Lebesgue points of ‘ 1 -Cesa`ro summability of d -dimensional Fourier series

We generalize the classical Lebesgue’s theorem and prove that the ‘ 1 -Cesa`ro means of the Fourier series of the multi-dimensional function f 2 L 1 ð T d Þ converge to f at each strong x -Lebesgue point.

Here we partly characterize the set of this convergence. We generalize the Lebesgue points and introduce a new type Lebesgue points, the so called strong x-Lebesgue points. In [24], we verified that almost every point is a strong x-Lebesgue where r a n f denotes the nth ' 1 -Cesàro mean of the Fourier series of f. A similar result was shown for the ' 1 -h-means of Fourier transforms in [25]. However, the hsummability does not contain the Cesàro summability and in this paper we consider Fourier series. So in the present proof, we have to use new ideas and the parts that are similar to [25] are omitted.

Hardy-Littlewood maximal function and strong x-Lebesgue points
Let us fix d ! 3, d 2 N. For a set Y 6 ¼ ;, let Y d be its Cartesian product Y Â Á Á Á Â Y taken with itself d times. We briefly write L p ðT d Þ instead of the L p ðT d ; kÞ space equipped with the norm with the usual modification for p ¼ 1, where k is the Lebesgue measure. We identify the torus T with ½Àp; p. By a diagonal, we understand any diagonal of the two-dimensional faces of the cube ½0; p 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 þ1 h if the kth side is parallel to an axis and ffiffi ffi 2 where the supremum is taken over all parallelepipeds P 2 i 1 h;...;2 i d h ði 2 N d ; h [ 0Þ just defined. Taking the supremum over all parallelepipeds whose sides are parallel to the axes and x ¼ 0, we obtain the strong Hardy-Littlewood maximal function, and, if in addition i 1 ¼ Á Á Á ¼ i d , the usual Hardy-Littlewood maximal function (for more about these maximal functions see e.g. Feichtinger and Weisz [4] and the references therein). We have proved in [24] that and, for 1\p 1, In this paper the constants C and C p may vary from line to line. Based on the Hardy-Littlewood maximal function M x , we introduced the following type of Lebesgue points in [25]. Let where the supremum is taken over all parallelepipeds mentioned above. For Taking the supremum in U x r f over all parallelepipeds whose sides are parallel to the axes and x ¼ 0, we obtain the strong Lebesgue points. Moreover, if in addition Note that every strong x 2 -Lebesgue point is a strong x 1 -Lebesgue point ð0\x 2 \x 1 \1Þ, because of U x 1 r f U x 2 r f . If f is continuous at x, then x is a strong x-Lebesgue point of f. The next theorem was proved in [25].

The kernel functions
For f 2 L 1 ðT d Þ and n 2 N, the nth ' 1 -partial sum s n f of the Fourier series of f and the nth ' 1 -Dirichlet kernel D n are given by respectively. It is known (see e.g. Grafakos [11] or Weisz [23]) that for f 2 L p ðT d Þ, 1\p\1, lim n!1 s n f ¼ f in the L p ðT d Þ-norm and a.e.
Since this convergence does not hold for p ¼ 1, we consider the Cesàro summation. For a 6 ¼ À1; À2; . . . and n 2 N, let A a n :¼ n þ a n ¼ ða þ 1Þða þ 2Þ Á Á Á ða þ nÞ n! : Then A a 0 ¼ 1, A 0 n ¼ 1 and A 1 n ¼ n þ 1 ðn 2 NÞ. Let f 2 L 1 ðT d Þ, n 2 N and a ! 0. The nth ' 1 -Cesàro means r a n f of the Fourier series of f are introduced by r a n f ðxÞ :¼ If a ¼ 0, we get s n f , if a ¼ 1, then the ' 1 -Fejér means It is easy to see that r a n f ðxÞ ¼ dt; where the ' 1 -Cesàro kernel is given by K a n ðtÞ :¼ nÀ1Àj D j ðtÞ: ð3:1Þ It is clear that jD n ðtÞj Cn d ; jK a n ðtÞj Cn d ðt 2 T d Þ: ð3:2Þ The next two lemmas can be found in Zygmund [27]. : Lemma 3 was proved in Weisz [21,22].
Using (3.4), we can easily prove the next lemma by induction (see also [21,25]). For k ¼ 0 the equation is the same as (3.4).
We estimate the Cesàro kernel K a n as follows.

Convergence at strong x-Lebesgue points
Now we are ready to prove our main theorem.
Theorem 2 Suppose that 0\a\1, 0\x\ minða; 1Þ=d and M x f ðxÞ is finite. If f 2 L 1 ðT d Þ is periodic with respect to p and x is a strong x-Lebesgue point of f, then lim n!1 r a n f ðxÞ ¼ f ðxÞ: Proof By Lemma 2, it is enough to prove the theorem for 0\a 1. Since 1 ð2pÞ d=2 Z T d K a n ðtÞ dt ¼ 1; we have r a n f ðxÞ À f ðxÞ 1 jK a n ðtÞ dt: It is enough to integrate on the set We introduce the sets and there exists 1 j d À i þ 1 such that t j À t jþiÀ1 2=n É nA 0 for i ¼ 2; . . .; d À 1, Since x is a strong x-Lebesgue point of f, we can fix a number r\p=2 such that U x r f ðxÞ\. Let us introduce the cubes and suppose that 8=n\r=2. Instead of S, we will integrate on the sets A 0 and where H 0 :¼ H and H 1 :¼ H c denotes the complement of the set H. (3.2) and the periodicity of f imply Z A 0 f ðx À tÞ À f ðxÞ j jK a n ðtÞ dt If t 2 T r=2 , then each t k [ p=2, hence we have to use only the second summand of (3.5). Henceforth, for ðt i l À t j l Þ À1 dt:
If m ¼ d À 1, then the integrals on A a 1 ;...;a m 2 \ B c 1 \ B 2 \ S c r=2 \ T r=2 and A a 1 ;...;a m 2 \ B k 1 1 \ B 2 \ S c r=2 \ T c r=2 are similar to the integrals above. Suppose that 1 m d À 2 and a m þ 1\d. Then let n j ¼ a j þ 1, j ¼ 1; . . .; m and by (4.1)  ðt i l À t j l Þ À1 dt: Here, we integrate first as follows: exceeds the permitted use, you will need to obtain permission directly from the copyright holder.