Harmonic Analysis, Maximal Functions, and the Singular Integrals¶

Job: develop the Hardy–Littlewood maximal / weak (1,1) / covering, the singular integrals / Calderón–Zygmund / Cotlar / T(1) / T(b), the Littlewood–Paley / Besov / Triebel–Lizorkin, the BMO / John–Nirenberg / Carleson, and the Muckenhoupt $A_p$ / weighted — with the Hardy–Littlewood maximal weak (1,1) via the Vitali / the CZ decomposition as the worked example.

Core¶

The Hardy–Littlewood maximal and the weak (1,1)¶

Maximal / weak (1,1) / covering / differentiation	Description
Hardy–Littlewood maximal $Mf$	$Mf(x)=\sup_{r>0}\frac1{
Weak (1,1)	$\mu\{Mf>\lambda\}\le C\\|f\\|_1/\lambda$ (the Vitali / the Besicovitch covering)
Lebesgue differentiation	$Mf\to f$ a.e. (the density, the ergodic / density)

The singular integrals and the Calderón–Zygmund¶

Singular integrals / CZ / Cotlar-T(1)-T(b)	Description
Singular integrals	$Tf=\mathrm{p.v.}\int K(x-y)f(y)\,dy$ (the principal value, the kernel $K$ the singular, the Hilbert / the Riesz)
Calderón–Zygmund	$T$ the weak (1,1) + the $L^p$ for $1<p<\infty$ (the functional analysis / harmonic)
Cotlar / $T(1)$ / $T(b)$	the Cotlar inequality, the $T(1)$ / the $T(b)$ theorem (the David–Journé–Semmes)

The Littlewood–Paley and the Besov / the Triebel¶

LP / Besov / Triebel / embeddings	Description
Littlewood–Paley	the dyadic $f=\sum\Delta_j f$, the square function $Sf=(\sum
Besov / Triebel–Lizorkin	$B^s_{p,q}$ / $F^s_{p,q}$ (the smoothness spaces, the LP / the dyadic, the functional analysis / function spaces)
Hölder / Sobolev / Besov	$C^s$ / $W^{k,p}$ / $B^s_{p,q}$ (the scales, the embeddings)

The BMO and the Carleson¶

BMO / John–Nirenberg / Carleson	Description
BMO	$\|f\|_{\mathrm{BMO}}=\sup_Q\frac1{
John–Nirenberg	the BMO the exponential distribution (the John–Nirenberg, the functional analysis / harmonic)
Carleson measure	the $\mu$ on $\mathbb{R}^{n+1}_+$ the Carleson ($\sup_Q\mu(Q\times(0,r))/

The Muckenhoupt $A_p$ and the weighted inequalities¶

$A_p$ / weighted / Hunt–Muckenhoupt–Wheeden / weighted CZ	Description
$A_p$ weights (Muckenhoupt)	$\sup_Q(\frac1{
Weighted inequalities	$T:L^2(w)\to L^2(w)$ (the $A_p$ the necessary / the sufficient, the Hunt–Muckenhoupt–Wheeden)
Weighted CZ	the weighted $T$ the $L^p(w)$ (the $A_p$ the condition)

See it¶

Hardy–Littlewood maximal weak (1,1) via the Vitali covering and the Calderón–Zygmund decomposition. The maximal $Mf(x)=\sup_{r>0}\frac1{|B_r|}\int_{B_r}|f|$ (the averaging). The weak (1,1): the Vitali / the Besicovitch covering (the fine cover of $\{Mf>\lambda\}$ by balls, the bounded overlap, the $\mu\{Mf>\lambda\}\le C\|f\|_1/\lambda$). The Calderón–Zygmund decomposition (the $f=g+\sum b_j$ the good + the bad, the atoms $b_j$ the mean-zero on the dyadic cubes, the CZ). The singular integrals: the $T$ the weak (1,1) via the CZ decomposition (the $T(g)$ the $L^\infty$ the good, the $T(b_j)$ the bad the $L^1$ the weak, the Calderón–Zygmund). The key point: the maximal the weak (1,1) (the Vitali covering), the CZ the singular integrals (the decomposition, the atoms) — the harmonic analysis the measure-theoretic / the covering.

Check¶

Show that the Hardy–Littlewood maximal the weak (1,1) via the Vitali covering, and the $L^p$ boundedness.

The Hardy–Littlewood maximal $Mf(x)=\sup_{r>0}\frac1{|B_r|}\int_{B_r}|f|$. The weak (1,1): the set $\{Mf>\lambda\}$ (the $x$ with the average $>\lambda$), the fine cover by balls $B_i$ (the $x\in B_i$ with $\frac1{|B_i|}\int_{B_i}|f|>\lambda$), the Vitali / the Besicovitch covering (the bounded overlap: the $\sum|B_i|\le C|\{Mf>\lambda\}|$ the Vitali the subcover, the $|\{Mf>\lambda\}|\le C\sum|B_i|\le C/\lambda\int|f|$ the $\int_{B_i}|f|>\lambda|B_i|$ the $\sum|B_i|\le\|f\|_1/\lambda$). The $\mu\{Mf>\lambda\}\le C\|f\|_1/\lambda$ (the weak (1,1)). The $L^p$: the weak (1,1) + the $L^\infty$ ($Mf\le\|f\|_\infty$) the Marcinkiewicz interpolation $\Rightarrow$ the $L^p$ for $1<p<\infty$ (the $\|Mf\|_p\le C_p\|f\|_p$). The Vitali / the Besicovitch the covering (the $\mathbb{R}^n$ the bounded overlap), the maximal the weak (1,1) + the $L^p$.

Explain the Calderón–Zygmund decomposition and the singular integral the weak (1,1), and the Cotlar / T(1).

The Calderón–Zygmund decomposition: for $f\in L^1$ and $\lambda>0$, the $f=g+\sum b_j$ (the good $g$ the $L^\infty$ $\|g\|_\infty\le C\lambda$, the bad $b_j$ the atoms the mean-zero $\int_{Q_j}b_j=0$ the supported on the dyadic cubes $Q_j$, the $\sum|Q_j|\le\|f\|_1/\lambda$). The singular integral $T$ the weak (1,1): the $T(f)=T(g)+\sum T(b_j)$, the $T(g)$ the $L^\infty$ (the $g$ the $L^\infty$, the $T$ the $L^2$ the bounded $\Rightarrow$ the $T(g)$ the $L^2$ the $L^\infty$), the $T(b_j)$ the bad the weak (the $b_j$ the mean-zero, the $T(b_j)$ the $L^1$ the weak, the kernel the singular the decay), the $\mu\{|Tf|>\lambda\}\le C\|f\|_1/\lambda$ (the weak (1,1), the Calderón–Zygmund). The Cotlar: the Cotlar inequality (the $T$ the $L^p$ the Cotlar the sum of the almost-orthogonal pieces). The $T(1)$ / the $T(b)$: the David–Journé–Semmes (the $T(1)\in\mathrm{BMO}$ / the $T(b)\in\mathrm{BMO}$ the $T$ the $L^2$ the bounded, the $T(1)$ / the $T(b)$ the criterion). So the CZ the decomposition (the atoms), the singular the weak (1,1), the Cotlar / the $T(1)$ / $T(b)$ the criterion.

Pitfalls¶

The maximal the weak (1,1) (the Vitali / the Besicovitch covering, the bounded overlap) — the Vitali the subcover (the bounded overlap), the Besicovitch the $\mathbb{R}^n$ the specific.
The CZ the singular integrals (the decomposition, the atoms, the principal value) — the CZ the $f=g+\sum b_j$ (the good + the bad, the atoms the mean-zero), the singular the weak (1,1) via the CZ.
The $A_p$ the Muckenhoupt (the $A_p$ the necessary / the sufficient, the weighted the Hunt–Muckenhoupt–Wheeden) — the $A_p$ the condition (the weighted the necessary / the sufficient), the Hunt–Muckenhoupt–Wheeden the theorem.
The BMO the John–Nirenberg (the BMO not $L^\infty$, the Carleson the interpolation) — the BMO the bounded mean oscillation (not the $L^\infty$), the John–Nirenberg the exponential, the Carleson the measure / the interpolation.
The Littlewood–Paley / the Besov / the Triebel (the dyadic / the square function, the smoothness spaces, the functional analysis / function spaces) — the LP the dyadic, the Besov / the Triebel the smoothness.

Refs¶

Hardy–Littlewood, "A maximal theorem with function-theoretic applications" (1930); the maximal / the weak (1,1)
Calderón–Zygmund, "On the existence of certain singular integrals" (1952); the CZ decomposition / the singular
David–Journé–Semmes, "Calderón–Zygmund operators" (1985); the $T(1)$ / the $T(b)$; Muckenhoupt, "Weighted norm inequalities" (1972)
John–Nirenberg, "On functions of bounded mean oscillation" (1961); Carleson, "Interpolation by bounded analytic functions" (1962)

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