Celebratio Mathematica

I’ve de­cided to write this es­say about “square func­tions” for two reas­ons. First, their de­vel­op­ment has been so in­ter­twined with the sci­entif­ic work of A. Zyg­mund that it seems highly ap­pro­pri­ate to do so now on the oc­ca­sion of his 80th birth­day. Also these func­tions are of fun­da­ment­al im­port­ance in ana­lys­is, stand­ing as they do at the cross­ing of three im­port­ant roads many of us have trav­elled by: com­plex func­tion the­ory, the Four­i­er trans­form (or or­tho­gon­al­ity in its vari­ous guises), and real-vari­able meth­ods. In fact, the more re­cent ap­plic­a­tions of these ideas, de­scribed at the end of this es­say, can be seen as con­firm­a­tion of the sig­ni­fic­ance Zyg­mund al­ways at­tached to square func­tions.

This is go­ing to be a partly his­tor­ic­al sur­vey, and so I hope you will al­low me to take the usu­al liber­ties as­so­ci­ated with this kind of en­ter­prise: I will break up the ex­pos­i­tion in­to cer­tain “his­tor­ic­al peri­ods”, five to be pre­cise; and by do­ing this I will be able to sug­gest my own views as to what might have been the key in­flu­ences and ideas that brought about these de­vel­op­ments.

One word of ex­plan­a­tion about “square func­tions” is called for. A deep concept in math­em­at­ics is usu­ally not an idea in its pure form, but rather takes vari­ous shapes de­pend­ing on the uses it is put to. The same is true of square func­tions. These ap­pear in a vari­ety of forms, and while in spir­it they are all the same, in ac­tu­al prac­tice they can be quite dif­fer­ent. Thus the meta­morph­os­is of square func­tions is all im­port­ant.

It ap­pears that square func­tions arose first in an ex­pli­cit form in a beau­ti­ful the­or­em of Kaczmarz and Zyg­mund deal­ing with the al­most every­where sum­mab­il­ity of or­tho­gon­al ex­pan­sions. The the­or­em was proved in 1926 as the cul­min­a­tion of sev­er­al pa­pers each had writ­ten at about that time. The the­or­em it­self was an out­growth of what cer­tainly was one of the main pre­oc­cu­pa­tions of ana­lysts at that time, namely the ques­tion of con­ver­gence of Four­i­er series. The prob­lem was the fol­low­ing. Sup­pose $f=f(\theta )$ is a con­tinu­ous func­tion on the circle, $0\le \theta \le 2\pi$, or more gen­er­ally as­sume that $f$ is in $L^2(0,2\pi )$ or even that $f$ is merely in­teg­rable; then does its Four­i­er series $$\begin{array}{}\text{(1)}& \sum a_n{e}^{in\theta }\text{with}{a}_n=\frac{1}{2\pi }{\int }_0^{2\pi }f(\theta )e^{-in\theta }d\theta ,\end{array}$$ con­verge al­most every­where?

A re­lated par­al­lel is­sue was the cor­res­pond­ing ques­tion for a gen­er­al or­thonor­mal ex­pan­sion, but now lim­ited to $f\in L^2$. Thus if $\{{\phi }_n\}$ is an or­thonor­mal sys­tem, and if $$f\sim \sum a_n{\phi }_n\text{with}{a}_n=\int f{\bar{\phi }}_n,$$ where $\sum |a_n{|}^2<\mathrm{\infty }$, then what could be said about the con­ver­gence al­most every­where of $$\begin{array}{}\text{(2)}& \sum _{n=1}^{\mathrm{\infty }}{a}_n{\phi }_n(x)?\end{array}$$

The peri­od we are deal­ing with (1922–1926) was marked by sev­er­al strik­ing achieve­ments in this area, whose es­sen­tial in­terest is not di­min­ished even when viewed from the dis­tant per­spect­ive of more than a half cen­tury. The first res­ult to men­tion was the con­struc­tion by Kolmogorov in 1923 [e3] of an $L^1$ func­tion whose Four­i­er series $\text{(1)}$ di­verged al­most every­where.1 This con­struc­tion made even more press­ing the ques­tion of wheth­er the Four­i­er series $\text{(1)}$ con­verges al­most every­where when (say) $f$ be­longs to $L^2$, a prob­lem that was not solved till more than forty years later. We shall turn to that in a mo­ment, but now we point out that Kolmogorov’s ex­ample put in­to sharp­er re­lief the $L^2$ res­ults for gen­er­al or­thonor­mal de­vel­op­ments that had been ob­tained (in 1922 and 1923) by Rademach­er and Men­shov. They showed that if $$\begin{array}{}\text{(3)}& \sum |a_n{|}^2(\log n{)}^2<\mathrm{\infty }\end{array}$$ then the series $\text{(2)}$ con­verges a.e.

Moreover the con­di­tion $\text{(3)}$ is best pos­sible in the sense that if $\{{\lambda }_n\}$ is mono­ton­ic and $${\lambda }_n/\log n\to 0,$$ then there ex­ists an or­thonor­mal sys­tem $\{{\phi }_n\}$ and ex­pan­sion $\text{(2)}$ which di­verged a.e., while $$\sum |a_n{|}^2{\lambda }_n^2<\mathrm{\infty }.$$

For or­din­ary Four­i­er series it was proved2 that the con­di­tion $\text{(3)}$ could be re­laxed and be re­placed by $$\begin{array}{}\text{(4)}& \sum _{-\mathrm{\infty }}^{\mathrm{\infty }}|a_n{|}^2\log (|n|+2)<\mathrm{\infty }.\end{array}$$

This last res­ult stood un­sur­passed for forty years un­til Car­leson in 1966 showed that in­deed the Four­i­er series of an $L^2$ func­tion con­verged al­most every­where. It may be in­ter­est­ing to note here that the ba­sic tools re­quired for Car­leson’s the­or­em — the prop­er­ties of the Hil­bert trans­form and their re­la­tion with par­tial sums of Four­i­er series — were first brought to light in this early peri­od: Kolmogorov’s proof of the weak-type (1, 1) prop­erty in 1925; M. Riesz’s pa­per of 1927 [e12] con­tain­ing the $L^p$ in­equal­it­ies for con­jug­ate func­tions and par­tial sums; and Be­sicov­itch’s work (in 1923 [e2] and 1926 [e6]) which began the de­vel­op­ment of “real-vari­able” meth­ods for Hil­bert trans­forms.

Against this back­ground we can now state the idea of Kaczmarz and Zyg­mund. It as­serts as a gen­er­al prin­ciple that for an $L^2$ or­thonor­mal ex­pan­sion (i.e., one where $\sum |a_n{|}^2<\mathrm{\infty }$), at al­most all points the sum­mab­il­ity of the series $\sum a_n{\phi }_n(x)$ by one meth­od one has as a con­sequence the sum­mab­il­ity by any oth­er meth­od which is es­sen­tially stronger than con­ver­gence. A spe­cial (but typ­ic­al) case is as fol­lows:

Re­call that the series is Abel sum­mable at $x$ if $$\underset{r\to 1}{lim}-\sum a_n{r}^n{\phi }_n(x)\text{exists}.$$ In ad­di­tion, set­ting $$\begin{array}{rl}{s}_n& =\sum _{k\equiv 0}^n{a}_k{\phi }_k\text{and}\\ {\sigma }_n& =(s_0+s_1+\cdots +s_{n-1})/n,\end{array}$$ the Cesàro sum­mab­il­ity at $x$ means the ex­ist­ence of the lim­it $$\underset{n\to \mathrm{\infty }}{lim}{\sigma }_n(x).$$

If a series is Cesàro sum­mable it is auto­mat­ic­ally Abel sum­mable (an ex­er­cise!), but the con­verse is in gen­er­al not true. To gain a bet­ter idea of the scope of The­or­em 1 let us point out that $${\sigma }_n(x)=\sum _{k=0}^n(1-\frac{k}{n})a_k{\phi }_k(x)$$ and a res­ult sim­il­ar to The­or­em 1 holds when ${\sigma }_n(x)$ is re­placed by $${\sigma }_n^{\epsilon }(x)=\sum _{k=0}^n(1-\frac{k}{n}{)}^{\epsilon }{a}_k{\phi }_k(x)\text{with }\epsilon >0$$ (which cor­res­ponds es­sen­tially to $(C,\epsilon )$ sum­mab­il­ity), but not for $\epsilon =0$ which of course would give the usu­al con­ver­gence.

For the proof of The­or­em 1 Kaczmarz and Zyg­mund used a square func­tion which they in­tro­duced for this pur­pose, namely $$\begin{array}{}\text{(5)}& K(f)=(\sum _{n=2}^{\mathrm{\infty }}n|{\sigma }_n-{\sigma }_{n-1}{|}^2{)}^{1/2}\end{array}$$ with $f\sim \sum a_n{\phi }_n$. The ba­sic fact was the $L^2$ in­equal­ity.

Clearly $${\sigma }_n={\sigma }_{n-1}=\frac{1}{n(n-1)}\sum _{k=0}^{n-1}k{a}_k{\phi }_k,$$ so $$\parallel {\sigma }_n-{\sigma }_{n-1}{\parallel }_2^2\le \frac{c}{n^4}\sum _{k<n}{k}^2|a_k{|}^2,n\ge 2,$$ and thus $$\sum _2^{\mathrm{\infty }}n\parallel {\sigma }_n-{\sigma }_{n-1}{\parallel }_2^2\le c^{\prime }\sum |a_k{|}^2=c^{\prime }\parallel f{\parallel }_2^2,$$ which proves the lemma.

To prove the the­or­em one in­vokes a vari­ant of the clas­sic­al Tauberi­an ar­gu­ment, namely, if $\sum A_n$ is Abel sum­mable and $\sum n{A}_n^2<\mathrm{\infty }$, then $\sum A_n$ con­verges. Now set $A_n={\sigma }_n-{\sigma }_{n-1}$; then the Abel sum­mab­il­ity of $\sum A_n$ fol­lows from the cor­res­pond­ing Abel sum­mab­il­ity of $\sum a_n{\phi }_n$. The Tauberi­an con­di­tion holds at al­most all points be­cause of the lemma, and hence one ob­tains a.e. the con­ver­gence of $\sum ({\sigma }_n-{\sigma }_{n-1})$, prov­ing the the­or­em.

We have seen the first ex­ample of a square func­tion, namely $\text{(5)}$. While here it plays a minor role, its ba­sic char­ac­ter is already re­vealed: Be­cause of the agil­ity of its quad­rat­ic nature it can ex­ploit eas­ily any situ­ation in which or­tho­gon­al­ity might be im­port­ant.

Our scene shifts now from the Con­tin­ent to Eng­land, and to the work of Lit­tle­wood and Pa­ley. Our at­ten­tion will be fo­cused on two im­port­ant series of con­nec­ted pa­pers: three jointly by Lit­tle­wood and Pa­ley 1931–1938 [e14], [e17], [e19], and two by Pa­ley 1932 [e15], [e16]. The in­vest­ig­a­tions de­scribed in these pa­pers were ini­ti­ated sim­ul­tan­eously (the first pa­per in each series was sub­mit­ted in April 1931), but be­cause of Pa­ley’s death in 1933 the fi­nal ver­sions of sev­er­al of the pa­pers were prob­ably Lit­tle­wood’s work alone. It is also in­ter­est­ing to note that no ref­er­ence is made in these pa­pers to the res­ults de­scribed above, and so it is a reas­on­able guess that they were not aware of the pos­sible rel­ev­ance of the ideas of Kaczmarz and Zyg­mund.

The main theme of the Lit­tle­wood–Pa­ley work was to con­sider the “dy­ad­ic de­com­pos­i­tion” of Four­i­er series, namely $$f(\theta )=\sum _{k=0}^{\mathrm{\infty }}{\mathrm{\Delta }}_k(\theta ),$$ with $$\begin{array}{rl}& {\mathrm{\Delta }}_k(\theta )=\sum _{2^{k-1}\le |n|<2^k}{a}_n{e}^{in\theta },k\ge 1;\\ & {\mathrm{\Delta }}_0=a_0.\end{array}$$

Their ba­sic res­ult was that the $L^p$ norm of a func­tion was equi­val­ent with the $L^p$ norm of the square func­tion as­so­ci­ated with its dy­ad­ic de­com­pos­i­tion.

To prove this the­or­em they needed and thus for­mu­lated an “abeli­an” ana­logue, where par­tial sums are re­placed by Abel means, i.e., the Pois­son in­teg­ral of $f=u(r,\theta )$. Thus giv­en $f$, let $\mathrm{\Phi }$ be the holo­morph­ic func­tion in the unit disc with $\mathrm{Re}(\mathrm{\Phi })=u$, and $\mathrm{Im}(\mathrm{\Phi }(0))=0$. They defined an­oth­er square func­tion the “$g$-func­tion” of $f$ by $$g(f)(\theta )=({\int }_0^1(1-r)|{\mathrm{\Phi }}^{\prime }(r‘e^{i\theta }){|}^{2}dr{)}^{1/2}$$ and proved the fol­low­ing

Pa­ley sought a bet­ter un­der­stand­ing of the nature of these prob­lems by con­sid­er­ing vari­ants of The­or­em 2 where the Four­i­er series ex­pan­sion is re­placed by the Walsh–Pa­ley ex­pan­sion. The Walsh–Pa­ley func­tions (called Walsh–Kaczmarz func­tions at that time) are now usu­ally de­scribed as fol­lows. We identi­fy the in­ter­val $[0,1]$ with the com­pact group con­sist­ing of an in­fin­ite product of cop­ies of the two-ele­ment group (via the usu­al bin­ary ex­pan­sion). The char­ac­ters of that group are the Walsh–Pa­ley func­tions. Writ­ing each in­teger as a sum of powers of 2 gives a nat­ur­al enu­mer­a­tion of the char­ac­ters $\{{\phi }_n{\}}_{n=0}^{\mathrm{\infty }}$. If we set $$\begin{array}{rl}& f\sim \sum a_n{\phi }_n\text{and}\\ & {\mathrm{\Delta }}_k=s_{2^k}-s_{2^{k-1}}=\sum _{2^{k-1}<n\le 2^k}{a}_n{\phi }_n\text{with}\\ & {\mathrm{\Delta }}_0=a_0,\end{array}$$ then Pa­ley’s the­or­em reads as

What makes the proof of The­or­em 4 easi­er than that of The­or­em 2 are the vari­ous sim­pli­fic­a­tions in­her­ent in the fact that $\{s_{2^k}(f)\}$ is a mar­tin­gale se­quence. The name “mar­tin­gale” had not yet been coined. Moreover, a sys­tem­at­ic ex­ten­sion of The­or­em 4 from the point of view of mar­tin­gales, and its fur­ther ex­plor­a­tion in the ma­gic­al world of Browni­an mo­tion — all these came much later, as we shall see. However in Pa­ley’s time some of the ar­gu­ments typ­ic­al of mar­tin­gale the­ory were already un­der­stood. Thus it had been ob­served that $s_{2^k}(f)$ was con­stant on each $2^k$ in­ter­vals (of length $2^{-k}$) of the form $$((l-1)/2^k,l/2^k),l=1,\dots ,2^k,$$ and that the value of $s_{2^k}(f)$ on each of these in­ter­vals was the mean-value of $f$ there. From this it is ob­vi­ous when $f\in L^p$, $1\le p\le \mathrm{\infty }$, then $\{s_{2^k}(f)\}$ are bounded in $L^p$ norm; the ana­logue for Four­i­er series is def­in­itely nonob­vi­ous when $1<p<\mathrm{\infty }$, and in fact false when $p=1$ or $p=\mathrm{\infty }$.

We shall now de­scribe the main device Pa­ley used in his proof of The­or­em 4. Pa­ley was, from what one can learn about his life, a man of cour­age and al­most reck­less dar­ing. A hint of that spir­it can be found in his ap­proach to dif­fi­cult math­em­at­ic­al prob­lems. When faced by the proof of an in­equal­ity like $$\begin{array}{}\text{(7)}& \int (\sum |{\mathrm{\Delta }}_k{|}^2{)}^{p/2}dx\le A_p^p\int |f{|}^{p}dx\end{array}$$ where $p$ is e.g. an even in­teger $2r$, he in­stinct­ively sought to face the prob­lem head-on by mul­tiply­ing out the $r$ in­fin­ite sums, and then com­ing to grips dir­ectly with the res­ult­ing mul­ti­tude of terms. This kind of au­da­cious at­tack is not so com­mon in our time when it is easi­er to rely on a vari­ety of soph­ist­ic­ated gad­gets which are house­hold items for the work­ing ana­lyst. But giv­en Pa­ley’s re­source­ful­ness this ap­proach worked mar­velously well. His key ob­ser­va­tion was that $$\begin{array}{}\text{(8)}& \sum _{i_r}\int {\mathrm{\Delta }}_{i_1}^2{\mathrm{\Delta }}_{i_2}^2\cdots {\mathrm{\Delta }}_{i_r}^{2}dx\le \int {\mathrm{\Delta }}_{i_1}^2\cdots {\mathrm{\Delta }}_{i_{r-1}}^2{f}^{2}dx\end{array}$$ where the sum­ma­tion is taken over those $i_r$ for which $i_r>max(i_1,\dots$, $i_{r-1})$, which in turn fol­lows from the mar­tin­gale prop­erty that $$\begin{array}{}\text{(9)}& \int g(x){\mathrm{\Delta }}_k(x)dx=0\end{array}$$ whenev­er $g$ is “meas­ur­able with re­spect to the past”. From $\text{(8)}$ Pa­ley was able to achieve the proof of $\text{(7)}$ in a few strokes.

The same idea in­spired Lit­tle­wood and Pa­ley’s proof of The­or­em 3, al­though the ex­e­cu­tion is more com­plic­ated; a more re­con­dite form of $\text{(8)}$ must be proved, and here noth­ing as simple as $\text{(9)}$ holds. The ap­pro­pri­ate sub­sti­tute must be fash­ioned with care out of Green’s the­or­em in con­junc­tion with the iden­tity $$\mathrm{\Delta }(|\mathrm{\Phi }{|}^2)=4|{\mathrm{\Phi }}^{\prime }{|}^2.$$ With The­or­em 3 proved, Lit­tle­wood and Pa­ley were able to de­duce The­or­em 4, but here also the steps re­quired were not easy. It was only after their the­ory was reex­amined by Zyg­mund and his stu­dent Mar­cinkiewicz, that a clear­er and broad­er view of the whole sub­ject began to emerge. To this we shall now turn.

There are two sig­ni­fic­ant events that marked the peri­od we are now con­cerned with. The first, which even pred­ated the Lit­tle­wood–Pa­ley col­lab­or­a­tion, was the in­tro­duc­tion by Lus­in in 1930 [e13] of his “area in­teg­ral”. The idea of Lus­in seems to have sparked no fur­ther in­terest un­til Mar­cinkiewicz and Zyg­mund took up the sub­ject again about 8 years later. There began a brief but very cre­at­ive peri­od of work by them — a flower­ing of the the­ory where con­nec­tions with a vari­ety of oth­er ideas were brought to light. The second event, a tra­gic one, fol­lowed soon there­after with the death of Mar­cinkiewicz in 1940, and it was left to Zyg­mund alone to re­solve some of the is­sues that their work had led them to.

It may help to cla­ri­fy the de­scrip­tion of the prin­cip­al ideas that Mar­cinkiewicz and Zyg­mund con­trib­uted to the study of square func­tions if we or­gan­ize our present­a­tion in terms of the four main lines along which their work pro­ceeded.

The first sub­ject we shall treat (and the only one that was, strictly speak­ing, joint work) deals with the area in­teg­ral of Lus­in. The defin­i­tion of this is as fol­lows. Sup­pose $\mathrm{\Phi }(z)$ is holo­morph­ic in the unit disc and define $A(\mathrm{\Phi })(\theta )$ by $$\begin{array}{}\text{(10)}& (A(\mathrm{\Phi })(\theta ){)}^2={\int }_{\mathrm{\Gamma }(\theta )}|{\mathrm{\Phi }}^{\prime }(z){|}^{2}dxdy\end{array}$$ with $\mathrm{\Gamma }(\theta )$ a stand­ard “tri­angle” (nontan­gen­tial ap­proach re­gion) in the unit disc with ver­tex at $e^{i\theta }$. Ob­serve that the ex­pres­sion rep­res­ents the area of the im­age of $\mathrm{\Gamma }(\theta )$ un­der the map­ping $z\to \mathrm{\Phi }(z)$, with points coun­ted ac­cord­ing to their mul­ti­pli­city. Lus­in’s dis­cov­ery was that if $\mathrm{\Phi }$ is bounded, then $A(\mathrm{\Phi })(\theta )$ is fi­nite for al­most any $\theta$; more gen­er­ally that $$\begin{array}{}\text{(11)}& \parallel A(\mathrm{\Phi })(\theta ){\parallel }_2\simeq \parallel \mathrm{\Phi }{\parallel }_2\text{if }\mathrm{\Phi }(0)=0.\end{array}$$

Mar­cinkiewicz and Zyg­mund real­ized that on the one hand there was a close ana­logy between the Lit­tle­wood–Pa­ley $g$-func­tion and $A(\mathrm{\Phi })$ (in fact $A$ is a point­wise ma­jor­ant of $g$, and the same kind of $L^p$ in­equal­it­ies held for $A$ as for $g)$; but on the oth­er hand they sur­mised that the par­al­lel between these two square func­tions should not be pushed too far. The main res­ult they ob­tained for $A$ was a loc­al­ized ver­sion of Lus­in’s res­ult. This can be stated as fol­lows. Let $${\mathrm{\Phi }}^{\ast }(\theta )=\underset{z\in \mathrm{\Gamma }(\theta )}{sup}|\mathrm{\Phi }(z)|.$$

The con­verse was proved five years later by Spen­cer,3 namely

A cor­res­pond­ing con­verse for $g$-func­tions is false, and so the area in­teg­ral $A$ has some spe­cial af­fin­it­ies with the bound­ary be­ha­vi­or of $\mathrm{\Phi }$, go­ing bey­ond what it shares with $g$.

The second line of in­vest­ig­a­tion was Zyg­mund’s reex­am­in­a­tion of the Lit­tle­wood–Pa­ley the­or­em for the dy­ad­ic de­com­pos­i­tion of Four­i­er series. His ana­lys­is led him to re­cast and sim­pli­fy the ideas of the proof. These sim­pli­fic­a­tions had im­port­ant con­sequences for later work, as we shall see; but their im­me­di­ate in­terest was that it al­lowed him to con­nect the square func­tion $(\sum |{\mathrm{\Delta }}_k{|}^2{)}^{1/2}$ with the one he and Kaczmarz had con­sidered a dozen years earli­er in their study of sum­mab­il­ity of or­tho­gon­al series (see $\text{(5)}$). We sup­pose that we take the Four­i­er ex­pan­sion and set $$f(\theta )\sim \sum _{n\ge 0}{a}_n{e}^{in\theta },$$ $f\in L^p$, so that $f\in H^p$. If we write as be­fore $$K(f)(\theta )=(\sum _{n\ge 1}n|{\sigma }_n(\theta )-{\sigma }_{n-1}(\theta ){|}^2{)}^{1/2}$$ where $${\sigma }_n(\theta )=\sum _{0\le k<n}(1-\frac{k}{n})a_k{e}^{ik\theta },$$ then we can state the fol­low­ing the­or­em:

The proof of this the­or­em re­quired two steps. First, like that of The­or­em 2, one needed the $L^p$ in­equal­it­ies for the $g$-func­tion (see $\text{(6)}$). Here the ma­jor sim­pli­fic­a­tion was made by Zyg­mund some years later5 and it came in the proof of the fact that $$\parallel g(f){\parallel }_p\le A_p\parallel f{\parallel }_p,$$ when $p>2$. (The case $p=2$ was easy, and the range $p<2$ was re­du­cible to $p=2$ by the ar­ti­fice stand­ard in those days of us­ing Blasch­ke product de­com­pos­i­tions for $H^p$ func­tions.) For the dif­fi­cult case $p>2$ a “square du­al­ity” was used. An in­geni­ous ar­gu­ment shows that whenev­er $\phi \ge 0$, $$\begin{array}{}\text{(12)}& \int g(f{)}^2\phi d\theta \le c\{\int g(f)g(\phi )M(f)d\theta +\int |f{|}^2\phi d\theta \}\end{array}$$ where $M$ is the Hardy–Lit­tle­wood max­im­al func­tion. For $p\ge 4$, $\text{(12)}$ then gives the de­sired res­ult as a con­sequence of the case $p\le 2$ ap­plied to $g(\phi )$. In­cid­ent­ally, the no­tion of square du­al­ity which seems to have ori­gin­ated in this con­text con­tin­ues to find oth­er ap­plic­a­tions of in­terest.

The second sim­pli­fic­a­tion Zyg­mund made was in the man­ner in which one could re­duce the $L^p$ con­trol of $(\sum |{\mathrm{\Delta }}_k{|}^2{)}^{1/2}$ to that of the $g$-func­tion; and in fact a whole list of oth­er square func­tions (in par­tic­u­lar, $(\sum _n|{\sigma }_n-{\sigma }_{n-1}{|}^2{)}^{1/2}$) could be handled in the same way.6 This stream­lin­ing of the proof he found can be said to have led dir­ectly to the “Mar­cinkiewicz mul­ti­pli­er the­or­em”.

In its one-di­men­sion­al form the cel­eb­rated the­or­em that bears Mar­cinkiewicz’s name can be stated as fol­lows. Sup­pose we con­sider a trans­form­a­tion $T$ giv­en by a mul­ti­pli­er se­quence $\{{\lambda }_n{\}}_{-\mathrm{\infty }}^{\mathrm{\infty }}$, defined by $$Tf\sim \sum {\lambda }_n{a}_n{e}^{in\theta }\text{ whenever }f\sim \sum a_n{e}^{in\theta }.$$ Then $T$ is bounded on $L^p,1<p<\mathrm{\infty }$, if (i) the se­quence $\{{\lambda }_n\}$ is bounded, and (ii) if it var­ies boundedly over each dy­ad­ic block; more pre­cisely, $$\sum _{2^k\le |j|<2^{k+1}}|{\lambda }_j-{\lambda }_{j-1}|\le M.$$ (Note that the spe­cial case when the se­quence is con­stant on each dy­ad­ic block is an im­me­di­ate con­sequence of The­or­em 2.) In one di­men­sion the the­or­em’s greatest mer­it is, I be­lieve, in its for­mu­la­tion rather than its proof; the lat­ter is much the same as that of The­or­em 6.

It is in the pas­sage to high­er di­men­sions, however, that one finds the great sig­ni­fic­ance of Mar­cinkiewicz’s work on mul­ti­pli­ers. Its im­port­ance was not only the fact that one could use hitherto one-di­men­sion­al meth­ods to prove $n$-di­men­sion­al res­ults; even more pro­found were the ap­plic­a­tions to oth­er ques­tions, such as es­tim­ates for par­tial dif­fer­en­tial equa­tions, already en­vis­aged at that time. We can now see in ret­ro­spect that Mar­cinkiewicz thus an­ti­cip­ated some of the ba­sic in­equal­it­ies later proved by the the­ory of sin­gu­lar in­teg­rals.7 For sim­pli­city of nota­tion we shall state the Mar­cinkiewicz mul­ti­pli­er the­or­em in the case of two di­men­sions. Con­sider the mul­ti­pli­er op­er­at­or $T$ giv­en by $$Tf\sim \sum {\lambda }_{nm}{a}_{nm}{e}^{i(n\theta +m\phi )}\text{for }f\sim \sum a_{nm}{e}^{i(n\theta +m\phi )}.$$ Let $I_k$ de­note the dy­ad­ic in­ter­val $$\{n\mid 2^{k-1}\le |n|<2^k\}\text{and}{J}_l=\{m\mid 2^{l-1}\le |m|<2^l\}.$$ Write $$\begin{array}{rl}& {\mathrm{\Delta }}_1{\lambda }_{n,m}={\lambda }_{n+1,m}-{\lambda }_{n,m},\\ & {\mathrm{\Delta }}_2{\lambda }_{n,m}={\lambda }_{n,m+1}-{\lambda }_{n,m},\text{ and}\\ & {\mathrm{\Delta }}_{1,2}={\mathrm{\Delta }}_1\cdot {\mathrm{\Delta }}_2.\end{array}$$ Now as­sume the fi­nite­ness of the fol­low­ing four quant­it­ies:

1. $\underset{n,m}{sup}|{\lambda }_{n,m}|$;

2. $\underset{k,m}{sup}\sum _{n\in I_k}|{\mathrm{\Delta }}_1{\lambda }_{n,m}|$, and $\underset{m,l}{sup}\sum _{m\in J_l}|{\mathrm{\Delta }}_2{\lambda }_{n,m}|$; and

3. $\underset{k,l}{sup}\sum _{n\in I_k}\sum _{n\in J_l}|{\mathrm{\Delta }}_1{\mathrm{\Delta }}_2{\lambda }_{n,m}|$.

The last of the four ma­jor lines of in­vest­ig­a­tion con­cern­ing square func­tions that Mar­cinkiewicz and Zyg­mund un­der­took dealt with the at­tempt to find a com­pletely “real-vari­able” ana­logue of the func­tions of Lus­in and Lit­tle­wood–Pa­ley. Start­ing with a func­tion $f$ on the circle, the area in­teg­ral and $g$-func­tions are defined in terms of holo­morph­ic (or har­mon­ic) func­tions whose bound­ary val­ues are re­lated to $f$. Also the dy­ad­ic square func­tion of The­or­em 2 re­quires the Four­i­er ex­pan­sion of $f$. What was de­sired was a vari­ant that could be defined more dir­ectly in terms of the ba­sic real-vari­able op­er­a­tions such as in­teg­ra­tion, dif­fer­en­ti­ation, etc.

After some ex­per­i­ment­a­tion Mar­cinkiewicz hit upon the idea of con­sid­er­ing $$\begin{array}{}\text{(13)}& \mu (F)(x)=({\int }_0^{\pi }|F(x+t)+F(x-t)-2F(x){|}^2\frac{dt}{t^3}{)}^{1/2}\end{array}$$ with $$F(x)={\int }^{x}f(t)dt.$$

It was not dif­fi­cult to see that $$\parallel \mu (F){\parallel }_{L^2}\simeq \parallel f{\parallel }_{L^2}\text{ if }{\int }_0^{2\pi }f(x)dx=0.$$ With this, and us­ing the real-vari­able tools he had already de­veloped, he was able to prove the ana­logue of the the­or­em he and Zyg­mund had found for the area in­teg­ral (The­or­em 5a). The res­ult was as fol­lows.

The ques­tions that arose were first, wheth­er some of the oth­er prop­er­ties of the area in­teg­ral or $g$-func­tion held as well for $\mu$; and, more in­ter­est­ingly, what was the real sig­ni­fic­ance of the Mar­cinkiewicz func­tion. Zyg­mund found an an­swer to the first ques­tion in 1944 [5] when he proved

The ar­gu­ment he de­veloped to show this was not an easy one. He was re­quired to in­voke the most ar­cane of the square func­tions, the func­tion $g^{\ast }$, which Lit­tle­wood and Pa­ley had also stud­ied. He es­tab­lished the $L^p$ in­equal­it­ies for it and showed that it ac­tu­ally was a ma­jor­ant of the Mar­cinkiewicz func­tion. In­cid­ent­ally $g^{\ast }$ is defined by $$(g^{\ast }(\mathrm{\Phi })(\theta ){)}^2={\int }_0^1{\int }_0^{2\pi }|{\mathrm{\Phi }}^{\prime }(r{e}^{i(\theta +\phi )}){|}^2|\frac{1-r}{1-r{e}^{i\phi }}{|}^{2}d\phi dr,$$ and so ma­jor­izes also of the area in­teg­ral $\text{(10)}$, but it takes in­to ac­count “the tan­gen­tial” ap­proach to the bound­ary.8 The prob­lem that re­mained was to dis­cov­er wheth­er there was a con­verse to the loc­al res­ult giv­en by The­or­em 8a, or to put the ques­tion more broadly, to find the mean­ing of the Mar­cinkiewicz func­tion. It was to be al­most twenty more years be­fore an an­swer to that ques­tion would be found.

Start­ing about 1950 a new dir­ec­tion of con­sid­er­able im­port­ance began to emerge in force. Hin­ted at in earli­er work (of Be­sicov­itch and Mar­cinkiewicz, among oth­ers), its thrust was the de­vel­op­ment of “real-vari­able” meth­ods to re­place com­plex func­tion the­ory — that favored ally of one-di­men­sion­al Four­i­er ana­lys­is. What made this new em­phas­is par­tic­u­larly timely, in fact in­dis­pens­able, was that only with tech­niques com­ing from real-vari­able the­ory could one hope to come to grips with many in­ter­est­ing $n$-di­men­sion­al ana­logues of the one-di­men­sion­al the­ory.

The math­em­atician an­im­at­ing this de­vel­op­ment was Ant­oni Zyg­mund. In many ways he set the broad out­lines of the ef­fort, he mastered by his work some of the cru­cial dif­fi­culties, and was throughout the source of in­spir­a­tion for his stu­dents and col­lab­or­at­ors.

A pi­on­eer­ing res­ult in this new dir­ec­tion was Calderón’s ex­ten­sion to ${\mathbf{R}}^n$ of the the­or­em of Mar­cinkiewicz and Zyg­mund con­cern­ing the area in­teg­ral, a sub­ject he had taken up at the sug­ges­tion of Zyg­mund. The set­ting for this is as fol­lows. We let $${\mathbf{R}}_{+}^{n+1}=\{(x,y),x=(x_1,\dots ,x_n)\in {\mathbf{R}}^n,y\in {\mathbf{R}}^{+}\}$$ be the up­per half-space, and sup­pose that $u(x,y)$ is har­mon­ic (with re­spect to the $n+1$ vari­able $x_1,\dots ,$ $x_n,y$). Some­times we shall as­sume that $u$ is in fact the Pois­son in­teg­ral of an ap­pro­pri­ate func­tion $f$ defined on ${\mathbf{R}}^n$, and then we shall write $u=\mathrm{PI}(f)$. We let $\mathrm{\Gamma }=\{(x,y),$ $|x|<y\}$ be a stand­ard cone with ver­tex at the ori­gin, ${\mathrm{\Gamma }}^{\prime }$ its trun­cated ver­sion, ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }\cap \{y<1\}$. For any $\overline{x}\in {\mathbf{R}}^n$, $\mathrm{\Gamma }(\overline{x})$ and ${\mathrm{\Gamma }}^{\prime }(\overline{x})$ will be the cor­res­pond­ing cones with ver­tices at $\overline{x}$. The area in­teg­ral of $u$ is defined by $$\begin{array}{}\text{(14)}& (A(u)(\overline{x}){)}^2={\int }_{\mathrm{\Gamma }(\overline{x})}|\mathrm{\nabla }u{|}^2{y}^{1-n}dxdy\end{array}$$ where $|\mathrm{\nabla }u{|}^2=|\partial u/\partial y{|}^2+\sum _{j=1}^n|\partial u/\partial x_{\partial }{|}^2$.

Sim­il­arly for the loc­al the­ory one needs the ana­logue of $\text{(14)}$ where $\mathrm{\Gamma }(\overline{x})$ is re­placed by ${\mathrm{\Gamma }}^{\prime }(\overline{x})$; this defines $A_{\mathrm{loc}}(u)(\overline{x})$. The max­im­al func­tion $u^{\ast }$ is defined by $$u^{\ast }(\overline{x})=\underset{(x,y)\in \mathrm{\Gamma }(\overline{x})}{sup}|u(x,y)|,$$ and its loc­al ana­logue $u_{\mathrm{loc}}^{\ast }$ is giv­en by re­pla­cing $\mathrm{\Gamma }(\overline{x})$ by ${\mathrm{\Gamma }}^{\prime }(\overline{x})$ in the defin­i­tion.

Calderón’s proof of this the­or­em was pub­lished at the same time (1950) as an­oth­er im­port­ant res­ult he found, namely the ex­ten­sion of Privalov’s the­or­em: $u$ has a nontan­gen­tial lim­it at al­most every $\overline{x}\in {\mathbf{R}}^n$, where $u_{\mathrm{loc}}^{\ast }(\overline{x})<\mathrm{\infty }$. We shall dis­cuss the ideas be­hind the proof of The­or­em 9a later when we take up its con­verse. Now we turn to the “glob­al” ver­sion, i.e., the high­er-di­men­sion­al ana­logue of the Lit­tle­wood–Pa­ley the­or­em (The­or­em 3).

It would be dif­fi­cult after 25 years to re­call the pre­cise thoughts that mo­tiv­ated the proof of The­or­em 9b, nor would it be easy now for one to ap­pre­ci­ate the dif­fi­culties that seemed then to stand in the way. But I do re­mem­ber that those of us who were gradu­ate stu­dents of Zyg­mund in the middle 1950’s were shaped by the event, akin to the Cre­ation, which ap­peared to some of us to be the be­gin­ning of everything im­port­ant: the 1952 Acta pa­per which de­veloped via the Calderón–Zyg­mund lemma, the real vari­able meth­ods giv­ing the ex­ten­sion of the Hil­bert trans­form to $n$-di­men­sions. What was more nat­ur­al, there­fore, than to at­tempt to prove the $L^p$ bounded­ness of $f\to A(u)$ by ad­apt­ing these meth­ods? This idea in­deed worked, al­though the ini­tial com­plic­ated proofs were later much sim­pli­fied. The ana­lys­is suc­ceeded as well for the Mar­cinkiewicz func­tion $\text{(13)}$, and proved also that the map­pings $f\to A(u)$ and $f\to \mu (F)$ were of weak-type (1, 1).

We turn now to the proof of The­or­em 9a. Its one-di­men­sion­al ver­sion (The­or­em 5a) had been done by us­ing com­plex func­tion the­ory, in par­tic­u­lar con­form­al map­pings. So a com­pletely dif­fer­ent ap­proach was needed. The idea be­hind it can be un­der­stood by ex­amin­ing the case $p=2$ of The­or­em 9b, which has an easy proof. A dir­ect cal­cu­la­tion shows that $$\begin{array}{}\text{(15)}& {\int }_{{\mathbf{R}}^n}{A}^2(u)dx=c{\int }_{{\mathbf{R}}_{+}^{n+1}}y|\mathrm{\nabla }u{|}^{2}dxdy,\end{array}$$ where $c$ is the volume of the unit ball. Next we can use the fact that $$|\mathrm{\nabla }u{|}^2=\frac{1}{2}\mathrm{\Delta }(|u{|}^2),$$ and so by Green’s the­or­em $$\begin{array}{rl}{\int }_{{\mathbf{R}}^n}{A}^2(u)dx& =\frac{c}{2}{\iint }_{{\mathbf{R}}_{+}^{n+1}}y\mathrm{\Delta }(|u{|}^2)dxdy\\ & =\frac{c}{2}\int |u(x,0){|}^{2}dx,\end{array}$$ which proves The­or­em 9b for $p=2$, since $u(x,0)=f(x)$. Thus in or­der to con­trol $A_{\mathrm{loc}}(u)(x)$ on a set $E$, it is nat­ur­al to con­sider $${\int }_E{A}_{\mathrm{loc}}^2(u)(x)dx$$ which in turn is dom­in­ated by $$c{\int }_{R(E)}y|\mathrm{\nabla }u{|}^{2}dxdy,$$ where $R(E)$ is a stand­ard “saw­tooth” re­gion in ${\mathbf{R}}_{+}^{n+1}$ based on $E$. At this stage (which is the turn­ing point of the proof) Calderón in­voked Green’s the­or­em for an­oth­er re­gion con­tain­ing $R(E)$, whose Green’s func­tion he could es­sen­tially bound from be­low by $c^{\prime }y$.

To prove the con­verse of The­or­em 9a along these lines ap­peared to re­quire, among oth­er things, ap­pro­pri­ate bounds from above for Green’s func­tion for such re­gions, and that seemed much bey­ond what could be done then.9 What turned out to be the right course of ac­tion was to fin­esse the prob­lem of Green’s func­tion and to pro­ceed dir­ectly with es­tim­ates that fol­lowed from the fi­nite­ness of $${\int }_{R(E)}y|\mathrm{\nabla }u{|}^{2}dxdy.$$ These ar­gu­ments also proved to be use­ful in oth­er situ­ations, as we shall see later. The res­ult ob­tained was

I re­mem­ber quite vividly the ex­cite­ment sur­round­ing the events at the time of this work. It was March 1959, and I had re­turned to the Uni­versity of Chica­go the fall be­fore. Fre­quently I met with my friends Guido Weiss and Mary Weiss, and to­geth­er we of­ten found ourselves in Zyg­mund’s of­fice (Eck­hart 309, two doors from mine). With our teach­er our con­ver­sa­tions ranged over a wide vari­ety of top­ics (not all math­em­at­ic­al) and more than once the sub­ject of square func­tions arose. When this happened the mood would change, if only slightly, as if in de­fer­ence to their spe­cial status, and the en­igma that sur­roun­ded them. I had an idea which seemed prom­ising. But be­fore we could see where it might lead came the spring break. Fur­ther work would have to be held in abey­ance since we were each go­ing our own ways: Zyg­mund trav­elled to Bo­ston to vis­it Calderón; Guido and Mary Weiss, hav­ing bor­rowed my Chev­ro­let, drove to Vir­gin­ia for a va­ca­tion trip; and I went to New York to be mar­ried.

In­flu­enced by the re­newed in­terest in area in­teg­rals, and en­cour­aged by some re­cent work he had done with Mary Weiss,10 Zyg­mund re­turned to the study of the Mar­cinkiewicz in­teg­ral $\text{(13)}$ and the prob­lem of find­ing a con­verse to The­or­em 8a. He was con­vinced that now (more than 20 years after Mar­cinkiewicz’s ori­gin­al work) the time was ripe to see mat­ters to a con­clu­sion. He sug­ges­ted to me that we work on the prob­lem to­geth­er, and of course I was very happy to ac­cept his of­fer. For me this was a unique and re­ward­ing col­lab­or­a­tion — not just be­cause of the spe­cial sat­is­fac­tion one de­rives when ac­cep­ted as an equal by one’s teach­er — but also be­cause as it turned out he did most of the work that really coun­ted!

We real­ized first that The­or­em 8a it­self could be some­what strengthened; what was re­quired was the no­tion of the de­riv­at­ive $F^{\prime }(x)$ ex­ist­ing (at $x$) “in the $L^2$ sense”. Thus $F^{\prime }(x)$ ex­is­ted in this gen­er­al­ized sense if11 $$\begin{array}{}\text{(16)}& \frac{1}{h}{\int }_0^h|\frac{F(x+t)-F(x)}{t}-F^{\prime }(x){|}^{2}dt\to 0,\text{ as }h\to 0.\end{array}$$

The finer ver­sion of The­or­em 8a was then: If $F\in L^2$ had a de­riv­at­ive in the sense of $\text{(16)}$ at each $x\in E$, then $\mu (F)(x)<\mathrm{\infty }$ for al­most every $x\in E$. It was in this form that one might seek a con­verse. The ba­sic plan was to try to make mat­ters turn on the ana­log­ous situ­ation which held for the area in­teg­ral, where one can pass from the fi­nite­ness of a quad­rat­ic ex­pres­sion to the ex­ist­ence of a lim­it. After a series of re­duc­tions we were able to show that at each point $x$ where $\mu (F)(x)<\mathrm{\infty }$ one had $$\begin{array}{}\text{(17)}& {\int }_{|t|\le y}|\frac{{\partial }^{2}u}{\partial y^2}(x+t,y)+\frac{{\partial }^{2}u}{\partial y^2}(x-t,y){|}^{2}dtdy<\mathrm{\infty }\end{array}$$ with $u=\mathrm{PI}(F)$. On the oth­er hand we could show (us­ing The­or­em 5b) that at al­most every $x$ where $$\begin{array}{}\text{(18)}& {\int }_{|t|\le y}|\frac{{\partial }^{2}u}{\partial y^2}(x+t,y){|}^{2}dtdy<\mathrm{\infty }\end{array}$$ the con­clu­sion $\text{(16)}$ ac­tu­ally held.

The ba­sic dif­fi­culty, the pas­sage from $\text{(17)}$ to $\text{(18)}$, was over­come by Zyg­mund us­ing a clev­er “desym­met­riz­a­tion” ar­gu­ment; sev­er­al weeks later he presen­ted me with an es­sen­tially fi­nal draft of the pa­per which he had typed him­self!

There were sev­er­al vari­ants of the fi­nal res­ult — in­volving ex­ten­sions to $n$-di­men­sions, or high­er de­riv­at­ives, or even frac­tion­al de­riv­at­ives. The simplest ver­sion, however, was the fol­low­ing:

We have traced the de­vel­op­ment of square func­tions from their be­gin­nings to a stage where their nature was much bet­ter un­der­stood, in terms of a series of deep the­or­ems that had been ob­tained. Yet it is only more re­cently that their cent­ral role in sev­er­al fields of ana­lys­is has be­come more ap­par­ent. I shall try to de­scribe this very briefly in terms of three spe­cif­ic areas: $H^p$ spaces, sym­met­ric dif­fu­sion semig­roups, and dif­fer­en­ti­ation the­ory in ${\mathbf{R}}^n$.

Be­gin­ning in about 1966 two sep­ar­ate dir­ec­tions of re­search in­volving square func­tions were un­der­taken, and when brought to­geth­er these ul­ti­mately led to a rich har­vest in the the­ory of $H^p$ spaces. The first star­ted with Burk­hold­er’s [e31] ex­ten­sion of Pa­ley’s the­or­em (The­or­em 4 for Walsh–Pa­ley series) to gen­er­al mar­tin­gales. He ob­served that Pa­ley’s ar­gu­ment ex­ten­ded to this gen­er­al set­ting, but also found his own ap­proach which was very dif­fer­ent. He showed that if $$E_k=E(\cdot \mid {\mathcal{F}}_k)$$ are the con­di­tion­al ex­pect­a­tions for an in­creas­ing se­quence of $\sigma$-fields $\{{\mathcal{F}}_k{\}}_{k=0}^{\mathrm{\infty }}$, then with $E_{-1}(f)\equiv 0$, $$\begin{array}{}\text{(19)}& \Vert (\sum _{k=0}^{\mathrm{\infty }}|(E_k-E_{k-1})(f){|}^2{)}^{1/2}{\Vert }_p\simeq \underset{k\to \mathrm{\infty }}{lim}\parallel E_k(f){\parallel }_p,1<p<\mathrm{\infty }.\end{array}$$

Next, in work with Gundy, and later also with Sil­ver­stein, the fol­low­ing ad­vances were made:12 It was shown that $\text{(19)}$ ex­ten­ded to $p\le 1$ if $\underset{k\to \mathrm{\infty }}{lim}\parallel E_k(f){\parallel }_p$ was re­placed with $\parallel \underset{k}{sup}{E}_k(f){\parallel }_p$, for a large class of mar­tin­gales. This class in­cid­ent­ally in­cludes those oc­cur­ring for the Walsh–Pa­ley series, but more im­port­antly these res­ults went over to the (con­tinu­ous para­met­er) mar­tin­gales arising from Browni­an mo­tion ap­plied to har­mon­ic func­tions. To be more pre­cise, let $z_t(\omega )$ de­note the stand­ard Browni­an mo­tion in the com­plex $z$-plane, start­ing at the ori­gin and stopped when reach­ing the unit circle. Here $0\le t<\mathrm{\infty }$ is the time para­met­er, and $\omega$ la­bels the Browni­an path, with $\omega \in \mathrm{\Omega }$, $\mathrm{\Omega }$ be­ing the prob­ab­il­ity space. If $u$ is har­mon­ic in the unit disc, $t\to u(z_t(\omega ))$ is a con­tinu­ous-time mar­tin­gale. Let $$M_B(u)(\omega )=\underset{0\le t<\mathrm{\infty }}{sup}|u(z_t(\omega ))|$$ be the Browni­an max­im­al func­tion, and $S(u)(\omega )$ the mar­tin­gale square func­tion, $$S(u)(\omega )=({\int }_0^{\mathrm{\infty }}|\mathrm{\nabla }u(z_t(\omega )){|}^{2}dt{)}^{1/2}.$$ Their res­ult then was that $$\begin{array}{}\text{(20)}& \parallel Su{\parallel }_{L^p(\mathrm{\Omega })}\simeq \parallel M_B(u){\parallel }_{L^p(\mathrm{\Omega })},0<p<\mathrm{\infty },\end{array}$$ whenev­er $u(0)=0$.

The most strik­ing ap­plic­a­tion of this circle of ideas was a con­clu­sion drawn from $\text{(20)}$, to wit, whenev­er $F=u+iv$ is holo­morph­ic in the unit disc, then $F\in H^p$ if and only if $u^{\ast }\in L^p,0<p<\mathrm{\infty }$.

The second line of re­search began when a more dir­ect con­nec­tion between stand­ard mul­ti­pli­er op­er­at­ors and square func­tion was dis­covered. The res­ult was easy to state. Whenev­er $T$ is a mul­ti­pli­er op­er­at­or of the Mar­cinkiewicz type on ${\mathbf{R}}^n$ (more pre­cisely one that sat­is­fies the kind of con­di­tions put in Hörmander’s ver­sion of that mul­ti­pli­er the­or­em), then the area in­teg­ral cor­res­pond­ing to $T(f)$ is point­wise dom­in­ated by a $g^{\ast }$ func­tion of $f$, i.e., $$\begin{array}{}\text{(21)}& A(Tf)(x)\le c{g}_{\lambda }^{\ast }(f)(x),\end{array}$$ where $$g_{\lambda }^{\ast }(f)(x)=(\int |\mathrm{\nabla }u(x-t,y){|}^2(\frac{y}{y+|t|}{)}^{n\lambda }{y}^{1-n}dydt{)}^{1/2},$$ and $\lambda$ is a para­met­er which de­pends on the nature of the mul­ti­pli­er. An $H^p$ the­ory in ${\mathbf{R}}^n$ had already been ini­ti­ated sev­er­al years be­fore (by the ef­forts of G. Weiss and oth­ers), and us­ing it and $\text{(21)}$ it fol­lowed that these mul­ti­pli­ers also ex­ten­ded to bounded op­er­at­ors on $H^p$.

From these con­sid­er­a­tions it might be guessed that a ba­sic tool for $H^p$ the­ory is the re­la­tion between square func­tions and max­im­al prop­er­ties of (har­mon­ic) func­tions. Here im­port­ant con­tri­bu­tions were made by C. Fef­fer­man. One of the res­ults ob­tained in this dir­ec­tion was the fol­low­ing the­or­em:

In­cid­ent­ally it should be re­marked that the proof used the same ap­proach as its “loc­al” ana­logue, The­or­em 9c, but ad­di­tion­al ar­gu­ments of a quant­it­at­ive nature were of course needed. More re­cently some of these res­ults for square func­tions have been ex­ten­ded to product do­mains, and in this con­text gen­er­al­iz­a­tions of The­or­ems 9 and 11 have been found.13

The semig­roups which are the sub­ject of the title are a fam­ily of op­er­at­ors $\{T^t{\}}_{t\ge 0}$, each bounded and sel­fad­joint on $L^2$, with $T^t$ hav­ing norm $\le 1$ on every $L^p$, $1\le p\le \mathrm{\infty }$, and $$T^{t_1+t_2}=T^{t_1}{T}^{t_2},$$ with $$\underset{t\to 0}{lim}{T}_f^t=f$$ for $f\in L^2$. Some­times the ad­di­tion­al hy­po­theses are made that $T^t(1)=1$, and $T^t$ is pos­it­iv­ity-pre­serving.

The sig­ni­fic­ance of this no­tion de­rives from the many im­port­ant ex­amples of such semig­roups in ana­lys­is, and the many rich prop­er­ties that they share. In fact some of the ba­sic res­ults dis­cussed above have ses­sions val­id in this con­text. Here we men­tion two, a max­im­al the­or­em, and a mul­ti­pli­er the­or­em in the spir­it of Mar­cinkiewicz’s the­or­em (The­or­em 7).

To for­mu­late the mul­ti­pli­er the­or­em we write $T^t$ in terms of its spec­tral de­com­pos­i­tion, $$T^t={\int }_0^{\mathrm{\infty }}{e}^{-\lambda t}dE(\lambda ),$$ where $E(\lambda )$ is a spec­tral res­ol­u­tion on $L^2$. For each bounded Borel meas­ur­able func­tion $m$ on $(0,\mathrm{\infty })$, con­sider the “mul­ti­pli­er” op­er­at­or $T_m$ giv­en by $$T_m={\int }_0^{\mathrm{\infty }}m(\lambda )dE(\lambda ).$$ Here we as­sume that $m$ is of the form $$m(\lambda )=\lambda {\int }_0^{\mathrm{\infty }}M(s)e^{-\lambda s}ds,$$ with $M$ a bounded func­tion.

A key tool used for the proof of both these the­or­ems are the Lit­tle­wood–Pa­ley type func­tions $$g_k(f)(x)=({\int }_0^{\mathrm{\infty }}{t}^{2k-1}|\frac{{\partial }^k}{\partial t^k}{T}^t(f){|}^{2}dt{)}^{1/2}\text{with }k=1,2,\dots .$$ Also for $T_m$ a re­la­tion of the same kind as $\text{(21)}$ holds.14

Prob­ably the most dra­mat­ic ap­plic­a­tions of square func­tions oc­cur in dif­fer­en­ti­ation the­ory. The gen­er­al prob­lem here is to prove that $$\begin{array}{}\text{(22)}& \underset{\mathrm{diam}R\to 0}{lim}\frac{1}{\mu (R)}{\int }_{R}f(x-y)d\mu (y)=f(x)\text{ a.e.}\end{array}$$ where $R$ ranges over a suit­able col­lec­tion $\mathcal{R}$ of sets “centered” at the ori­gin. The clas­sic­al ex­amples of these are (i) where $\mathcal{R}$ is the col­lec­tion of all balls (or cubes) con­tain­ing the ori­gin, and (ii) where $\mathcal{R}$ is the col­lec­tion of all rect­angles con­tain­ing the ori­gin, with sides par­al­lel to the axes. For each of these res­ults a Vi­tali-type cov­er­ing the­or­em has played a de­cis­ive res­ult. Thus it may seem sur­pris­ing that the ali­en no­tion of square func­tions would turn out to be the ap­pro­pri­ate idea in re­lated situ­ations, where cov­er­ing ar­gu­ments were un­avail­ing. In for­mu­lat­ing the res­ults ob­tained this way we shall, as is usu­al, deal with the cor­res­pond­ing max­im­al func­tion $$M_{\mathcal{R}}(f)(x)=\underset{R\in \mathcal{R}}{sup}\frac{1}{\mu (R)}|{\int }_{R}f(x-y)d\mu (y)|,$$ and the pos­sib­il­ity of as­sert­ing in­equal­it­ies of the type $$\begin{array}{}\text{(23)}& \parallel M_{\mathcal{R}}(f){\parallel }_p\le A_p\parallel f{\parallel }_p.\end{array}$$

The proof of each part of this the­or­em re­quires its own square func­tion. We shall not de­scribe these rather com­plic­ated quad­rat­ic func­tions here, but refer the read­er to the lit­er­at­ure for fur­ther de­tails.15

Since the ori­gin­al draft of this es­say was writ­ten two new res­ults were found which use square func­tions in a de­cis­ive way.

The first is the solu­tion of the prob­lem of Cauchy’s in­teg­ral for Lipschitz curves by Coi­f­man, McIn­tosh, and Mey­er [e50]. It is to be noted that in Calderón’s ini­tial work on this prob­lem (1965), square func­tions were already used in a cru­cial way. In par­tic­u­lar the in­equal­ity $$c\parallel F{\parallel }_{H^p}\le \parallel A(F){\parallel }_p,p\le 1,$$ was proved there for this pur­pose.

The second res­ult deals with the stand­ard max­im­al func­tion in ${\mathbf{R}}^n$ $$M_n(f)(x)=\underset{r>0}{sup}\frac{1}{c_n{r}^n}|{\int }_{|y|\le r}f(x-y)dy|,$$ where $c_n$ is the volume of the unit ball in ${\mathbf{R}}^n$.

The ques­tion that arises is, how does the $L^p$ norm of $M_n$ be­have for large $n$? The best that can be proved by the usu­al Vi­tali cov­er­ing ar­gu­ments gives $$\parallel M_n(f){\parallel }_p\le A(p,n)\parallel f{\parallel }_p,1<p,$$ with $A(p,n)\le A(p)2^{n/p}$, which is a large growth as $n\to \mathrm{\infty }$. However much more can be said.

The idea of the proof is to con­sider in ${\mathbf{R}}^m$ the max­im­al func­tions $M_{m,k}$ defined by $$M_{m,k}(f)(x)=\underset{r>0}{sup}\frac{|{\int }_{|y|\le r}f(x-y)|y{|}^{k}dy|}{{\int }_{|y|\le r}|y{|}^{k}dy},k\ge 0.$$ Then if $m$ is so large that $p>m/(m-1)$,

$$\begin{array}{}\text{(24)}& \parallel M_{m,k}(f){\parallel }_p\le A_{p,m}\parallel f{\parallel }_p\end{array}$$

with $A_{p,m}$ in­de­pend­ent of $k$, $k\ge 0$. This fol­lows from The­or­em 12, Part (a). From this The­or­em 13 is ob­tained by lift­ing the $m$-di­men­sion­al res­ult $\text{(24)}$ in­to ${\mathbf{R}}^n$, where $n\ge m$ (and $k=n-m$), by in­teg­rat­ing over the Grass­man­ni­an of $m$-planes in ${\mathbf{R}}^n$ through the ori­gin.