Calderon-Zygmund operators map $H^1$ boundedly into $L^1$

Given: Countable Choice, a fixed sequence ${\delta }_j$ realizing $W$, $n\ge 1$, $0<\delta \le 1$, the fixed $H^1$ kernel $\phi$ and atomic order $N$, the kernel $k$, the principal-value distribution $W$, the operator $T$ and the constants as in the statement.

The Holder bound makes $k$ continuous at each nonzero point: take $y\to 0$ with $2\mathrm{\mid }y\mathrm{\mid }\le \mathrm{\mid }x\mathrm{\mid }$ in the stated difference bound. Thus $k$ is Borel, and its size bound gives integrability on compact sets away from zero. Truncations: for $f\in L^p$, $1\le p<\mathrm{\infty }$, and $0<\epsilon <\mathrm{\infty }$, $T_{\epsilon }f(x)={\int }_{\mathrm{\mid }y\mathrm{\mid }>\epsilon }k(y)f(x-y)dy$ converges absolutely at every $x$; the maximal truncations obey the weak $(1,1)$ bound $\mathrm{\mid }\{T^{\ast }f>\lambda \}\mathrm{\mid }\le C_{n,\delta }(A_1+A_2^{\mathrm{\prime }}+A_3+B){\lambda }^{-1}\mathrm{\parallel }f{\mathrm{\parallel }}_1$ for $f\in L^1$ and the strong $L^p$ bounds for $1<p<\mathrm{\infty }$ (Maximal truncated singular integrals, Maximal truncations: weak (1,1) and strong Lp bounds, Standard Hölder kernels satisfy the Hörmander condition, Standard (Hölder) Calderón–Zygmund kernels).

Fix a sequence ${\delta }_j\downarrow 0$ realizing $W$. For $g\in C_c^{\mathrm{\infty }}$, the definition of $W$ applied to the Schwartz test $g(x-\cdot )$ gives $T_{{\delta }_j}g(x)\to (W\ast g)(x)$ at every $x$. This extends to a.e. sequential convergence for every $f\in L^1$: for $g$ approximating $f$ in $L^1$, the tail oscillation of $(T_{{\delta }_j}f)$ is at most $2T^{\ast }(f-g)$. For every $\eta >0$, [F1] therefore bounds the measure of the set where that oscillation exceeds $\eta$ by $2C{\eta }^{-1}\mathrm{\parallel }f-g{\mathrm{\parallel }}_1$. Density (Complex finite-simple and smooth compact-support density for finite p) makes this zero. Taking a countable sequence of $\eta$ shows that the scalar sequence is Cauchy, hence convergent, a.e. The same argument uses the strong $L^2$ bound for $f\in L^2$. On $C_c^{\mathrm{\infty }}$, dominated convergence with majorant $T^{\ast }g\in L^2$ gives $T_{{\delta }_j}g\to Tg$ in $L^2$; density and the uniform $L^2$ bound of $T^{\ast }$ extend this to all $L^2$. If the principal-value integrals converge along all radii on Schwartz tests, the identical oscillation argument over $0<\epsilon <\eta$ gives the full a.e. limit. The measurable suprema can be reduced to rational radii by absolute convergence away from zero.

Atomic characterisation: every $f\in H^1$ has a representation $f={\sum }_j{\lambda }_j{a}_j$ in ${\mathcal{S}}^{\mathrm{\prime }}$ with $(1,\mathrm{\infty },0)$-atoms $a_j$ and $({\lambda }_j)\in {\mathrm{\ell }}^1$; the series also converges in the $H^1$ norm and one may choose ${\sum }_j\mathrm{\mid }{\lambda }_j\mathrm{\mid }\le C_{n,N,\phi }\mathrm{\parallel }f{\mathrm{\parallel }}_{H^1}$ (Atomic characterisation of real $H^p$ for $0<p\le 1$). A $(1,\mathrm{\infty },0)$-atom is supported in a cube $Q$, satisfies $\mathrm{\mid }a\mathrm{\mid }\le \mathrm{\mid }Q{\mathrm{\mid }}^{-1}$ and $\int a=0$ ($H^p$ atoms with a prescribed moment order).

Complex $L^1$ is complete under Countable Choice (Complex Lp completeness and almost-everywhere subsequences). Norm convergence implies convergence in measure (Convergence in $L^p$ implies convergence in measure). A weak $(1,1)$ difference estimate also gives convergence in measure directly, since $\mathrm{\mid }\{\mathrm{\mid }{u}_j-u\mathrm{\mid }>\eta \}\mathrm{\mid }\le C{\eta }^{-1}\mathrm{\parallel }{f}_j-f{\mathrm{\parallel }}_1\to 0$. Limits in measure are unique: $\{\mathrm{\mid }u-v\mathrm{\mid }>\eta \}$ lies in the union of the two error sets at threshold $\eta \mathrm{/}2$, whose measures tend to zero.

Under Countable Choice, an $L^2$-norm convergent sequence has a subsequence of representatives converging almost everywhere to a representative of its limit (Assuming Countable Choice, $L^p$-convergent sequences have almost-everywhere convergent subsequences).

Tonelli's theorem permits interchanging the integrals of nonnegative measurable functions on sigma-finite product measure spaces (Tonelli's theorem for nonnegative measurable functions on a sigma-finite product).

The Calderon-Zygmund kernel and operator conventions are those of Calderón–Zygmund kernels and their associated operators: conditions (1) and (2) are the annular size and Hormander conditions, and condition (3) is the off-support representation by the kernel.

Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Under Countable Choice, a closed cube of side length $L$ in ${\mathbb{R}}^n$ is Lebesgue measurable and has measure $L^n$: its volume is the product of its side lengths (A box in ${\mathbb{R}}^n$ with parameters $a_i\le b_i$ is Lebesgue measurable of measure ${\prod }_{i<n}(b_i-a_i)$, whichever of its faces are included, Axis-parallel rectangles in ${\mathbb{R}}^m$ and their volume).

Proof technique: the near/far atom estimate, the a.e. sequential limit of the truncations, and summation over an atomic representation.