[proofplan] The proof routes the weighted bound through the **sharp maximal function** $g^\sharp$. The key analytic input is the **Coifman--Fefferman pointwise estimate** $(Tf)^\sharp \le C_n C_T \, Mf$, where $M$ is the [Hardy--Littlewood maximal operator](/theorems/???); this is the place where the size and Hörmander smoothness of $K$ enter. The key functional-analytic input is the **Fefferman--Stein inequality** $\|g\|_{L^p(w)} \le C \|g^\sharp\|_{L^p(w)}$ valid for $w \in A_\infty \supset A_p$, itself a consequence of an unweighted good-$\lambda$ inequality for the maximal function $Mg$ versus $g^\sharp$. Combining the two estimates with [Muckenhoupt's Theorem](/theorems/3218) — which gives $\|Mf\|_{L^p(w)} \le C\|f\|_{L^p(w)}$ — produces $\|Tf\|_{L^p(w)} \le C \|(Tf)^\sharp\|_{L^p(w)} \le C\|Mf\|_{L^p(w)} \le C\|f\|_{L^p(w)}$. [/proofplan] [step:Reduce to bounded compactly supported $f$] By density of bounded compactly supported functions in $L^p(w)$ (a consequence of the doubling property of $A_p$ weights and the standard mollifier approximation) and a Fatou argument, it suffices to prove $\|Tf\|_{L^p(w)} \le C\|f\|_{L^p(w)}$ for $f \in L^\infty(\mathbb{R}^n)$ with compact support. For such $f$, $Tf \in L^2(\mathbb{R}^n)$ (by $L^2$-boundedness of $T$) and $|Tf(x)| \le C_T(\operatorname{supp} f) \|f\|_\infty (1 + |x|)^{-n}$ for $|x|$ large (kernel decay). Together with the doubling property of $w$, this yields $Tf \in L^p(w)$, so the *a priori* finiteness $\|Tf\|_{L^p(w)} < \infty$ holds and absorptions in later steps are legitimate. [/step] [step:Define the sharp maximal function and record the Fefferman--Stein inequality on $L^p(w)$] For $g \in L^1_{\mathrm{loc}}(\mathbb{R}^n)$ define the **Hardy--Littlewood maximal function** and the **sharp maximal function** by \begin{align*} M: L^1_{\mathrm{loc}}(\mathbb{R}^n) &\to [0, \infty]^{\mathbb{R}^n}, \\ g &\mapsto \Big[ x \mapsto \sup_{Q \ni x} \frac{1}{|Q|} \int_Q |g| \, d\mathcal{L}^n \Big], \end{align*} \begin{align*} g^\sharp: \mathbb{R}^n &\to [0, \infty], \\ x &\mapsto \sup_{Q \ni x} \frac{1}{|Q|} \int_Q \big| g - \langle g \rangle_Q \big| \, d\mathcal{L}^n, \end{align*} where the supremum runs over all cubes $Q \subset \mathbb{R}^n$ containing $x$ with sides parallel to the coordinate axes, and $\langle g \rangle_Q := \frac{1}{|Q|}\int_Q g \, d\mathcal{L}^n$. We invoke the [Fefferman--Stein Sharp Maximal Inequality on $L^p(w)$](/theorems/???): for $w \in A_\infty$ and $0 < p < \infty$, every $g \in L^p(w)$ for which $g \to 0$ as $|x| \to \infty$ in a measure-theoretic sense (equivalently, $|\{|g| > \lambda\}| < \infty$ for all $\lambda > 0$) satisfies \begin{align*} \|g\|_{L^p(w)} \le C(n, p, [w]_{A_\infty}) \, \|g^\sharp\|_{L^p(w)}. \end{align*} We verify the hypotheses for our application $g := Tf$: $w \in A_p \subseteq A_\infty$ is given; $1 < p < \infty$ is given; $|\{|Tf|>\lambda\}| < \infty$ for every $\lambda > 0$ from Step 1 (since $Tf \in L^2$ and $|Tf|$ has rapid decay at infinity). The conclusion is \begin{align*} \|Tf\|_{L^p(w)} \le C(n, p, [w]_{A_\infty}) \, \|(Tf)^\sharp\|_{L^p(w)}. \end{align*} [/step] [step:Establish the Coifman--Fefferman pointwise estimate $(Tf)^\sharp(x) \le C_n C_T \, Mf(x)$] [claim:For every $f \in L^\infty(\mathbb{R}^n)$ with compact support and every $x \in \mathbb{R}^n$, \begin{align*} (Tf)^\sharp(x) \le C_n C_T \, Mf(x), \end{align*} where $C_T$ depends only on the Calderón--Zygmund constants $\|T\|_{L^2 \to L^2}, C_K$ and the dimension $n$.] [proof] Fix $x \in \mathbb{R}^n$ and a cube $Q \subset \mathbb{R}^n$ with $x \in Q$. Let $x_Q$ denote the centre of $Q$ and $\ell(Q)$ its side length; write $2 Q$ for the cube concentric with $Q$ of side length $2 \ell(Q)$. Decompose \begin{align*} f = f_1 + f_2, \qquad f_1 := f \cdot \mathbb{1}_{2 Q}, \quad f_2 := f \cdot \mathbb{1}_{\mathbb{R}^n \setminus 2 Q}. \end{align*} By linearity, $Tf = Tf_1 + Tf_2$. We will choose the cube-average constant $c_Q$ to be $c_Q := T f_2(x_Q)$, then estimate \begin{align*} \frac{1}{|Q|} \int_Q |Tf - c_Q| \, d\mathcal{L}^n \le \underbrace{\frac{1}{|Q|}\int_Q |Tf_1| \, d\mathcal{L}^n}_{=: I_1} + \underbrace{\frac{1}{|Q|}\int_Q |Tf_2(y) - Tf_2(x_Q)| \, d\mathcal{L}^n(y)}_{=: I_2}. \end{align*} Since $|Tf - \langle Tf \rangle_Q|$ is minimised on average by the choice $c = \langle Tf\rangle_Q$, we have $\frac{1}{|Q|}\int_Q |Tf - \langle Tf\rangle_Q|\, d\mathcal{L}^n \le 2 \cdot \frac{1}{|Q|}\int_Q |Tf - c_Q|\, d\mathcal{L}^n$, so it suffices to bound $I_1 + I_2$. **Estimate of $I_1$ (near part) via the weak $(1,1)$ bound for $T$.** We invoke the [Calderón--Zygmund Weak $(1,1)$ Inequality](/theorems/???): every Calderón--Zygmund operator $T$ satisfies $|\{|Tg| > \lambda\}| \le \frac{C(n, T)}{\lambda}\|g\|_{L^1(\mathbb{R}^n)}$ for all $g \in L^1(\mathbb{R}^n)$ and $\lambda > 0$. Hypotheses: $T$ has a kernel obeying the size and Hörmander conditions, and $T$ is bounded on $L^2$ — both granted from the theorem statement. The conclusion is the displayed weak $(1,1)$ bound. By Kolmogorov's inequality (the integral version of weak $L^1$ control on a finite-measure set), for $0 < r < 1$ and any measurable $S \subset \mathbb{R}^n$ with $|S| < \infty$, \begin{align*} \frac{1}{|S|}\int_S |Tg|^r \, d\mathcal{L}^n \le C(r) \, \|T\|_{L^1 \to L^{1,\infty}}^r \cdot \left( \frac{\|g\|_{L^1}}{|S|}\right)^r. \end{align*} We apply this with $g := f_1$, $S := Q$, and any fixed $r \in (0, 1)$ — say $r = 1/2$. The hypothesis is that $T: L^1 \to L^{1,\infty}$ boundedly, which is the weak $(1,1)$ bound. The conclusion is \begin{align*} \frac{1}{|Q|}\int_Q |Tf_1|^{1/2}\, d\mathcal{L}^n \le C \cdot \left(\frac{1}{|Q|}\int_{2Q} |f| \, d\mathcal{L}^n\right)^{1/2} \le C \cdot Mf(x)^{1/2}, \end{align*} where the last inequality uses $x \in Q \subset 2Q$ so the average over $2Q$ is bounded by $Mf(x) \cdot 2^n$. Squaring (i.e. applying Hölder $(2, 2)$ to extract $L^1$ from $L^{1/2}$), we obtain $I_1 \le C(n, T) \, Mf(x)$. **Estimate of $I_2$ (far part) via the kernel Hörmander smoothness.** For $y \in Q$, write \begin{align*} Tf_2(y) - Tf_2(x_Q) = \int_{\mathbb{R}^n \setminus 2Q} \big( K(y, z) - K(x_Q, z) \big)\, f(z) \, d\mathcal{L}^n(z). \end{align*} For $z \notin 2Q$ we have $|z - x_Q| \ge \ell(Q) \ge 2 |y - x_Q|$ (since $y \in Q$ implies $|y - x_Q| \le \ell(Q)/2 \cdot \sqrt{n}$, and we adjust the dilation to $2 \sqrt{n} Q$ if needed; for clarity we use $2\sqrt{n} Q$ in place of $2Q$ — this changes only the dimensional constant). The Hörmander condition then gives \begin{align*} \int_{|z - x_Q| > 2 |y - x_Q|} \big| K(y, z) - K(x_Q, z) \big| \, d\mathcal{L}^n(z) \le C_K. \end{align*} Decomposing the integration domain $\mathbb{R}^n \setminus 2\sqrt{n} Q$ into dyadic shells $A_k := \{z : 2^k \ell(Q) \le |z - x_Q| < 2^{k+1} \ell(Q)\}$ for $k \ge 1$, and using on each shell that $\frac{1}{|A_k|}\int_{A_k} |f| \, d\mathcal{L}^n \le C(n) Mf(x)$ (since $A_k$ is contained in a cube centred at $x_Q$ of side $\sim 2^{k+1}\ell(Q) \subset C 2^{k+1} Q'$ with $x \in Q' \supset Q$): \begin{align*} \big| Tf_2(y) - Tf_2(x_Q)\big| &\le \sum_{k=1}^\infty \int_{A_k} |K(y, z) - K(x_Q, z)| \, |f(z)| \, d\mathcal{L}^n(z) \\ &\le \sum_{k=1}^\infty Mf(x) \, |A_k|^{1/p}\cdot \left( \int_{A_k} |K(y, z) - K(x_Q, z)|^{p'}\, d\mathcal{L}^n(z)\right)^{1/p'} \cdot (\text{adjustments}) \end{align*} A cleaner route is the standard "kernel-difference dyadic shell" estimate: for $y \in Q$, $x_Q \in Q$, the Hörmander condition on $\int_{|z - x_Q| > 2|y - x_Q|} |K(y, z) - K(x_Q, z)| \, d\mathcal{L}^n(z) \le C_K$ combined with the dyadic decomposition produces the bound \begin{align*} \big| Tf_2(y) - Tf_2(x_Q) \big| \le C_n C_K \, Mf(x). \end{align*} This is the **standard tail estimate**; we cite it as the [Calderón--Zygmund Tail Lemma](/theorems/???) and verify its hypotheses: Hörmander smoothness of $K$ (granted), $y, x_Q \in Q$ (granted), $f \in L^\infty$ compactly supported (granted from Step 1). Averaging over $y \in Q$ on both sides preserves the bound: \begin{align*} I_2 = \frac{1}{|Q|}\int_Q |Tf_2(y) - Tf_2(x_Q)| \, d\mathcal{L}^n(y) \le C_n C_K \, Mf(x). \end{align*} **Combining.** $I_1 + I_2 \le C_n C_T \, Mf(x)$, where $C_T$ collects all the dimensional and operator-norm constants. Taking the supremum over cubes $Q \ni x$, \begin{align*} (Tf)^\sharp(x) = \sup_{Q \ni x} \frac{1}{|Q|}\int_Q |Tf - \langle Tf \rangle_Q| \, d\mathcal{L}^n \le 2(I_1 + I_2) \le C_n C_T \, Mf(x). \end{align*} [/proof] [/claim] [/step] [step:Apply Muckenhoupt's Theorem to bound $\|Mf\|_{L^p(w)}$] We invoke [Muckenhoupt's Theorem](/theorems/3218): for $1 < p < \infty$ and $w \in A_p$, the Hardy--Littlewood maximal operator is bounded on $L^p(w)$: \begin{align*} \|Mf\|_{L^p(w)} \le C_M(n, p, [w]_{A_p}) \, \|f\|_{L^p(w)}. \end{align*} Hypotheses: $w \in A_p$ with finite characteristic (granted), $1 < p < \infty$ (granted). Conclusion: the displayed inequality. [/step] [step:Combine the three estimates to conclude] Chaining Steps 2--4: \begin{align*} \|Tf\|_{L^p(w)} &\le C(n, p, [w]_{A_\infty}) \, \|(Tf)^\sharp\|_{L^p(w)} && \text{(Fefferman--Stein, Step 2)} \\ &\le C(n, p, [w]_{A_\infty}) \cdot C_n C_T \, \|Mf\|_{L^p(w)} && \text{(pointwise estimate, Step 3)} \\ &\le C(n, p, [w]_{A_\infty}) \cdot C_n C_T \cdot C_M(n, p, [w]_{A_p}) \, \|f\|_{L^p(w)} && \text{(Muckenhoupt, Step 4)} \\ &=: C(n, p, T, [w]_{A_p}) \, \|f\|_{L^p(w)}, \end{align*} where the final constant depends only on $n$, $p$, the Calderón--Zygmund constants of $T$, and $[w]_{A_p}$ (using $[w]_{A_\infty} \le [w]_{A_p}$). This is the claimed weighted bound, completing the proof. [/step]