A Variant of Yano's Extrapolation Theorem on Hardy Spaces

In this note we prove a variant of Yano's classical extrapolation theorem for sublinear operators acting on analytic Hardy spaces over the torus.


Introduction
Let pX, µq and pY, νq be two finite measure spaces. If T : pX, µq Ñ pY, νq is a sublinear operator such that there exist constants C 0 , r ą 0 satisfying sup }g} L p pXq "1 }T pgq} L p pY q ď C 0 pp´1q´r, (1.1) for every 1 ă p ď 2, then a classical theorem of S. Yano [17] asserts that }T pf q} L 1 pY q ď A`B ż X |f pxq| log r p1`|f pxq|qdµpxq (1.2) for all simple functions f on X, where A, B ą 0 are constants depending only on C 0 , r, µpXq, and νpY q. See also Theorem 4.41 in Chapter XII of [19]. In this note we prove a version of the aforementioned extrapolation theorem of Yano for sublinear operators acting on functions belonging to analytic Hardy spaces over the torus. Namely, we prove that if T is a sublinear operator acting on functions defined over the torus such that its operator norm from pH p pTq, }¨} L p pTq q to pL p pTq, }¨} L p pTq q behaves like pp´1q´r as p Ñ 1`, then T satisfies an inequality analogous to (1.2) for functions in Hardy spaces, see Theorem 1 below. Here, for 1 ď p ď 8, H p pTq denotes the analytic Hardy space H p on T given by H p pTq " tf P L p pTq : p f pnq " 0 for n ă 0u.
The study of such variants of Yano's extrapolation theorem in the present paper is motivated by some classical results of S. Pichorides [14] and A. Zygmund [18,Theorem 8] on mapping properties of the Littlewood-Paley square function S "near" H 1 pTq. Recall that given a trigonometric polynomial f on T, the classical Littlewood-Paley square function Spf q of f is defined as where ∆ 0 pf qpθq :" p f p0q, θ P T and for k P N, for θ P T. In [18], Zygmund showed that for every function f P H 1 pTq one has and in [14], Pichorides showed that sup gPH p pTq: as p Ñ 1`. Both of these results were originally proved by using one-dimensional complex-analytic techniques such as the analytic factorisation of Hardy spaces on the torus. Recently, in [2] the aforementioned results of Pichorides and Zygmund were extended to higher dimensions in a unified way using "real-variable" techniques. In particular, it is shown in [2] that both (1.3) and (1.4) can be obtained using a result of T. Tao and J. Wright on endpoint mapping properties of Marcinkiewicz multiplier operators [16] together with a "Marcinkiewicz-type" interpolation argument for H p pTq spaces due to S. Kislyakov and Q. Xu [9], see also [4]. Therefore, motivated by the above remarks, one is naturally led to ask whether it is possible to deduce the result of Zygmund (1.3) directly from that of Pichorides (1.4) by using some "real-variable" approach or, even more generally, whether a general extrapolation theorem for sublinear operators acting on H p pTq spaces holds true. In this note we prove that this is indeed the case, namely we have the following result.
Theorem 1. Let T be a sublinear operator acting on functions defined over T, namely for all measurable functions f, g on T and each complex number α one has |T pf`gq| ď |T pf q|`|T pgq| and |T pαf q| " |α||T pf q|.
If there exist constants C 0 , r ą 0 such that sup gPH p pTq: for every 1 ă p ď 2, then there exists a constant D ą 0, depending only on C 0 , r, such that }T pf q} L 1 pTq ď D}f } L log r LpTq (1.6) for every analytic trigonometric polynomial f on T.
Note that if T satisfies the assumptions of Theorem 1, then it is easy to see that for every analytic trigonometric polynomial f one has }T pf q} L 1 pTq À }f } L log r`1 LpTq . Indeed, if P denotes the Riesz projection onto non-negative frequencies, namely P is the multiplier operator on T with symbol χ N0 , then it is a standard fact that }P } L p pTqÑL p pTq À pp´1q´1 as p Ñ 1`. Hence, if T is a sublinear operator satisfying (1.5), then }T˝P } L p pTqÑL p pTq À pp´1q´p r`1q as p Ñ 1`. It thus follows from Yano's extrapolation theorem applied to T˝P that T satisfies the aforementioned weaker version of (1.6) where }f } L log r LpTq is replaced by }f } L log r`1 LpTq , f being an analytic trigonometric polynomial. Therefore, for sublinear operators satisfying (1.5), Theorem 1 improves the trivial exponent s " r`1 in L log s LpTq to the optimal one; s " r, see discussion in Subsection 3.1.
The proof of our result is based on Yano's original argument [17] combined with some well-known techniques on interpolation between H p pTq spaces, see [5], [7]. At this point, it is worth mentioning that the main idea in Yano's paper is to decompose a given function f as where f 0 " χ t|f |ă1u f and f n " χ t2 n´1 ď|f |ă2 n u f (for n P N) and then apply the assumption (1.1) to each f n separately for p " p n " 1`1{pn`1q, noting that f n P L pn pTq for n P N 0 . Unfortunately, given a function f P H 1 pTq, if one defines f n as above, then f n are not necessarily in the analytic Hardy space H pn pTq anymore and hence, one cannot apply (1.5) to f n . To surpass this difficulty, the idea is to use an "analytic decomposition of unity" of f " ř n r f n which is due to Kislyakov [7], see also J. Bourgain's paper [5]. In particular, each r f n may be regarded as an appropriate "H 8 -replacement" of f n in the sense that r f n P H 8 pTq, f " ř n r f n and each r f n "essentially behaves like" f n , f n being as above. The proof of Theorem 1 is given in Section 2. In Section 3 we briefly discuss about some further remarks related to the present work.
Notation. We denote the set of integers by Z. The set of natural numbers is denoted by N and the set of non-negative integers is denoted by N 0 .
We identify functions over the torus T with functions defined on the set r0, 1q. If f P L 1 pTq is such that suppp p f q is finite, then f is said to be a trigonometric polynomial on T. If f is a trigonometric polynomial on T such that suppp p f q Ă N 0 , then we say that f is an analytic trigonometric polynomial on T.
For r ą 0, L log r LpTq denotes the class of all measurable functions f on T satisfying where Φ r pxq " xpr1`logpx`1qs r´1 q px ě 0q, then }¨} L log r LpTq is a norm on L log r LpTq and, moreover, pL log r LpTq, }¨} L log r LpTq q is a Banach space. For more details on Orlicz spaces, see [10]. If x is a real number, then txu denotes its integer part. As usual, log x denotes the natural logarithm of a positive real number x. The logarithm of x ą 0 to the base 2 is denoted by log 2 pxq.
Given two positive quantities X and Y , if there exists a positive constant C ą 0 such that X ď CY we shall write X À Y . If the constant C depends on some parameters s 1 ,¨¨¨, s n then we shall also write X À s1,¨¨¨,sn Y . Moreover, if X À Y and Y À X we write X " Y .

Proof of Theorem 1
Let T be a sublinear operator satisfying (1.5) and let f be a fixed analytic trigonometric polynomial on T. We shall prove that where A, B ą 0 depend only on C 0 , r and not on f . Towards this aim, following [8] (see also [7]), for λ ą 0 consider the function and then define F λ pθq :" 1 a λ pθq`iHpa λ qpθq and G λ pθq :" 1´p1´rF λ pθqs 4 q 4 . Here, H denotes the periodic Hilbert transform. It follows that F λ , G λ P H 8 pTq, see e.g. the proof of [13,Lemma 7.4 where A 0 ą 0 is an absolute constant. In particular, |G λ f | À λ on T and so, in order to define an appropriate "bounded analytic replacement" of χ tλď|f |ă2λu f , one is led to consider functions of the form pG 2λ´Gλ qf . More precisely, arguing as in the proof of [8,Lemma 4.2], consider the functions p r f n q nPN0 in H 8 pTq given by r f 0 :" G 1 f and r f n :" pG 2 n´G 2 n´1 qf for n P N. Note that there exists an N P N, depending on f , such that r f k " 0 for all k ě N . Indeed, since f is an analytic trigonometric polynomial, if we take N P N such that 2 N´1 ą }f } L 8 pTq , then for every k ě N one has |f pθq| ď 2 k´1 for all θ P T. Hence, for every k ě N one has a 2 k´1 " a 2 k " 1 on T. Therefore, G 2 k´1 " G 2 k " 1 on T and we thus deduce that r f k " pG 2 k´G 2 k´1 qf " 0 on T whenever k ě N . So, one has the decomposition f pθq " N ÿ n"0 r f n pθq for all θ P T. Next, as in Yano's paper [17], using the sublinearity of T and then Hölder's inequality, one deduces that where p n " 1`1{pn`1q. Hence, using our assumption (1.5) for each 0 ď n ď N , one gets where A 1 0 ą 0 is an absolute constant, independent of f and n. Hence, we have and we thus deduce that where the implied constant is independent of n. Using now the elementary inequality t pn`1q{pn`2q ď e r`2 t`pn`1q´p r`2q which is valid for all t ą 0 and n ě 0, we obtain where the implied constant depends only on r and not on f, n. Moreover, note that by using (2.4) for n " 0, one has } r f 0 } L 1 pTq À 1. (2.6)
Therefore, to prove (2.1) it suffices to show that N ÿ n"0 pn`1q r } r f n } L 1 pTq À r 1`ż T |f pθq| log r p1`|f pθq|qdθ. (2.7) To this end, the idea is to write Hence, by using Fubini's theorem, we obtain N ÿ n"1 pn`1q r I p2q n À ż T´2`t log 2 p|f pθq|`1qu ÿ n"1 2 n pn`1q r¯d θ À r 1`ż T |f pθq| log r p1`|f pθq|qdθ, as desired. It remains to prove (2.8) for i " 1. Towards this aim, note that since G 2 n´G 2 n´1 " p1´rF 2 n´1s 4 q 4´p 1´rF 2 n s 4 q 4 and |F 2 n´1| ď 1, |F 2 n | ď 1 on T, it follows that on T, where the implied constant is independent of n. Hence, we have To prove (2.9), we shall again make use of arguments from [8] (see also [7]); since the periodic Hilbert transform H of any constant function on T is identically 0, one may write I p1,αq n " 2 n´1 ż t|f |ă2 n´1 uˇp 1´a 2 n pθqq`iHp1´a 2 n qpθq a 2 n pθq`iHp1´a 2 n qpθqˇˇˇ2 dθ and since a 2 n " 1 on t|f | ă 2 n´1 u, it follows that Hence, by using the L 2 -boundedness of H, one gets and this completes the proof of (2.9). The proof of (2.10) is completely analogous. It thus follows from (2.9) and (2.10) that I p1q n À 2 n{3 ż t|f |ě2 n u |f pθq| 2{3 dθ and hence, by using Fubini's theorem, we have Therefore, the proof of (2.8) for i " 1 is complete. So, (2.8) holds for i " 1, 2 and hence, (2.7) is also true. We have thus shown that (2.1) holds for every analytic trigonometric polynomial on the torus. To complete the proof of Theorem 1, note that (1.6) follows directly from (2.1) using a simple scaling argument. Indeed, if f is an analytic trigonometric polynomial on T with }f } L log r LpTq " 1, then ş T |f pθq| log r p1`|f pθq|qdθ À r 1. Hence, one deduces from (2.1) that if }f } L log r LpTq " 1 then }T pf q} L 1 pTq ď D, where D ą 0 is an absolute constant depending only on C 0 , r. In the general case, if f is a non-zero analytic trigonometric polynomial on T then, using the previous implication and the scaling invariance of T , (1.6) follows.

Some Further Remarks
3.1. Sharpness of Theorem 1. As mentioned in the introduction, Zygmund's inequality (1.3) on the classical Littlewood-Paley square function can now be obtained as a corollary of Pichorides's result (1.4) via Theorem 1. Moreover, the exponent r " 1 in the Orlicz space L log r LpTq in (1.3) cannot be improved, see [2]. Therefore, the example of the Littlewood-Paley square function S shows that Theorem 1 is in general sharp in the following sense; there exists a sublinear operator T and an r " rpT q ą 0 such that }T } pH p pTq,}¨} L p pTq qÑpL p pTq,}¨} L p pTq q " pp´1q´r as p Ñ 1`and }T pf q} L 1 pTq À }f } L log s LpTq holds for every analytic trigonometric polynomial f when s " r but it does not hold for any exponent s ă r. In particular, for the Littlewood-Paley square function S one has r " rpSq " 1.
Note that if one removes the analyticity assumptions, then the behaviour of S "near" L 1 pTq is different than the one mentioned above. In particular, Bourgain showed in [6] that the L p pTq Ñ L p pTq operator norm of S behaves like pp´1q´3 {2 as p Ñ 1`. Moreover, f P L log 3{2 LpTq implies that Spf q P L 1 pTq and the exponent r " 3{2 in L log 3{2 LpTq is best possible, see [1].
Another example illustrating Theorem 1 and its sharpness (in the sense discussed above) is given by some classical results of Y. Meyer [12] and A. Bonami [3]. More specifically, consider the following subset of positive integers Λ " t3 k´3m : 0 ď m ď k´1 and k, m P Zu.
Then [3,Corollaire 4] asserts that there exists an absolute constant C ą 0 such that for every q ą 2 one has for all trigonometric polynomials h on T with suppp p hq Ă Λ. Hence, if we consider the multiplier operator T Λ on T with symbol χ Λ , namely for every trigonometric polynomial f one has T Λ pf qpθq " ÿ nPΛ p f pnqe i2πnθ for all θ P T, then it follows from (3.1) and duality that T Λ satisfies sup gPL p pTq: }g} L p pTq "1 }T Λ pgq} L p pTq À pp´1q´1 pas p Ñ 1`q (3.2) and However, if we restrict ourselves to analytic Hardy spaces on the torus, then it follows from parts (a) and (c) of Théorème 1 on p. 549-550 in [12] that one has the improved bounds sup gPH p pTq: and }T Λ pf q} L 1 pTq À }f } L log 1{2 LpTq pf P H 1 pTqq, (3.5) respectively. In particular, the linear operator T Λ satisfies (1.5) and (1.6) in Theorem 1 for r " 1{2. Furthermore, the exponents r " 1{2 in pp´1q´1 {2 in (3.4) and r " 1{2 in L log 1{2 LpTq in (3.5) cannot be improved. Indeed, to see that (3.5) is sharp, note that, by using (3.1) and [15, (1.4.1)], one has }T Λ pf q} L 1 pTq ď }T Λ pf q} L 2 pTq À }T Λ pf q} L 1 pTq (3.6) for every trigonometric polynomial f on T. Assume now that for some r ą 0 one has }T Λ pf q} L 1 pTq À }f } L log r LpTq pf P H 1 pTqq.
To show that r ě 1{2, for a large N P N that will eventually be sent to infinity, consider the function β N P H 8 pTq given by where V n " 2K 2n`1´Kn is the de la Vallée Poussin kernel of order n P N and K n denotes the n-th Féjer kernel, i.e. K n pθq " ř |j|ďn r1´|j|{pn`1qse i2πjθ , θ P T. Since }β N } L log r LpTq À N r and x β N pjq " 1 for all j P N with 3 N ď j ď 3 N`1`2 , it follows from (3.7) and (3.6) that and so, by taking N Ñ 8, one deduces that r ě 1{2, i.e. (3.5) is sharp. Similarly, one shows that the exponent r " 1{2 in pp´1q´1 {2 in (3.4) as well as the exponents r " 1 in pp´1q´1 in (3.2) and r " 1 in L log LpTq in (3.3) are best possible.

3.2.
Extension of Theorem 1 to Hardy-Orlicz spaces. If T satisfies the assumptions of Theorem 1 and, moreover, |T pf q´T pgq| ď |T pf´gq| holds, then one can easily show that (1.6) can be extended to all functions f such that }f } L log r LpTq ă 8 and suppp p f q Ă N 0 . To this end, one adapts e.g. the density argument on p. 120 in Chapter XII of [19]. More precisely, fix a function f with }f } L log r LpTq ă 8 and suppp p f q Ă N 0 . By [11,Proposition 3.4], there exists a sequence of analytic trigonometric polynomials pφ n q nPN that converges to f in pL log r LpTq, }¨} L log r LpTq q. It thus follows from (1.6) that the sequence pT pφ n qq nPN is Cauchy in pL 1 pTq, }¨} L 1 pTq q and so, it converges to some g P L 1 pTq and g is independent of the choice of pφ n q nPN . Hence, if we set T pf q :" g, we deduce that (1.6) holds for f . Therefore, following the terminology of [11], T is uniquely extended as a bounded operator from the Orlicz-Hardy space H Φr pTq to L 1 pTq, where Φ r pxq " xpr1`logpx`1qs r´1 q, x ě 0.

3.3.
Higher-dimensional variants. In [2], both the results of Zygmund and Pichorides were extended to higher dimensions. Hence, in analogy to the onedimensional case, one is naturally led to ask whether versions of Theorem 1 involving Hardy spaces of several variables hold as well. However, it should be mentioned that it does not seem that the methods of this paper can be extended in a straightforward way to operators acting on Hardy spaces in polydiscs.