Weighted L-p bounds for standard Calderon-Zygmund maximal truncations

A priori finiteness on the dense class. Fix $g\in C_c^{\mathrm{\infty }}$ supported in $B(0,R_g)$. In a double truncation, split at radius one. On the part below one, subtract $g(x)$: [F8] bounds the resulting integral by $A_1{L}_g{\int }_{\mathrm{\mid }z\mathrm{\mid }<1}\mathrm{\mid }z{\mathrm{\mid }}^{1-n}dz=A_1{L}_g\mathrm{\mid }{S}^{n-1}\mathrm{\mid }$, while cancellation bounds the constant term by $A_3\mathrm{\parallel }g{\mathrm{\parallel }}_{\mathrm{\infty }}$. On the part above one, use the original integrand and $\mathrm{\mid }k(z)\mathrm{\mid }\le A_1$ to obtain the bound $A_1\mathrm{\parallel }g{\mathrm{\parallel }}_1$. Hence $T^{\ast \ast }g$ is uniformly bounded. If $\mathrm{\mid }x\mathrm{\mid }>2R_g+1$, the original size estimate gives $T^{\ast \ast }g(x)\le 2^n{A}_1\mathrm{\parallel }g{\mathrm{\parallel }}_1\mathrm{\mid }x{\mathrm{\mid }}^{-n}$, so $T^{\ast \ast }g\le C_g(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-n}$. Each finite truncation is continuous by dominated convergence, so its supremum has open level sets; the decay makes these bounded and proper. By [F3], $T^{\ast \ast }g\le 2^n{C}_{g}M({\mathbf{1}}_{B(0,1)})$. The maximal bounds in [F3] therefore give finite $L^p(w)$ norm for $w\in A_p$, $p>1$, and finite weak $L^1(w)$ norm for $w\in A_1$.

Fix $p>1$, $w\in A_p$ and $g\in C_c^{\mathrm{\infty }}$, with $A_1+A_2^{\mathrm{\prime }}+A_3>0$; if this sum vanishes the truncations are zero and the strong bound is immediate. Write $D(\mu )=w(\{T^{\ast \ast }g>\mu \})$. For every $0<\gamma <{\gamma }_0$, splitting the level set and applying [F2] gives $D(2\mu )\le C_1{\gamma }^{{\delta }^{\mathrm{\prime }}}D(\mu )+w(\{Mg>\gamma \mu \})$. By layer cake [F4], $\mathrm{\parallel }{T}^{\ast \ast }g{\mathrm{\parallel }}_{L^p(w)}^p=2^p{\int }_0^{\mathrm{\infty }}p{\mu }^{p-1}D(2\mu )d\mu$. The second distribution term integrates exactly to ${\gamma }^{-p}\mathrm{\parallel }Mg{\mathrm{\parallel }}_{L^p(w)}^p$ by the substitution $t=\gamma \mu$; it is finite by [F3].

Weak endpoint on the dense class. Let $w\in A_1$ and $g\in C_c^{\mathrm{\infty }}$. If $A_1+A_2^{\mathrm{\prime }}+A_3=0$ the kernel vanishes and the bound is immediate. Otherwise, since $A_1\subseteq A_{\mathrm{\infty }}$, [F2] applies; for every $\lambda >0$ the same splitting with the weak bound of [F3] in place of the strong one gives $w(\{T^{\ast \ast }g>2\lambda \})\le C_1{\gamma }^{{\delta }^{\mathrm{\prime }}}w(\{T^{\ast \ast }g>\lambda \})+5^n[w{]}_{A_1}(\gamma \lambda {)}^{-1}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(w)}$. Multiplying by $2\lambda$, taking the supremum over $\lambda >0$ (finite by step 1.1) and choosing $\gamma =\frac{1}{2}\min \{{\gamma }_0,(4C_1{)}^{-1\mathrm{/}{\delta }^{\mathrm{\prime }}}\}$ gives $2C_1{\gamma }^{{\delta }^{\mathrm{\prime }}}\le \frac{1}{2}$ and ${\gamma }^{-1}\le C(n,\delta ,[w{]}_{A_1})S$, and yields $\mathrm{\parallel }{T}^{\ast \ast }g{\mathrm{\parallel }}_{L^{1,\mathrm{\infty }}(w)}\le C(n,\delta ,[w{]}_{A_1})(A_1+A_2^{\mathrm{\prime }}+A_3+B)\mathrm{\parallel }g{\mathrm{\parallel }}_{L^1(w)}$; the weak $(1,1)$ bound for $T^{\ast }$ follows from $T^{\ast }\le T^{\ast \ast }$.

If $A_1+A_2^{\mathrm{\prime }}+A_3=0$, both maximal truncations vanish and the bounds are immediate. Otherwise, substituting step 2.1 into the layer-cake identity and using the finiteness of $\mathrm{\parallel }{T}^{\ast \ast }g{\mathrm{\parallel }}_{L^p(w)}$ from step 1.1 gives $\mathrm{\parallel }{T}^{\ast \ast }g{\mathrm{\parallel }}_{L^p(w)}^p\le 2^p{C}_1{\gamma }^{{\delta }^{\mathrm{\prime }}}\mathrm{\parallel }{T}^{\ast \ast }g{\mathrm{\parallel }}_{L^p(w)}^p+2^p(C_{n,p}[w{]}_{A_p}^{1\mathrm{/}(p-1)}{)}^p{\gamma }^{-p}\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p(w)}^p$; choosing $\gamma =\frac{1}{2}\min \{{\gamma }_0,(2^{p+1}{C}_1{)}^{-1\mathrm{/}{\delta }^{\mathrm{\prime }}}\}$ gives $2^p{C}_1{\gamma }^{{\delta }^{\mathrm{\prime }}}\le \frac{1}{2}$, and since ${\gamma }_0=c_0(A_1+A_2^{\mathrm{\prime }}+A_3{)}^{-1}$ and $C_1\le C_w{C}_n^{{\delta }^{\mathrm{\prime }}}(A_1+A_2^{\mathrm{\prime }}+A_3+B{)}^{{\delta }^{\mathrm{\prime }}}$ one has ${\gamma }^{-1}\le C^{\mathrm{\prime }}(n,p,\delta ,[w{]}_{A_p})(A_1+A_2^{\mathrm{\prime }}+A_3+B)$; hence $\mathrm{\parallel }{T}^{\ast \ast }g{\mathrm{\parallel }}_{L^p(w)}\le C(n,p,\delta ,[w{]}_{A_p})(A_1+A_2^{\mathrm{\prime }}+A_3+B)\mathrm{\parallel }g{\mathrm{\parallel }}_{L^p(w)}$, and $T^{\ast }$ inherits the bound because $T^{\ast }\le T^{\ast \ast }\le 2T^{\ast }$ by [F7].

Extension to $f\in L^p(w)$. Let $w\in A_p$, $1<p<\mathrm{\infty }$, $f\in L^p(w)$, and choose $g_m\in C_c^{\mathrm{\infty }}$ with $\mathrm{\parallel }{g}_m-f{\mathrm{\parallel }}_{L^p(w)}\to 0$ by [F5]. For every $x$ and every fixed pair $0<\epsilon <N$, [F1] gives $\mathrm{\mid }{T}^{(\epsilon ,N)}f(x)-T^{(\epsilon ,N)}{g}_m(x)\mathrm{\mid }\le C(n,p,[w{]}_{A_p})A_{1}w(Q(x,\epsilon ){)}^{-1\mathrm{/}p}\mathrm{\parallel }f-g_m{\mathrm{\parallel }}_{L^p(w)}\to 0$, so $\mathrm{\mid }{T}^{(\epsilon ,N)}f(x)\mathrm{\mid }={\lim }_m\mathrm{\mid }{T}^{(\epsilon ,N)}{g}_m(x)\mathrm{\mid }\le {\mathrm{lim\hspace{0.17em}inf}}_m{T}^{\ast \ast }{g}_m(x)$ for each fixed pair and, taking the supremum over all pairs, $T^{\ast \ast }f\le {\mathrm{lim\hspace{0.17em}inf}}_m{T}^{\ast \ast }{g}_m$ pointwise. Fatou's lemma [F4] and step 3.1 then give $\mathrm{\parallel }{T}^{\ast \ast }f{\mathrm{\parallel }}_{L^p(w)}^p\le {\mathrm{lim\hspace{0.17em}inf}}_m\mathrm{\parallel }{T}^{\ast \ast }{g}_m{\mathrm{\parallel }}_{L^p(w)}^p\le C(n,p,\delta ,[w{]}_{A_p}{)}^p(A_1+A_2^{\mathrm{\prime }}+A_3+B{)}^p\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}^p$, in particular $T^{\ast \ast }f<\mathrm{\infty }$ $w$-almost everywhere, and $T^{\ast }f\le T^{\ast \ast }f\le 2T^{\ast }f$ carries the same conclusions. If instead $w\in A_1$ and $f\in L^1(w)$, the same approximation gives $T^{\ast \ast }f\le {\mathrm{lim\hspace{0.17em}inf}}_m{T}^{\ast \ast }{g}_m$ pointwise, hence $\{T^{\ast \ast }f>\lambda \}\subseteq {\mathrm{lim\hspace{0.17em}inf}}_m\{T^{\ast \ast }{g}_m>\lambda \}$ for every $\lambda >0$ and $w(\{T^{\ast \ast }f>\lambda \})\le {\mathrm{lim\hspace{0.17em}inf}}_{m}w(\{T^{\ast \ast }{g}_m>\lambda \})\le C(n,\delta ,[w{]}_{A_1})(A_1+A_2^{\mathrm{\prime }}+A_3+B){\lambda }^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_{L^1(w)}$ by step 2.2, so again $T^{\ast \ast }f<\mathrm{\infty }$ $w$-a.e.; $T^{\ast }$ inherits both bounds by the pointwise comparison.

Final clause. Let $\mathcal{D}\subseteq L^p(w)$ be a dense subspace on which $T_{\epsilon }$ converges almost everywhere as $\epsilon \downarrow 0$, and let $f\in L^p(w)$. For $g\in \mathcal{D}$ and $\epsilon ,{\epsilon }^{\mathrm{\prime }}>0$ one has $\mathrm{\mid }{T}_{\epsilon }f-T_{{\epsilon }^{\mathrm{\prime }}}f\mathrm{\mid }\le 2T^{\ast }(f-g)+\mathrm{\mid }{T}_{\epsilon }g-T_{{\epsilon }^{\mathrm{\prime }}}g\mathrm{\mid }$ pointwise, so for every $\eta >0$ the set where ${\mathrm{lim\hspace{0.17em}sup}}_{\epsilon ,{\epsilon }^{\mathrm{\prime }}\downarrow 0}\mathrm{\mid }{T}_{\epsilon }f-T_{{\epsilon }^{\mathrm{\prime }}}f\mathrm{\mid }>4\eta$ has $w$-measure at most $w(\{T^{\ast }(f-g)>\eta \})\le {\eta }^{-p}\mathrm{\parallel }{T}^{\ast }(f-g){\mathrm{\parallel }}_{L^p(w)}^p\le C(n,p,\delta ,[w{]}_{A_p}{)}^p(A_1+A_2^{\mathrm{\prime }}+A_3+B{)}^p{\eta }^{-p}\mathrm{\parallel }f-g{\mathrm{\parallel }}_{L^p(w)}^p$ by Chebyshev [F6] and step 4.1. For every $\eta >0$, take the infimum of this bound over $g\in \mathcal{D}$; density makes the infimum zero, so the exceptional set is null; intersecting the resulting full-measure sets over $\eta =1\mathrm{/}m$, $m\ge 1$, shows that $(T_{\epsilon }f)$ is $w$-almost everywhere Cauchy as $\epsilon \downarrow 0$, so the limit $Tf$ exists $w$-a.e. and is measurable as an a.e. limit of measurable functions [F1]. It obeys $\mathrm{\mid }Tf\mathrm{\mid }\le T^{\ast }f$ pointwise, hence $\mathrm{\parallel }Tf{\mathrm{\parallel }}_{L^p(w)}\le \mathrm{\parallel }{T}^{\ast }f{\mathrm{\parallel }}_{L^p(w)}\le C(n,p,\delta ,[w{]}_{A_p})(A_1+A_2^{\mathrm{\prime }}+A_3+B)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}$ by step 4.1.