Hilbert and Riesz transforms are bounded on weighted L-p

Kernel data in normalized form. For the Hilbert kernel, $\mathrm{\mid }k(x)\mathrm{\mid }={\pi }^{-1}\mathrm{\mid }x{\mathrm{\mid }}^{-1}$ gives the pointwise size constant $A_1=1\mathrm{/}\pi$, the difference estimate of [F1] is the standard $1$-Hölder condition with $A_2^{\mathrm{\prime }}=2\mathrm{/}\pi$, and oddness gives ${\int }_{r<\mathrm{\mid }x\mathrm{\mid }<R}k(x)dx=0$ for all $0<r<R$, so the cancellation constant is $A_3=0$; the $L^2$ norm bound is $B=1$, and $H$ is the convolution operator with the principal-value distribution of [F1] satisfying the off-support representation with kernel $k$. For each Riesz kernel, [F2] gives the size constant $A_1=c_n$, the standard $1$-Hölder constant $A_2^{\mathrm{\prime }}=C_n=c_n{2}^{n+1}(3n+4)$, vanishing annulus integrals and hence $A_3=0$, and $B=1$, together with the principal value on Schwartz functions. Hence both families meet the hypotheses of the weighted theorem [F3] with the constants displayed in the statement.

Weighted bounds for the maximal truncations. By [F3] applied to the Hilbert kernel and to each Riesz kernel: for $f\in L^p(w)$ one has $\mathrm{\parallel }{H}^{\ast \ast }f{\mathrm{\parallel }}_{L^p(w)}\le C(n,p,[w{]}_{A_p})(1\mathrm{/}\pi +2\mathrm{/}\pi +0+1)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}$ and $\mathrm{\parallel }{R}_j^{\ast \ast }f{\mathrm{\parallel }}_{L^p(w)}\le C(n,p,[w{]}_{A_p})(c_n+C_n+0+1)\mathrm{\parallel }f{\mathrm{\parallel }}_{L^p(w)}$, with the same bounds for $H^{\ast }$ and $R_j^{\ast }$ since $T^{\ast }\le T^{\ast \ast }\le 2T^{\ast }$; for $w\in A_1$ the theorem gives the weighted weak $(1,1)$ bounds for the maximal truncations with the corresponding constants.

Almost-everywhere convergence and the bounded extension. Let $1<p<\mathrm{\infty }$ and $w\in A_p$. The subspace $C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ is dense in $L^p(w)$ by [F4], and for every $g\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ — a Schwartz function — the truncations converge almost everywhere by [F5]. The final clause of [F3] applied to the Hilbert kernel and to each Riesz kernel therefore gives, for every $f\in L^p(w)$, an almost-everywhere limit $Hf$ or $R_{j}f$ satisfying the displayed $L^p(w)$ bounds; these limits define the stated bounded extensions. For $w\in A_1$ and $f\in L^1(w)$ the same closure argument applies with the weak $(1,1)$ bound in place of the strong bound: for $g\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$, ${\mathrm{lim\hspace{0.17em}sup}}_{\epsilon ,{\epsilon }^{\mathrm{\prime }}\downarrow 0}\mathrm{\mid }{T}_{\epsilon }f-T_{{\epsilon }^{\mathrm{\prime }}}f\mathrm{\mid }\le 2T^{\ast }(f-g)$ off the null set where the truncations of $g$ converge, and $w(\{T^{\ast }(f-g)>\eta \})\le C(n,\delta ,[w{]}_{A_1})(A_1+A_2^{\mathrm{\prime }}+A_3+B){\eta }^{-1}\mathrm{\parallel }f-g{\mathrm{\parallel }}_{L^1(w)}$ tends to zero by density [F4], so the truncations converge $w$-almost everywhere and the limit obeys $\mathrm{\mid }Tf\mathrm{\mid }\le T^{\ast }f$.