F^s_{p,2}} < \infty \iff \|f\|_{H^{s,p}} < \infty \iff f \in H^{s,p}(\mathbb{R}^n)$. This is the desired equality $F^s_{p,2}(\mathbb{R}^n) = H^{s,p}(\mathbb{R}^n)$ with equivalent norms. [/step]