ON Λ-CONVERGENCE ALMOST EVERYWHERE OF MULTIPLE TRIGONOMETRIC FOURIER SERIES1

We consider one type of convergence of multiple trigonometric Fourier series intermediate between the convergence over cubes and the λ-convergence for λ > 1. The well-known result on the almost everywhere convergence over cubes of Fourier series of functions from the class L(ln L)d ln ln ln L([0, 2π)d) has been generalized to the case of the Λ-convergence for some sequences Λ.

Suppose that d is a natural number, T d = [−π, π) d is a d-dimensional torus, and ϕ : [0, +∞) → [0, +∞) is a nondecreasing function. Let ϕ(L)(T d ) be the set of all Lebesgue measurable real-valued functions f on the torus T d such that Let f ∈ L(T d ), k = (k 1 , k 2 , . . . , k d ) ∈ Z d , x = (x 1 , x 2 , . . . , x d ) ∈ R d , and kx = k 1 x 1 + k 2 x 2 + . . . + k d x d . Denote by the kth Fourier coefficient of the function f and by k∈Z d the multiple trigonometric Fourier series of the function f . Let n = (n 1 , n 2 , . . . , n d ) be a vector with nonnegative integer coordinates, and let S n (f, x) be the nth rectangular partial sum of series (1): everywhere. R. Hunt [5] generalized the statement about the almost everywhere convergence of the Fourier series to the class L(ln + L) 2 (T), particularly, to L p (T) with p > 1. P. Sjölin [6] generalized it to the wider class L(ln + L)(ln + ln + L)(T). In [7], the author showed that the condition f ∈ L(ln + L)(ln + ln + ln + L)(T) is also sufficient for the almost everywhere convergence of the Fourier series of the function f . At present, the best negative result in this direction belongs to S.V. Konyagin [8]: if a function ϕ(u) satisfies the condition ϕ(u) = o(u ln u/ ln ln u) as u → +∞, then, in the class ϕ(L)(T), there exists a function with the Fourier series divergent everywhere on T.
Let us now consider the case d ≥ 2, i.e., the case of multiple Fourier series. Let λ ≥ 1. A multiple Fourier series of a function f is called λ-convergent at a point x ∈ T d if there exists a limit lim min{n j :1≤j≤d}→+∞ considered only for vectors n = (n 1 , n 2 , . . . , n d ) such that 1/λ ≤ n i /n j ≤ λ, 1 ≤ i, j ≤ d. The λ-convergence is called the convergence over cubes (the convergence over squares for d = 2) in the case λ = 1 and the Pringsheim convergence in the case λ = +∞, i. e., in the case without any restrictions on the relation between coordinates of vectors n.
N.R. Tevzadze [9] proved that, if f ∈ L 2 (T 2 ), then the Fourier series of the function f converges over cubes almost everywhere. Ch. Fefferman [10] generalized this result to functions from L p (T d ), p > 1, d ≥ 2. P. Sjölin [11] showed that, if a function f is from the class L(ln + L) d (ln + ln + L)(T d ), d ≥ 2, then its Fourier series converges over cubes almost everywhere. The author [12] (see also [13]) proved the almost everywhere convergence over cubes of Fourier series of functions from the class L(ln + L) d (ln + ln + ln + L)(T d ). The best current result concerning the divergence over cubes on a set of positive measure of multiple Fourier series of functions from ϕ(L)(T d ), d ≥ 2, belongs to S.V. Konyagin [14]: for any function ϕ(u) = o(u(ln u) d−1 ln ln u) as u → +∞, there exists a function F ∈ ϕ(L)(T d ) with the Fourier series divergent over cubes everywhere.
On the other hand, Ch. Fefferman [15] constructed an example of a continuous function of two variables, i. e., a function from C(T 2 ) whose Fourier series diverges in the Pringsheim sense everywhere on T 2 . M. Bakhbukh and E.M. Nikishin [16] proved that there exists F ∈ C(T 2 ) such that its modulus of continuity satisfies the condition ω(F, δ) = O ln −1 (1/δ) as δ → +0 and its Fourier series diverges in the Pringsheim sense almost everywhere. A.N. Bakhvalov [17] established that, for m ∈ N and any λ > 1, there is a function F ∈ C(T 2m ) such that the Fourier series of F is λ-divergent everywhere and the modulus of continuity of F satisfies the condition Later on, Bakhvalov [18] proved the existence of a function F ∈ C(T 2m ) satisfying condition (2) and such that its Fourier series is λ-divergent for all λ > 1 simultaneously.
be a nonincreasing sequence of positive numbers. Assume that We will say that a multiple Fourier series of a function f ∈ L(T d ) is Λ-convergent at a point x ∈ T d if there exists a limit lim n∈Ω Λ , min{n j :1≤j≤d}→∞ S n (f, x).
Let us note that, if λ ν ≡ λ − 1 for some λ > 1, then the condition of Λ-convergence turns into the condition of λ-convergence defined above. And if λ ν → 0 as ν → ∞, then the condition of Λ-convergence is weaker than the condition of λ-convergence for any λ > 1.
In the present paper, we obtain the following statement that strengthens the result of [12].
Theorem 1. Assume that a nonincreasing sequence of positive numbers Λ = {λ ν } ∞ ν=1 satisfies the condition and a function ϕ for some u 0 ≥ 0 and any δ > 0. Assume that the trigonometric Fourier series of any function g ∈ ϕ(L)(T) converges almost everywhere on T. Then, for any d ≥ 2, the Fourier series of any function f from the class ϕ(L)(ln Theorem 1 and the result of paper [7] imply the following statement.

Theorem 2. Let a nonincreasing sequence of positive numbers
Then the Fourier series of any function f from the class Suppose that Under the conditions of the theorem (see [12, formula (3.1) and Lemma 3]), there are constants K d > 0 and y d ≥ 0 such that Using (4), we will prove that, for every y > y d and f ∈ ϕ d (L)(T d ), and, for every f ∈ ϕ d+1 (L)(T d ), where A d is independent of f and y; B d is independent of f .
The proof is by induction on d. Consider the base case, i. e., d = 1: statement (5) immediately follows from (4)  Let d ≥ 2. Suppose that statements (5) and (6) hold for d − 1 and let us show that the same is true for d.
First, let us prove the validity of (5). Let n = (n 1 , n 2 , . . . , n d ) ∈ Ω Λ . According to (3), there is an absolute constant C > 0 such that λ ν ν ≤ C for all natural numbers ν. Combining this with the definition of Ω Λ , we obtain that, for all i, j ∈ {1, 2, . . . , d}, Recall that, if n = (n 1 , n 2 , . . . , n d ), then the following representation holds for the nth rectangular partial sum of the Fourier series of the function f : where D n (t) = sin((n + 1/2)t)/(2 sin(t/2)) is the one-dimensional Dirichlet kernel of order n. Let us add to and subtract from the d-dimensional Dirichlet kernel d j=1 D n j (t j ) of order n the sum (here and in what follows, we suppose that all products with an upper index less than a lower one are equal to 1). Rearranging the terms, we obtain .

From this and (8), it follows that
Note that the latter term on the right hand side of (9) is the n 1 th cubic partial sum of the Fourier series of the function f . By (7), for all k ∈ {2, 3, . . . , d} and t ∈ T, we have |D n k (t) − D n 1 (t)| ≤ C.
Combining this with (9), we obtain Applying the definitions of M Λ (f, x) and M (f, x), from the latter estimate, we obtain where M k (f, x) denotes the kth term of the sum on the left hand side of the equality in (10). Let k ∈ {2, 3, . . . , d}. Consider M k (f, x). Denote by g k,t k the function of d − 1 variables that can be obtained from the function f by fixing the kth variable t k : Define Ω Λ as the set of m k = (m 1 , . . . , m k−1 , m k+1 , . . . , m d ) ∈ N d−1 such that m = (m 1 , . . . , m d ) ∈ Ω Λ . Note that, in view of the invariance of Ω Λ with respect to a rearrangement of variables, the set Ω Λ is independent of k. Suppose that n k = (n 1 , . . . , n 1 , n k+1 , . . . , n d ) ∈ N d−1 . Then and M Further, From this, applying the induction hypothesis (more precisely, statement (6) for the dimension d−1) to the inner integral on the right hand part of (11), we obtain According to (10), (13) Combining (13), (4) and (12), we obtain (5) with the constant Now, we only need to prove the validity of statement (6). To this end, let us use statement (5) proved above.
From (5), it follows that the majorant M Λ (f, x) is finite almost everywhere on T d for all f ∈ ϕ d (L)(T d ), in particular, for all f ∈ L 2 (T d ). Applying Stein's theorem on limits of sequences of operators [20, Theorem 1], we see that the operator M Λ (f, · ) is of weak type (2, 2), i.e., there is a constant A 2 d > 0 such that, for all y > 0 and f ∈ L 2 (T d ), Similarly, from [20, Theorem 3], we can obtain the following refinement of statement (5): there is a constantĀ d > 0 such that, for all y ≥ȳ d /2 =Ā d and f ∈ ϕ d (L)(T d ), Further, let f ∈ ϕ d (L)(T d ) and y > 0. Suppose that . Then From this, using the equality Taking into account that g ∈ ϕ d (L)(T d ) and h ∈ L ∞ (T d ) ⊂ L 2 (T d ) and applying estimate (15) to λ g (y/2) and estimate (14) to λ h (y/2), from (16), we obtain (17) Applying Fubibi's theorem to the integrals on the right hand side of (17), we conclude that hence, statement (6) follows easily. Finally, the Λ-convergence of the Fourier series of an arbitrary function from the class ϕ d (L)(T d ) can be obtained from (5) by means of standard arguments (see, for example, [12,Lemma 3]). Theorem 1 is proved.