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CES`ARO SUMMABILITY AND LEBESGUE OF HIGHER DIMENSIONAL FOURIER SERIES

. We give four generalizations of the classical Lebesgue’s theorem to multi-dimensional functions and Fourier series. We introduce four new con- cepts of Lebesgue points, the corresponding Hardy-Littlewood type maximal functions and show that almost every point is a Lebesgue point. For four dif- ferent types of summability and convergences investigated in the literature, we prove that the Ces`aro means σ αn f of the Fourier series of a multi-dimensional function converge to f at each Lebesgue point as n → ∞ .

1. Introduction. The almost everywhere convergence of the Fejér means [17] of trigonometric Fourier series of a one-dimensional integrable was proved by Lebesgue [34], more exactly, for almost every x ∈ T as n → ∞, where T denotes the torus and f (k) the kth Fourier coefficient. Moreover, he proved that the convergence holds at every Lebesgue point of f . Some years later M. Riesz [41] generalized this theorem for the so called Cesàro means of one-dimensional integrable functions. For two-dimensional functions, the Fejér means of a function f ∈ L 1 (log L)(T 2 ) converge almost everywhere to f as n → ∞, i.e., as n 1 , n 2 → ∞ (see Zygmund [60]). Considering the convergence on the diagonal or on a cone, only, we get better convergence results. Marcinkiewicz and Zygmund [37] proved that for f ∈ L 1 (T 2 ), σ n1,n2 f → f almost everywhere provided that n is in a cone and n → ∞, i.e., τ −1 ≤ n 1 /n 2 ≤ τ for some τ ≥ 1. The almost everywhere convergence was generalized for d-dimensional functions and for Cesàro means in [50], however, the set of convergence was unknown.
In this paper, we give four generalizations of the one-dimensional Lebesgue theorem to Cesàro means σ α n f of multi-dimensional functions. The Cesàro summability includes the Fejér summation (when α = 1) and was investigated in a great number of papers and books (see e.g. Gát [20,21,22], Goginava [23,24,25], Simon [43,44], Nagy, Persson, Tephnadze and Wall [39,40], Weisz [51,53] and Zygmund [60]). We generalize the Lebesgue points in four different ways and introduce the so called strong Lebesgue points and the (strong) ω-Lebesgue points, where ω > 0. For every type of Lebesgue point, we introduce a different maximal function and study its boundedness on L p spaces. This boundedness result shows that almost every point is a Lebesgue point of f . We consider four different types of Cesàro summability, the unrestricted and restricted rectangular summability, the ∞ and 1 -summability. The main result is that for each generalization lim for each corresponding Lebesgue point and f ∈ L 1 (T d ) (in case of the unrestricted rectangular summability f ∈ L 1 (log L) d−1 (T d )). This implies the convergence of the Cesàro means almost everywhere.
with the usual modification for p = ∞, where λ denotes the Lebesgue measure and we identify the torus T with [−π, π]. The measure of a set H is also denoted by |H|.
The kth Fourier coefficient and the nth partial sum of an integrable function f ∈ L 1 (T) are defined by and respectively. The following theorem due to Riesz [41] is a fundamental result and it can be found in most books about trigonometric Fourier series (e.g., Zygmund [60], Bary [2], Torchinsky [49] or Grafakos [26]).
Du Bois Reymond [11] and Fejér [18] proved the existence of a continuous function f ∈ C(T) and a point x 0 ∈ T such that the partial sums s n f (x 0 ) diverge as n → ∞. Kolmogorov gave an integrable function f ∈ L 1 (T), whose Fourier series diverges almost everywhere or even everywhere (see Kolmogorov [31,32], Zygmund [60] or Grafakos [26]). Theorems 2.1 and 2.2 do not hold for p = 1 or p = ∞. In these endpoint cases, we have to consider a summability method. The most known summability method is the Fejér summation. In this paper, we study its generalization, the so called Cesàro summability.
It is easy to see that A α 0 = 1, A 0 n = 1 and A 1 n = n + 1 (n ∈ N). For f ∈ L 1 (T), n ∈ N and α > 0, let us define the Cesàro means σ α n f of the Fourier series of f and the Cesàro kernel K α n by respectively. The Cesàro means are also called (C, α)-means. It is easy to see that If α = 1, then we get back the Fejér means: The next theorem extends Theorem 2.1. If f is a continuous function, then the theorem holds also for p = ∞.
In the proof, we use the important inequality K α n (t) ≤ C min n, 1 n α |t| α+1 =: g α n (t), (n ∈ N, t ∈ (−π, π), t = 0) (1) (see Lebesgue [34], Riesz [41] or Zygmund [60]). From this and Minkowki's inequality, it follows that which implies the proof. The almost everywhere convergence is much more complicated. To this end, first we introduce a maximal function. For an integrable function f of one variable, the Hardy-Littlewood maximal function is defined by It is well known (see e.g. Stein [47]) that the maximal operator M is of weak type (1, 1) and bounded on L p (T) (1 < p ≤ ∞), i.e., Now we introduce the concept of Lebesgue points. Theorem 3.2 implies that for almost every x ∈ T and f ∈ L 1 (T). Thus for almost every x ∈ T. Though the definition of Lebesgue points is a stronger condition, we can see in the next theorem that almost every point is a Lebesgue One can see easily that if x is a Lebesgue point of f ∈ L 1 (T), then f (x) and M f (x) are both finite. The next theorem can be found e.g. in Riesz [41] or Zygmund [60]. Here we give a simple proof for this theorem.
for all Lebesgue points of f ∈ L 1 (T).
Proof. First suppose that 0 < α ≤ 1. Set We fix this integer m. Since Using that g α n is non-increasing and its integral is independent of n (see (1)), we estimate A 1 (x) by On the other hand, (1) implies if n is large enough. The case 1 < α < ∞ can be handled in the usual way.
Since every point of continuity is a Lebesgue point, we get Now we generalize these results to higher dimensional Fourier series.

Higher dimensional Fourier series. For a set
consists all measurable functions f for which Obviously, we get back the L p (T d ) spaces if k = 0. We have for all k ∈ P and 1 ≤ p < r ≤ ∞ that We introduce the following notations. For and For higher dimensional integrable functions f , we define the Fourier coefficients by We generalize the one-dimensional partial sums for higher dimensional functions basically in two ways. In the first generalization, we take the sum over the indices k q ≤ n instead of k = −n, . . . , n, where 1 ≤ q ≤ ∞. These sums are called q -partial sums. In the second generalization, we take the sum in each dimension, i.e., over the indices |k 1 | ≤ n 1 , . . . , |k d | ≤ n d . Here, we call the sums rectangular partial sums. The most natural choices, q = 2, q = 1, q = ∞ and the rectangular partial sums are investigated in several papers and books (for q = 2, see e.g. Stein and Weiss [48,47], Davis and Chang [9], Grafakos [26,27,28], Lu and Yan [35], Feichtinger and Weisz [15,16], for q = 1, Berens, Li and Xu [5,6,4,58], Weisz [51,53], for q = ∞, Marcinkiewicz [36], Zhizhiashvili [59], Weisz [51,53], for the rectangular sums, Zygmund [60] and Weisz [51,53]). Since Cesàro summability is considered only for q = 1, q = ∞ and for the rectangular version, in this paper we focus on these cases (see Figure 1).
For f ∈ L 1 (T d ), the nth q -partial sum s q n f (1 ≤ q ≤ ∞) and the nth rectangular partial sum of the Fourier series of f are given by respectively. For n ∈ N d , n → ∞ means that n j → ∞ for each j = 1, . . . , d. If d ≥ 2 and q = 2, then the same result holds only for p = 2 (see Fefferman [14] or Grafakos [26, p. 743] or Lu and Yan [35, p. 743]).
The analogue of Carleson's theorem holds also for the q -partial sums with q = 1, ∞ in higher dimensions (see Fefferman [12,13] and Grafakos [26, p. 231]), but it does not hold for the rectangular partial sums. A counterexample, which proves the next result, can be found in Fefferman [13]. does not hold for any x ∈ T d .
Since Theorems 4.1 and 4.2 are not true for p = 1, we consider again Cesàro summability in the endpoint case.

5.
Higher dimensional Cesàro summability. For f ∈ L 1 (T d ), n ∈ N, α ≥ 0 and q = 1 or q = ∞, the nth q -Cesàro means σ q,α n f of the Fourier series of f and the corresponding q -Cesàro kernel K q,α n are introduced by respectively. We can show that If f ∈ L 1 (T d ), n = (n 1 , . . . , n d ) ∈ N d and α = (α 1 , . . . , α d ) ∈ R d + , we define the nth rectangular Cesàro mean σ α n f and the Cesàro kernel K α n by respectively. We can see that and K α n = K α1 n1 ⊗ · · · ⊗ K α d n d , where the functions K αi ni are the one-dimensional Cesàro kernels. If α = 1, resp. all α i = 1, then we get back the q -, resp. rectangular, Fejér means, resp.
We generalize Theorem 3.1 as follows.
If 0 < α < ∞ and q = 1 or q = ∞, then If f is a continuous function, then the theorem holds also for p = ∞.
The proof for the rectangular summability follows by iteration. Moreover, the proof for the q summation is again based on the inequality The proof of this inequality is much more complicated than in the one-dimensional case (see [51]). In the remaining part of this paper, we will generalize Lebesgue's theorem (Theorem 3.4) to higher dimensional Fourier series in four different ways. We study four different type of summability and for each summation we have to investigate different maximal operators and Lebesgue points.
6. Unrestricted rectangular Cesàro summability. As a first generalization of the maximal operator M , we introduce the strong Hardy-Littlewood maximal function by that M s f = ∞ almost everywhere (see Jessen, Marcinkiewicz and Zygmund [30] and Saks [42]). Thus, in contrary to the one-dimensional case, M s cannot be of weak type (1, 1) if d > 1. However, we know the following weak type inequality (see Chang and Fefferman [8]).
If we take the supremum in M s only over the diagonal, then M is again of weak type (1, 1). Unfortunately we cannot use this operator in the theory of Cesàro summability.
The convergence for f ∈ L 1 (log L) d−1 (T d ) and almost every x ∈ T d , follows from Theorem 6.1 by a usual density argument. Here h → 0 means that h j → 0 for all j = 1, . . . , d. Note that this result does not hold for al f ∈ L 1 (T d ) if d > 1 (see Jessen, Marcinkiewicz and Zygmund [30] and Saks [42]). Motivated by this convergence result, a point For the concept of the Lebesgue and strong Lebesgue points, see e.g. Feichtinger and Weisz [16] and the references therein.

This is not true for
Here is the first generalization of Lebesgue's theorem (see [57]). 7. Restricted rectangular Cesàro summability. Based on the result of Marcinkiewicz and Zygmund mentioned in the introduction, we can hope that we get convergence results for all integrable functions if we consider the Cesàro means and the convergence on a cone, only. To this, we have to find a maximal operator that is of weak type (1, 1). As we mentioned in the preceding section, M f is not suitable. So we start from the definition of the strong maximal operator M s f and introduce its weighted version. For some ω ≥ 0 and f ∈ L 1 (T d ), we define the Hardy-Littlewood type maximal function . Here 2 −ω i 1 is the weight, (2h) d 2 i 1 is the measure of the rectangle we integrate on. If ω = 0, we obtain the strong Hardy-Littlewood maximal function, moreover, if ω = 0 and i 1 = · · · = i d , then the Hardy-Littlewood maximal function M f . The operator M ω,1 is of weak type (1,1) if ω > 0 (see [52]). .
The definition of the strong Lebesgue points presented in the previous section is equivalent to Based on this observation and on the definition of M ω , let Different versions of Lebesgue points were considered in Gabisoniya [19] and Skopina [45,46] for two dimensions. If ω = 0, then we get back the definition of the strong Lebesgue points. Every ω 2 -Lebesgue point is an f . If f is continuous at x, then x is an ω-Lebesgue point of f . The next two theorem were proved in [52,55]. In this section, we investigate the convergence of σ α n f over a cone . . , d}, where τ ≥ 1 is fixed (see Figure 2).   8. ∞ -Cesàro summability. In this section, we generalize the maximal operator M ω,1 f investigated in the previous section. Under a diagonal, we understand a diagonal of the cube [0, π] d . Let us denote by P 2 i 1 h,...,2 i d h a parallelepiped, whose center is the origin and whose sides are parallel to the axes and/or to the diagonals and whose kth side length is 2 i k +1 h if the kth side is parallel to an axis and √ 22 i k +1 h if the kth side is parallel to a diagonal (i ∈ N d , h > 0, k = 1, . . . , d). More exactly, at least one side of P 2 i 1 h,...,2 i d h is parallel to one of the axes and the other sides are parallel to the axes and/or to the diagonals.
For ω > 0 and f ∈ L 1 (T d ), the Hardy-Littlewood type maximal function M ω f is given by  9. 1 -Cesàro summability. In this short section, we use the same maximal operator M ω f and the strong ω-Lebesgue points as in the previous section.