Two-dimensional Hardy-Littlewood theorem for functions with general monotone Fourier coefficients

We prove the Hardy-Littlewood theorem in two dimensions for functions whose Fourier coefficients obey general monotonicity conditions and, importantly, are not necessarily positive. The sharpness of the result is given by a counterexample, which shows that if one slightly extends the considered class of coefficients, the Hardy-Littlewood relation fails.


Introduction
Establishing interconnections between integrability of functions and summability of their Fourier coefficients is the problem which occupies a special place in harmonic analysis.The celebrated Parseval's identity enables us to reduce a wide class of problems concerning functions to those concerning their Fourier series, and vice versa.Although we do not have such equalities for the spaces L p , p ̸ = 2, we can still obtain equivalences of norms of functions and norms of their Fourier series if we impose some additional requirements.Results of this kind are important, in the first place, due to the fact that once such a relation is found, one becomes free to choose if it is handy to deal with functions or with coefficients in this or that case, as if having Parseval's identity (see, e.g.[4,[12][13] and [15,Sec. 7] for applications).The following result by Paley [26] can be considered the starting point for the research in this direction.
Theorem A (Paley, 1931).Let {ϕ n (x)} be an orthonormal system on Throughout the paper, for two functions f and g, the relation f ≳ g (or g ≲ f ) means that there exists a constant C such that f (x) ≥ Cg(x) for all x, and the relation f ≍ g is equivalent to f ≳ g ≳ f (if we write f ≳ a g, this means that the corresponding constant is allowed to depend on a).From now on, we discuss Fourier series only with respect to the trigonometric system.
The ranges of p in Theorem A are sharp, therefore to have both (1) and ( 2) true for all p ∈ (1, ∞), one has to impose some additional requirements.Hardy and Littlewood [18] showed that if we restrict ourselves to sine or cosine series with monotone tending to zero coefficients, then both relations (1) and ( 2) hold for all p ∈ (1, ∞).In this regard, a natural question to ask was: how much can we release the requirement of monotonicity to have ∞ n=1 |c n | p n p−2 ≍ p ∥f ∥ p p (3) still true?This question in turn motivated creation of various extentions of the class of monotone sequences satisfying (3).One of these classes, the so-called general monotone or just GM class [28,Th. 4.2], consists of all sequences {a n } fulfilling the condition for all n.Thus, now we dropped not only the monotonicity condition but even the basic requirement of positivity, keeping though some regularity of our sequences.One can see that GM class can yet be generalized (see [29,Th. 6.2(B)] and [32,Th. 1]) by putting a mean value on the right-hand side of (4) instead of |a n | as follows: for some λ > 1 (see also [14] for some properties of such sequences).Note that these classes and several other ones, defined as (4) but with some other majorants on the right-hand side, in different sources can be also called GM .For a comprehensive survey on the concept of general monotonicity, we refer the reader to [21].
One more direction of extending the obtained results (see [?, ?, 1, 32]) is proving them for weighted spaces.Define weighted Lebesgue spaces L q w(p,q) , p, q ∈ (0, ∞], on [−π, π], as the set of all measurable functions f with finite norm ∥f ∥ L q w(p,q) := The discrete weighted Lebesgue space l q w(p,q) is to be defined in the same way.Now, a weighted version of relation (3) is where p ′ stands for the conjugate to p, that is, 1/p+1/p ′ = 1.Note that if we put q = p, we get the standard Hardy-Littlewood relation (3).From now on, writing Hardy-Littlewood type relations we will omit the dependence on p of the corresponding constants, so this dependence will be taken for granted.The following theorem for weighted Lebesgue spaces was obtained by Sagher [27].
Theorem B (Sagher, 1976).If the sequences {a n } and {b n } are monotone and vanishing at infinity and the function f has the Fourier series .
It turns out that the same holds if we release the monotonicity condition in the theorem above to (5), thus withdrawing the requirement of positivity.This result, along with the similar statement proved for Lorentz spaces, was given by Dyachenko, Mukanov and Tikhonov [9].
So, in the one-dimensional case we have quite a complete picture.
The whole scenario becomes more complicated if we step out from the one-dimensional setting to the multidimensional one, and the first question we face is to determine what we should mean by monotonicity if we deal with multiple sequences.The usual one-dimensional monotonicity is characterized by the inequalities a n ≥ a n+1 , or equivalently, ∆a n := a n −a n+1 ≥ 0. These two ways of writing the same property give rise to the following fundamentally different multidimensional monotonicity concepts.Our focus will be on the two-dimensional case.
Monotonicity in each variable.Likewise a n ≥ a n+1 in one dimension, we can require coordinatewise monotonicity, that is, in two-dimensional case the condition will be It turns out, however, that for such sequence the Hardy-Littlewood relation (3) does not hold for some values of p > 1, namely, we have the following result proved by Dyachenko [6,8].

Monotonicity by Hardy.
The next approach to the multiple concept of monotonicity is to consider the monotonicity in the so-called sense of Hardy (or Hardy-Krause, see [16] and [19], where this concept initially arises).That is, to introduce the following differences ∆ 10 a mn := a mn − a m+1,n , ∆ 01 a mn := a mn − a m,n+1 , ∆ 11 a mn := ∆ 01 (∆ 10 a mn ) = ∆ 10 (∆ 01 a mn ) = a mn − a m+1,n − a m,n+1 + a m+1,n+1 , and recalling the one-dimentional condition ∆a n ≥ 0, generalize it in the following way ∆ 11 a mn ≥ 0 for all m, n.
Theorem D (Móricz, 1990).Let p ≥ 1 and the sequence {a mn } satisfy (8) and (10).a) If ∞ m,n=1 a p mn (mn) p−2 < ∞, then the double sine or cosine series with coefficients {a mn } is the Fourier series of its sum f and a p mn (mn) p−2 ≲ ∥f ∥ p p .
The reader can find Theorem D proved for Vilenkin systems (and hence for the Walsh system) in [30,Sec. 6.3] and [31].
Condition (10) is quite restrictive and one of the closest generalizations of it in, say, GM spirit is the following one Note that if the sequence satisfies (10), then the left-hand side above becomes just equal to a mn .The next result [10, Th. 6B] (see [11] for the proof) extends the one of Móricz.
Theorem E (Dyachenko, Tikhonov, 2007).If a nonnegative sequence {a mn } satisfy (8) and the so-called then the corresponding double sine, cosine, or exponential series converges everywhere on (0, 2π) 2 and is the Fourier series of its sum.Besides, for any It is worth mentioning that the ≳ part was proved without assuming a mn ≥ 0, moreover, it was shown that if However, in the proof of the counterpart the requirement of nonnegativity plays a crucial role.It was noted in [12,Th. 4.1] that following the lines of this proof one can adapt it for a more general class of sequences for which the right-hand side of ( 11) is replaces by ∞ m=⌈k/λ⌉ ∞ n=⌈l/λ⌉ |a mn |/mn, λ > 1.Further, it was shown [33] that some other GM type nonnegative sequences happen to obey the two-sided Hardy-Littlewood relation.We present the result from [33] for weighted spaces.
Theorem F (Yu, Zhou, Zhou, 2012).Let {a mn } be a nonnegative sequence satisfying (8) and the following GM type conditions for some λ ≥ 2, and let f (x, y) := ∞ m,n=1 a mn sin mx sin ny.Then, for any p ∈ [1, ∞), for any function ϕ ∈ Φ with either ϕ In the above result Φ stands for some class of power-like positive functions, which we are not going to specify here.A similar result with a more general GM type positive sequences and some other (not comparable) class of power-like functions was obtained in [5].
The main purpose of this work is to show that for some kinds of double GM sequences we can prove the Hardy-Littlewood theorem without restricting ourselves only to positive sequences.We present two GM type classes for which the two-sided Hardy-Littlewood inequality holds true.
We write that {a mn } ∈ GM c 1 if it satisfies (8) and and {a mn } ∈ GM c 2 , if it satisfies (8) and for all k, l ∈ N and some constant C depending only on the sequence {a mn }.We remark that the letter c in GM c comes from the word "corner", since a set of the kind ] generates a corner on the plane.Note that GM c 1 sequences obey the one-dimensional GM conditions (4) in each variable (see (14) in the proof of Lemma 1), while GM c 2 in one variable satisfy (4), and in another one, the "backward" GM condition.
Note that for [−π, π] 2 the L q w(p,q) -norms take the form From now on, for convenience, we adopt the following notation: using that (sin x) (1) = (sin x) ′ = cos x and (sin x) (0) = sin x, we will write a two-dimensional trigonometric series as a ij mn sin (i) mx sin (j) ny and we will say that {a ij mn } ∞ m,n=1 , i, j = 0, 1, is the sequence of its coefficients.The main result of the paper is the following.
|a ij mn | q (mn) then the corresponding trigonometric series converges everywhere on (0, 2π) 2 and is the Fourier series of its sum, moreover, Sharpness of Theorem 1 for GM c 2 sequences is provided by a counterexample in Theorem 2, which shows that if we restrict the sum on the left-hand side of ( 13) to the rectangle (that is, to the intersection and not the union of the two corresponding strips), which is one of the most natural generalizations of the left-hand side of the GM condition (4), then the ≳ part fails for p > 2 and q ≥ p.

Proof of the Hardy-Littlewood theorem for GM c sequences
For a sequence {a mn } ∞ m,n=1 , we define where c and v depend only on C and T .b) For any sequence where c and v depend only on C and T .
The same holds for ] on the plane and draw all the segments [(k, l), (k + 1, l)] such that a k,l−1 and a k,l have different signs and all the segments [(k, l), (k, l + 1)] such that a k−1,l and a k,l have different signs (call them marked segments).Then our rectangle R is divided by the marked segments into several connected parts corresponding to the terms of {a kl } of the same sign.The interior part of the union of their boundaries has at most b2 n−1 vertical marked segments and at most b2 m−1 horizontal ones.Take a positive integer u such that where τ := 4 T (C + 1) 2 + 1. Divide R into 2 2u equal rectangles of size 2 m−1−u × 2 n−1+u and consider a half of them in a checkerboard pattern.Suppose that there is no rectangle among them containing at most 2 n−1−u /τ vertical marked segments and at most 2 m−1−u /τ horizontal ones.Then we must have which contradicts (17).So, there is a rectangle r with at most 2 n−1−u /τ vertical marked segments and at most 2 m−1−u /τ horizontal ones inside it.Consider the parts corresponding to the terms of {a kl } of the same sign inside r.Call the parts whose boundaries intersect the boundary of r by A-parts, the other ones, by B-parts.Note that there is no marked segment of an A-part inside the rectangle r ]. Indeed, otherwise there would exist a broken line of marked segments with either at least 0.25(α 2 − α 1 ) = 2 m−3−u horizontal segments or at least 0.25(β 2 − β 1 ) = 2 n−3−u vertical ones.But this is impossible, since τ > 4. The area of all B-parts does not exceed 2 m+n−2−2u /τ 2 .Thus, there are at least 2 m+n−4−2u (1 − 4τ −2 ) terms of the same sign in r ′ , so the absolute value of the sum of the terms {a kl } in r ′ is at least which concludes the proof of Lemma 1a) with c := 2 −2u−5 α and v := u + 1.A similar argument is valid for Q m+1,n−1 in Lemma 1b), which completes the proof.□ Remark 1.In the proof of Lemma 1, for GM c 1 class we only used its one-dimensional GM properties (14), and for GM c 2 , the corresponding nonsymmetric relations (namely, (14) with a 2k,l in place of a k,l ).
Remark 2. The claim of Lemma 1a) is no longer true if we substitute the GM c 1 condition (12) Proof.Indeed, consider the sequence where f m (n) we define as follows: For such a sequence, condition (8) obviously holds.Consider a rectangle S mn of the form [m, 2m) × [n, 2n).The only nonzero ∆ 11 a kl in this rectangle are ∆ 11 a m ′ −1,n ′ and ∆ 11 a m ′ n ′ , where n ′ ∈ [n, 2n) : ⌊log 2 (n ′ )⌋ = ⌊log 2 (n ′ − 1)⌋ + 1, i.e. n ′ is a power of two, and Assume that the assertion of Lemma 1 holds.Then there must exist a constant c such that for at least cmn squares [k, k + 2) × [l, l + 2) in any S mn there holds Consider a rectangle S mn with where t > 4m is a positive integer.For any a kl in S mn , we have as m → ∞, which leads to a contradiction.□ Lemma 2. For a function f ∈ L(−π, π), given the representation Proof.The ≲ part is clear, so we have to prove the reverse.We start with the case q < ∞.Noting that for any pair of functions g 1 , g 2 there always holds and recalling that the weight is an even in each variable function, we obtain for i = 0, 1.Similarly, . For q = ∞, the claim follows from the equalities

□
Next we prove a two-dimensional analogue of [9, L. 2.2] (see also the one-dimensional result [27, Th. 2.4] for Lorentz spaces).Note that similar multidimensional results for Lorentz spaces were obtained in [24] and [25].
We start with the part a).Due to Lemma 3 there holds Denote Consider first GM c 1 sequences.Let us fix some T > 1.We call a pair (m, n) good (we write where B mn , (m, n) ∈ G, stands for the set of all pairs (k, l) / ∈ G such that k = m + t, l = n + t for some t ∈ N.
According to the one-dimentional Hardy-Littlewood theorem for GM sequences [9, Th. 1.2], we obtain where g(x) = π −π f (x, y) sin y dy.A similar estimate is valid for J 2 .Consider a pair (m, n) ∈ G∩N 2 .Denote the rectangles we constructed in Lemma 1a) [s 1 mn , s 2 mn ]× [t 1 mn , t 2 mn ], so we have Here we used the inequality valid for any x, y, z, t ∈ C. Hence, using ( 26), we obtain Finally, combining ( 27), the similar estimate for J 2 , and (28), we derive , which concludes the proof of the first part for the case of GM c 1 .

If we replace GM c
1 by GM c 2 , i.e. ( 12) by ( 13), we change the definition of a good pair of numbers to the following one: we call a pair (m, n) good, if either mn = 0 or A m+1,n−1 ≤ T A mn .The rest of the proof is the same in light of Lemma 1b) with the only changes: now B mn , (m, n) ∈ G, stands for the set of all pairs (k, l) / ∈ G such that k = m − t, l = n + t for some t ∈ N and P m−1,n−1 in (28) becomes P m+1,n−1 .
We will provide the proof only for the system {sin mx, sin ny}, the other cases will follow then from boundedness of Hilbert transform in weighted Lebesgue spaces.
For (x, y) Applying condition (12), we derive In turn, (13) yields Hence, in both cases we get Thus, for q < ∞, denoting α := 1 − q/p, we obtain Recall the Hardy-type inequalities for power weights (see, for instance, [20, (0.6), (0.10), (1.102)]) for q ≥ 1: and its dual, Using (30) in each variable we arrive at where we used inequality (15).The similar estimate holds for I 3 .And finally, due to (31) which completes the proof for the case q ∈ [1, ∞).For q = ∞, using (29) we can write sup (xy) We also have and the similar estimate for I 3 .Finally, (mn) which completes the proof of the theorem.□ Remark 3.For the spaces L q w(p,q) (0, 2π) in place of L q w(p,q) (−π, π), the assertion of Theorem 1 still holds for q ≤ p but fails for q > p.
Indeed, for q > p it suffices to consider the one-dimensional sine series We have However, for q ≤ p, there holds x q/p−1 ≳ 1, so that ∥f ∥ L q w(p,q) (0,2π) ≍ ∥f ∥ L q w(p,q) (0,π) + ∥f ∥ L q w(p,q) (π,2π) ≍ ∥f ∥ L q w(p,q) (0,π) ≍ ∥f ∥ L q w(p,q) (−π,π) .The reason of the failure of the Hardy-Littlewood relation here is that the function in case is supposed to be periodic, while a power weight is not.Thus, if one deals with weighted Lebesgue spaces on [0, 2π] 2 , it makes more sense to consider a weight of the type | sin x| α in place of |x| α , which was in fact done by many authors.Note that for a power weight, weighted integrability at 2π is equivalent to integrability at zero without weight, so, as in the example above, one has to additionally check integrability at zero.

Sharpness of the result
Theorem 2. For p > 2, q ≥ p, the claim of Theorem 1a) does not hold if we replace the GM c Proof.Assume that p > 2 and consider the sequence a mn := (−1) δm m γ g m (n), where γ > 0, δ m ∈ {0, 1} are to be chosen later, and g m (n) = g m (n, p ′ ) we define as follows In other words, the functions g m are constructed in the following way.First, we divide [1, ∞) into intervals I j , j = 0, 1, ..., so that I j := {x : 2p ′ j ≤ log 2 x < 2p ′ (j + 1)}.After that consider the lower-triangular infinite down and to the right matrix that is filled by all positive integers in increasing order going down and to the right.Next, for any j we asign it the integer i = i(j) if it is ith column that contains the element j.Fix some m and consider the values g m (1), g m (2), ....While i(j) ̸ = m, we have g m (n) = (−1) δm m −3 n −1/p ′ for n ∈ I j .Once i(j) becomes equal to m for the first time, that is, when log 2 n ≥ m(m + 1)p ′ for the first time, we get g m (n) = 2 −m 2 −3m and this value does not change till i(j) becomes equal to m again and n ∈ I j .When i(j) becomes equal to m for the (s + 1)th time, the value g m (n) changes for 2 −(m+s) 2 −3(m+s) (see Figure 1 for a scheme of changes of absolute values of g m (n)).

Figure 1.
Fix n ∈ I j for some j and consider g 1 (n), g 2 (n), ... Let k be such that g m (n) has type 1 if 1 ≤ m ≤ k and type 0 if m ≥ k + 1.Then Let us compare g k (n) and g k+1 (n).There are two cases.Case 1. m 0 = i(j + 1) = k + 1.Then Case 2. m 0 = i(j + 1) < k + 1.Then Thus, in both cases we obtain 0 , whence in light of (33), It remains to note that for a fixed m, we have for and for other n there holds g m (n) ≥ g m (n + 1 with (m, n) ∈ r kl .Note that it can be only of the following five types 0 0 0 0 where 0 stands for the terms with log 2 n < m(m + 1)p ′ , while 1, for those with log 2 n ≥ m(m + 1)p ′ .We will write (m, n) ∈ T i , i = 1, ..., 5, if the corresponding quadruple is of the ith type.Note that As for (m, n) ∈ T 2 ∩ r kl , they all belong to a strip [k ′ , k |a mn | q (mn) Note that our sequence generates the Fourier sine (or cosine) series of an odd (or even) function f that converges in the Pringsheim sense everywhere on (0, 2π) 2 to f according to [6, L. 4].To prove this, since the sequence fulfils (8), it suffices to show that the following sum is finite Let us stick to the case of an odd f , as for cosine series the argument is exactly the same.Denote for m, n ≥ 1,  c mn sin mx sin ny ∆ 01 c mn sin mxD n (y) Combining ( 35) and ( 36) we arrive at ∆ 11 b mn D m (x)D n (y) ∆ 01 c mn sin mxD n (y) L q w(p,q) =: S 1 + S 2 .

,b∆
and b mn := a mn − c mn .Then∥f ∥ L q w(p,q) ≤ ∞ m,n=1b mn sin mx sin ny mn sin mx sin ny = 01 b mn D n (y) + b mN D N