Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 32 (2016), ISSN

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1 Acta Mathematica Academiae Paedagogicae Nyíregyháziensis 32 (26), ISSN CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES FERENC WEISZ Dedicated to Professor Ferenc Schipp on the occasion of his 75th birthday, to Professor William Wade on the occasion of his 7th birthday to Professor Péter Simon on the occasion of his 65th birthday. Abstract. In this paper we present some results on convergence summability of one- multi-dimensional trigonometric Walsh-Fourier series. The Fejér Cesàro summability methods are investigated. We will prove that the maximal operator of the summability means is bounded from the corresponding classical or martingale Hardy space H p to L p for some p > p. For p = we obtain a weak type inequality by interpolation, which ensures the almost everywhere convergence of the summability means.. Introduction In this survey paper we will consider summation methods for one- multidimensional trigonometric Walsh-Fourier series. Two types of summability methods will be investigated, the Fejér Cesàro or (C, α) methods. The Fejér summation is a special case of the Cesàro method, (C, ) is exactly the Fejér method. In the multi-dimensional case two types of convergence maximal operators will be considered, the restricted (convergence over the diagonal or over a cone or over a cone-like set), the unrestricted (convergence over N d in Pringsheim s sense). We introduce three types of classical martingale Hardy spaces H p prove that the maximal operators of the summability means are bounded from the corresponding H p to L p whenever 2 Mathematics Subject Classification. Primary 42B8, 42C, Secondary 42B3, 42B25, 6G42. Key words phrases. Martingale classical Hardy spaces, p-atom, atomic decomposition, interpolation, Walsh functions, trigonometric functions, Fejér summability, (C, α) summability. This research was supported by the Hungarian Scientific Research Funds (OTKA) No K

2 278 FERENC WEISZ p > p for some p <. For p = we obtain a weak type inequality by interpolation, which implies the almost everywhere convergence of the summability means to the original function. The almost everywhere convergence the weak type inequality are proved usually with the help of a Calderon-Zygmund type decomposition lemma. However, this lemma does not work in higher dimensions. Our method, that can be applied in higher dimension, too, can be regarded as a new method to prove the almost everywhere convergence weak type inequalities. In this survey paper we summarize the results appeared in this topic in the last 2 years. This paper was the base of my talk given at the Conference on Dyadic Analysis Related Fields with Applications, June 24, in Nyíregyháza (Hungary). I would like to thank the referee for reading the paper carefully for his useful comments suggestions. 2. Trigonometric Walsh system We consider either the torus X = T or the unit interval X = [, ), both with the Lebesgue measure λ. We briefly write L p (X) instead of the real L p (X, λ) space equipped with the norm (or quasinorm) ( /p f p := f dλ) p ( < p ), X where λ is the Lebesgue measure. We use the notation I for the Lebesgue measure of the set I. The weak L p (X) space L p, (X) ( < p < ) consists of all measurable functions f for which f p, := sup ρλ( f > ρ) /p <. ρ> Note that L p, is a quasi-normed space. It is easy to see that L p (X) L p, (X) p, p for each < p <. The Rademacher functions are defined by {, if x [, r(x) := ); 2, if x [, ), 2 r n (x) := r(2 n x) (x [, ), n N). The product system generated by the Rademacher functions is the one-dimensional Walsh system: n w n := r k k (n N), where n = n k 2 k, ( n k < 2)

3 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 279 (see Figure ()). In what follows let φ n (x) denote the trigonometric system Figure. Walsh system. e 2πın x (n Z) defined on T or the Walsh system φ n (x) := w n (x) (n N) defined on the unit interval. For the Walsh system let φ n = if n Z \ N. In this paper the constants C p depend only on p may denote different constants in different contexts. 3. Partial sums of one-dimensional Fourier series For an integrable function f L (X) (X = T or X = [, )) its kth trigonometric or Walsh-Fourier coefficient is defined by f(k) := fφ k dλ (k Z). X The definition of the Fourier coefficients can be extended easily to distributions in case of the trigonometric system to martingales in case of the Walsh system (see Weisz [77, 82]). For f L (X) the nth partial sum s n f of the Fourier series of f is introduced by s n f(x) := f(k)φ k (x) = f(x u)d n (u) du (n N), where k n X D n (u) := φ k (u) k n

4 28 FERENC WEISZ is the nth trigonometric or Walsh-Dirichlet kernel (see Figure 2). In case of the Walsh system we use dyadic addition instead of addition /8 2/8 3/8 4/8 5/8 6/8 7/8 (a) The trigonometric Dirichlet kernel (b) The Walsh-Dirichlet kernel Figure 2. The Dirichlet kernels D n with n = 5. It is a basic question as to whether the function f can be reconstructed from the partial sums of its Fourier series. It can be found in most books about Fourier series (e.g., Zygmund [85], Bary [], Torchinsky [62], Grafakos [28], Schipp, Wade, Simon Pál [5]), that the partial sums converge to f in the L p -norm if < p <. Theorem. If f L p (T) for some < p <, then s n f p C p f p (n N) lim s nf = f n in the L p -norm. This theorem is due to Riesz [44] for trigonometric series to Paley [43] for Walsh-Fourier series. One of the deepest results in harmonic analysis is Carleson s result, i.e., the partial sums s n f of the Fourier series converge almost everywhere to f L p (X) ( < p < ). This result is due to Carleson [8] for trigonometric Fourier series for one-dimensional functions f L 2 (T). Later Hunt [3] extended this result to all f L p (T) spaces, < p <. Billard [4], Sjölin [58] Schipp [47, 5] generalized both results for Walsh-Fourier series. Theorem 2. If f L p (X) for some < p <, then sup s n f C p f p n N p lim s nf = f n a.e.

5 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 28 The inequalities of Theorems 2 do not hold if p = or p =, the almost everywhere convergence does not hold if p =. du Bois Reymond proved the existence of a continuous function f C(T) a point x T such that the partial sums s n f(x ) diverge as n. Kolmogorov gave an integrable function f L (T), whose Fourier series diverges almost everywhere or even everywhere (see Kolgomorov [32, 33], Zygmund [85] or Grafakos [28]). The analogous results for Walsh-Fourier series can be found in Schipp [45] Simon [53]. 4. Hardy spaces H p (X) To prove almost everywhere convergence of the summability means introduced in the next section, we will need the concept of Hardy spaces their atomic decomposition. First we consider the classical Hardy spaces for the trigonometric system then the dyadic Hardy spaces for the Walsh system. 4.. The H p (T) classical Hardy spaces. A distribution f is in the classical Hardy space H p (T) ( < p ) if f Hp := sup f P t <, <t p where P t (x) := k= r k e 2πıkx = r 2 + r 2 2r cos 2πx (r := e t, x T) is the periodic Poisson kernel. Since P t L (T), the convolution in the definition of the norms are well defined The H p [, ) dyadic Hardy spaces. By a dyadic interval we mean one of the form [k2 n, (k + )2 n ) for some k, n N, k < 2 n. Given n N x [, ) let I n (x) be the dyadic interval of length 2 n which contains x. The σ-algebra generated by the dyadic intervals {I n (x) : x [, )} will be denoted by F n (n N). It is easy to show that for a martingale f = (f n, n N) we have s 2 nf = f n. We investigate the class of martingales f = (f n, n N) with respect to (F n, n N). For < p the dyadic Hardy space H p [, ) consists of all martingales for which f Hp := sup n N f n <. p 4.3. Atomic decomposition of H p (X). The results of this subsection hold for both the classical the dyadic Hardy spaces. It is known (see e.g. Stein [59] or Weisz [77]) that H p (X) L p (X) ( < p ) H (X) L (X), where denotes the equivalence of spaces norms.

6 282 FERENC WEISZ The atomic decomposition provides a useful characterization of Hardy spaces. A function a L (T) is a classical p-atom if there exists an interval I T such that () supp a I, (2) a I /p, (3) I a(x)xk dx = for all k N with k /p, where denotes the integer part. While a function a L [, ) is called a dyadic p-atom if there exists a dyadic interval I [, ) such that (i), (ii) (iii) with k = hold. The Hardy space H p (X) has an atomic decomposition. In other words, every function (more exactly, distribution resp. martingale) from the Hardy space can be decomposed into the sum of atoms. A first version of the atomic decomposition was introduced by Coifman Weiss [] in the classical case by Herz [29] in the martingale case. The proof of the next theorem can be found in Latter [34], Lu [36], Wilson [83, 84], Stein [59] Weisz [66, 77]. Theorem 3. A distribution (resp. martingale) f is in H p (X) ( < p ) if only if there exist a sequence (a k, k N) of classical (resp. dyadic) p-atoms a sequence (µ k, k N) of real numbers such that µ k p < µ k a k = f in the sense of distributions (resp. martingales). Moreover, ( ) /p f Hp inf µ k p. The only if part of the theorem holds also for < p <. The following result gives a sufficient condition for an operator to be bounded from H p (X) to L p (X) (see Weisz [77, 8], for p =, Schipp, Wade, Simon Pál [5] Móricz, Schipp Wade [4]). For I T let I r be the interval having the same center as the interval I length 2 r I. If I [, ) is a dyadic interval then let I r be a dyadic interval, for which I I r I r = 2 r I (r N). Theorem 4. For each n N, let V n : L (X) L (X) be a bounded linear operator let V f := sup V n f. n N Suppose that X\I r V a p dλ C p for all classical (resp. dyadic) p -atoms a for some fixed r N < p, where the interval I is the support of the atom. If V is bounded from

7 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 283 L p (X) to L p (X) for some < p, then () V f p C p f Hp (f H p (X)) for all p p p. Moreover, if p <, then the operator V is of weak type (, ), i.e., if f L (X) then (2) sup ρ λ( V f > ρ) C f. ρ> Now we give a typical proof of this theorem. If, instead of V, the linear or sublinear operator V satisfies the condition X d \I r V a p dλ C p with p, then we can easily show that V a p C for all p -atoms a. We take an atomic decomposition of f: f = µ k a k, where each a k is a p -atom ( /p µ k ) p C p f. Hp Next (3) V f V f p p µ k V a k µ k p V a k p p C p f p H p. The problem is that this proof is falls because the inequality (3) does not necessarily hold. Indeed, Bownik [6] have given an operator V for which (3) does not hold. Moreover, though the L p -norms of V a are uniformly bounded, V is not bounded from H p (R) to L p (R). The correct proof of Theorem 4 can be found in Weisz [8]. Note that (2) can be obtained from () by interpolation. For the basic definitions theorems on interpolation theory see Bergh Löfström [3] Bennett Sharpley [2] or Weisz [66, 77]. The interpolation of martingale Hardy spaces was worked out in [66]. Theorem 4 can be regarded also as an alternative tool to the Calderon-Zygmund decomposition lemma for proving weak type (, ) inequalities. In many cases this theorem can be applied better more simply than the Calderon-Zygmund decomposition lemma.

8 284 FERENC WEISZ 5. Summability of one-dimensional Fourier series Though Theorems 2 are not true for p = p =, with the help of some summability methods they can be generalized for these endpoint cases. Obviously, summability means have better convergence properties than the original Fourier series. Summability is intensively studied in the literature. We refer at this time only to the books Stein Weiss [6], Butzer Nessel [7], Trigub Belinsky [63], Grafakos [28] Weisz [77, 82] the references therein. The best known summability method is the Fejér method. In 94 Fejér [4] investigated the arithmetic means of the partial sums, the so called Fejér means proved that if the left right limits f(x ) f(x + ) exist at a point x, then the Fejér means converge to (f(x ) + f(x + ))/2. One year later Lebesgue [35] extended this theorem obtained that every integrable function is Fejér summable at each Lebesgue point, thus almost everywhere. Some years later M. Riesz [44] proved that the Cesàro means of a function f L (T) converge almost everywhere to f (see also Zygmund [85, Vol. I, p.94]). In this paper we consider the Fejér Cesàro (or (C, α)) means defined by σ n f(x) := n s k f(x) = ( j ) f(j)φ j (x) = f(x u)k n (u) du n n X σnf(x) α := n A α n = A α n j n j n A α n k s kf(x) A α f(j)φ n j j (x) = where ( ) k + α A α k := = k the Fejér Cesàro kernels are given by K n (u) := ( j n K α n (u) := A α n j n j n (α + )(α + 2)... (α + k) k! X ) φ j (u) = n n f(x u)k α n (u) du, D k (u) A α n j φ j (u) = n A α A α n k D k(u) n (see Figure 3). It is known (Zygmund [85]) that A α k kα (k N). The Cesàro means are generalizations of the Fejér means, if α =, then we get back the Fejér means. We will suppose always that < α. The case α > can

9 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES /8 2/8 3/8 4/8 5/8 6/8 7/8 (a) The trigonometric Fejér kernel (b) The Walsh-Fejér kernel Figure 3. The Fejér kernels K n with n = 5. be led back to α =. The next result extends Theorem to the summability means (see Zygmund [85] Paley [43]). Theorem 5. If < α p, then σ α nf p C p f p Moreover, for all f L p (X) ( p < ), lim n σα nf = f (f L p (X), n N). in the L p -norm. The maximal operator of the Cesàro means are defined by σ α f := sup σnf α. n N Applying Theorem 4, we have extended the previous result to the L p (X) spaces ( < p < ) to the maximal operator in [67, 76, 77]. The first inequality of Theorem 6 was proved by Fujii [8] in the Walsh case for p = (see also Schipp Simon [49]). Theorem 6. If < α /(α + ) < p, then for f H /(α+) (X), σ α f p C p f Hp (f H p (X)) σ α f /(α+), = sup ρλ(σ α f > ρ) α+ C f H/(α+). ρ> The critical index is p = /(α+), if p is smaller than or equal to this critical index, then σ α is not bounded anymore (see Stein, Taibleson Weiss [6], Simon Weisz [57], Simon [55] Gát Goginava [23]). Theorem 7. The operator σ α L p (X) if < p /(α + ). ( < α ) is not bounded from H p (X) to

10 286 FERENC WEISZ We get the next weak type (, ) inequality from Theorem 6 by interpolation (Weisz [67, 76, 77], Zygmund [85] for the trigonometric system, for α = p = Móricz [39], for α = for the Walsh system Schipp [46] Simon [54]). Corollary. If < α f L (X) then sup ρλ(σ α f > ρ) C f. ρ> This weak type (, ) inequality the density argument of Marcinkiewicz Zygmund [37] imply the well known theorem of Fejér [4] Lebesgue [35] with α =. Riesz [44] proved it for other α s Fine [5], Schipp [46] Weisz [76] for the Walsh system. Corollary 2. If < α f L (X) then lim n σα nf = f a.e. With the help of the conjugate functions we ([77]) proved also Theorem 8. If < α /(α + ) < p then σ α nf Hp C p f Hp (f H p (X)). Corollary 3. If < α, /(α + ) < p < f H p (X) then lim n σα nf = f in the H p -norm. 6. Partial sums of multi-dimensional Fourier series Let us fix d, d N. For a set Y let Y d be its Cartesian product Y Y taken with itself d-times. The L p (X d ) spaces are defined in the usual way. The d-dimensional trigonometric Walsh system is introduced as a Kronecker product by φ k (x) := φ k (x ) φ kd (x d ), where k = (k,..., k d ) N d, x = (x,..., x d ) X d. The multi-dimensional Fourier coefficients of an integrable function f are defined by f(k) := fφ k dλ (k N d ). X d The definition of the Fourier coefficients can be again extended to distributions resp. to martingales (see Weisz [77, 82]). For f L (X d ) the nth rectangular partial sum s n f of the Fourier series of f is introduced by s n f(x) :=... f(k)φk (x) = f(x u)d n (u) du (n N d ), k n k d n X d d where D n (u) := d D nj (u j ) j=

11 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 287 is the nth multi-dimensional trigonometric or Walsh-Dirichlet kernel (see Figure 4). By iterating the one-dimensional result, we get easily the next theorem. Theorem 9. If f L p (X d ) for some < p <, then s n f p C p f p (n N d ) lim s nf = f n in the L p -norm /8 6/8 5/8 4/8 3/8 2/8 /8 /8 2/8 3/8 4/8 5/8 6/8 7/8 (a) The trigonometric Dirichlet kernel (b) The Walsh-Dirichlet kernel Figure 4. The Dirichlet kernels D n with n = 5, n 2 = 4. Other types of partial sums are considered e.g. in Weisz [77, 82]. The analogue of the Carleson s theorem is not true, i.e., s n f is not convergent (Fefferman [, 2]). However, investigating the partial sums over the diagonal, only, Carleson s theorem holds also for higher dimensions for the trigonometric system (see Fefferman [] Grafakos [28]), it holds for the Walsh system if p = 2 (see Móricz [38] or Schipp, Wade, Simon Pál [5]). Theorem. If f L p (X d ) for some < p <, then for the trigonometric Fourier series sup s n,...,n f C p f p n N p lim s n,...,nf = f a.e. n The same result holds for the Walsh-Fourier series if p = 2. It is an open question, whether this theorem holds for the Walsh system for p 2 (cf. Schipp, Wade, Simon Pál [5]). 7. Multi-dimensional Hardy spaces In this section we introduce three types of multi-dimensional classical Hardy spaces for the trigonometric system three types of multi-dimensional dyadic Hardy spaces for the Walsh system.

12 288 FERENC WEISZ 7.. Multi-dimensional classical Hardy spaces. A distribution f is in the classical Hardy space Hp (T d ), in the product Hardy space H p (T d ) in the hybrid Hardy space Hp(T i d ) ( < p ) if f H i p := f H p := sup f (P t P t ) <, t> p f Hp := sup f (P t P td ) < sup t k >,k=,...,d;k i t k >,k=,...,d p f (Pt P ti P ti+ P td ) respectively, where P t the one-dimensional Poisson kernel i =,..., d. p <, 7.2. Multi-dimensional dyadic Hardy spaces. By a dyadic rectangle we mean a Cartesian product of d dyadic intervals. For n N d x = (x,..., x d ) [, ) d let I n (x) := I n (x ) I nd (x d ) be a dyadic rectangle. The σ-algebra generated by the dyadic rectangles {I n (x) : x [, ) d } will be denoted again by F n (n N d ). For < p the martingale Hardy space Hp [, ) d, the product Hardy space H p [, ) d the hybrid Hardy space Hp[, i ) d consist of all d-parameter dyadic martingales f = (f n, n N d ) with respect to (F n, n N d ), for which f H p := sup f n,...,n <, n N p f Hp := sup f n,...,n d <, n N d p f H i p := sup En E ni E ni+ E nd f <, n k N,k i respectively, where E ni denotes the conditional expectation operator relative to F ni (i =,..., d). We can show again that for a martingale f = (f n, n N d ) we have s 2 n,...,2 n df = f n Atomic decomposition of H p (X d ). It is known again (see e.g. Stein [59] or Weisz [77, 82]) that i.e., H p (X d ) H p (X d ) H i p(x d ) L p (X d ) H i (X d ) L(log L) d (X d ) p ( < p ). (i =,..., d), f H i C + C f (log + f ) d (f L(log L) d (X d )) where log + u = {u>} log u.

13 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 289 To obtain some convergence results of the summability means over the diagonal or over a cone we consider the Hardy space Hp (X d ). Now the situation is similar to the one-dimensional case. A function a L (T d ) is a multidimensional classical p-atom if there exists a cube I T d such that () supp a I, (2) a I /p, (3) I a(x)xk dx = for all multi-indices k = (k,..., k d ) for which k 2 d(/p ). A function a L [, ) d is called a multi-dimensional dyadic p-atom if there exists a dyadic cube I [, ) d such that (i), (ii) (iii) with k = hold. The atomic decomposition holds for the multi-dimensional Hardy spaces, too (see Latter [34], Lu [36], Wilson [83, 84], Stein [59] Weisz [66, 77]). Theorem. A distribution (resp. martingale) f is in Hp (X d ) ( < p ) if only if there exist a sequence (a k, k N) of multi-dimensional classical (resp. dyadic) p-atoms a sequence (µ k, k N) of real numbers such that µ k p < µ k a k = f in the sense of distributions (resp. martingales). Moreover, ( ) /p f Hp inf µ k p. For a cube I = I I d X d let I r = I r I r d. For the proof of the next theorem see Weisz [77, 8]. Theorem 2. For each n N d, let V n : L (X d ) L (X d ) be a bounded linear operator let V f := sup n N d V n f. Suppose that X d \I r V a p dλ C p for all classical (resp. dyadic) p -atoms a for some fixed r N < p, where the cube I is the support of the atom. If V is bounded from L p (X d ) to L p (X d ) for some < p, then V f p C p f Hp (f H p (X d )) for all p p p. Moreover, if p <, then sup ρ λ( V f > ρ) C f (f L (X d )). ρ>

14 29 FERENC WEISZ 7.4. Atomic decomposition of H p (X d ). In the investigation of the convergence in the Prighheim s sense (i.e., over all n) we use the Hardy spaces H p (X d ). The atomic decomposition for H p (X d ) is much more complicated. One reason of this is that the support of an atom is not a rectangle but an open set. Moreover, here we have to choose the atoms from L 2 (X d ) instead of L (X d ). This atomic decomposition was proved by Chang Fefferman [9, 3] Weisz [72, 77]. For an open set F (X d ) denote by M(F ) the maximal dyadic subrectangles of F. A function a L 2 (X d ) is a classical H p -atom if (a) supp a F for some open set F X d, (b) a 2 F /2 /p, (c) a can be further decomposed into the sum of elementary particles a R L 2 (X d ), a = R M(F ) a R in L 2 (X d ), satisfying (a) supp a R R F, (b) for i =,..., d, k 2/p 3/2 R M(F ), we have a R (x)x k i dx i =, X (c) for every disjoint partition P l (l =, 2,...) of M(F ), 2 /2 a R F /2 /p. l R P l 2 We get the definition of dyadic H p -atoms if k = in (b). The analogue of Theorem holds in this case, too, however, the proof is much more complicated (see Chang Fefferman [9, 3] Weisz [72, 77]). Theorem 3. A distribution (resp. martingale) f is in H p (X d ) ( < p ) if only if there exist a sequence (a k, k N) of classical (resp. dyadic) H p -atoms a sequence (µ k, k N) of real numbers such that µ k p < µ k a k = f in the sense of distributions (resp. martingales). Moreover, ( ) /p f Hp inf µ k p. The corresponding result to Theorem 2 for the H p (X d ) space are much more complicated again. Since the definition of the H p -atom is very complex, to obtain a usable condition about the boundedness of the operators, we have to introduce simpler atoms. First suppose that d = 2. A function a is a simple H p -atom if () supp a R for some rectangle R T 2,

15 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 29 (2) a 2 R /2 /p, (3) T a(x)xk i dx i = for all i =, 2 k 2/p 3/2. A function a is called a simple dyadic H p -atom if there exists a dyadic rectangle R [, ) 2 such that (i), (ii) (iii) with k = hold. Note that there are not enough simple H p -atoms, more exactly, H p (X 2 ) cannot be decomposed into simple H p -atoms, a counterexample can be found in Weisz [66]. However, the following result says that for an operator V to be bounded from H p (X d ) to L p (X d ) ( < p ) it is enough to check V on simple H p -atoms the boundedness of V on L 2 (X d ). Theorem 4. For each n N 2, let V n : L (X 2 ) L (X 2 ) be a bounded linear operator V f := sup n N 2 V n f. Let d = 2 < p. Suppose that there exists η > such that for every simple H p -atom a for every r X 2 \R r V a p dλ C p 2 ηr, where R is the support of a. If V is bounded from L 2 (X 2 ) to L 2 (X 2 ), then V f p C p f Hp (f H p (X 2 )) for all p p 2. Moreover, if p <, then sup ρ λ( V f > ρ) C f H i (f H(X i 2 ), i =, 2). ρ> Theorem 4 for two-dimensional classical Hardy spaces is due to Fefferman [3] for martingale Hardy spaces to Weisz [7]. Journé [3] verified that the preceding result do not hold for dimensions greater than 2. So there are fundamental differences between the theory in the two-parameter three- or more-parameter cases. Now we present the analogous theorem for higher dimensions. If d 3, a function a L 2 (T d ) is called a simple H p -atom if there exist intervals I i T, i =,..., j for some j d, such that () supp a I I j A for some measurable set A T d j, (2) a 2 ( I I j A ) /2 /p, (3) T a(x)xk i dx i = a dλ = for all i =,..., j k 2/p 3/2. A If j = d, we may suppose that A = I d is also an interval. Of course if a L 2 (T d ) satisfies these conditions for another subset of {,..., d} than {,..., j}, then it is also called a simple H p -atom. If the intervals are dyadic k =, then we get the definition of simple dyadic H p -atoms. Note that H p (X d ) cannot be decomposed into simple p-atoms. The following result is due to the author [72, 77]. Let H c denote the complement of the set H.

16 292 FERENC WEISZ Theorem 5. For each n N d, let V n : L (X d ) L (X d ) be a bounded linear operator V f := sup V n f. n N d Let d 3 < p. Suppose that there exist η,..., η d > such that for every simple H p -atom a for every r..., r d (I r )c (I r j j )c A V a p dλ C p 2 η r 2 η jr j, where I I j A is the support of a. If j = d A = I d is an interval, then we also assume that V a p dλ C p 2 η r 2 η d r d. (I d ) c (I r )c (I r d d )c If V is bounded from L 2 (X d ) to L 2 (X d ), then V f p C p f Hp (f H p (X d )) for all p p 2. Moreover, if p <, then sup ρ λ( V f > ρ) C f H i ρ> (f H i (X d ), i =,..., d). In some sense the space H i (X d ) plays the role of the one-dimensional L (X) space. 8. Summability of multi-dimensional Fourier series The multi-dimensional Fejér Cesàro means of a distribution resp. martingale f are defined by n n d σ n f := d i= n... s k f = d (... k ) i f(k)φ k i n k = k d = k n k d n d i= i = f(x u)k n (u) du X d σ α nf := = = n d i= Aα i n i d i= Aα i n i k = n d... k n... k d = X d f(x u)k α n (u) du, A α j n j k j s k f k d n d ( d i= A α i n i k i ) f(k)φ k

17 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 293 where the Fejér Cesàro kernels (see Figure 5) are given by d d K n (u) := K nj (u j ) Kn(u) a := K α j n j (u j ). j= j= /8 6/8 5/8 4/8 3/8 2/8 /8 /8 2/8 3/8 4/8 5/8 6/8 7/8 (a) The trigonometric Fejér kernel (b) The Walsh-Fejér kernel Figure 5. The Fejér kernels K n with n = 5 n 2 = 4. Theorem 6. If < α j (j =,..., d) p, then σ α nf p C p f p (f L p (X d ), n N d ). Moreover, for all f L p (X d ) ( p < ), lim n σα nf = f in the L p -norm. This theorem can be found e.g. in Zygmund [85] Weisz [77]. Here the convergence is understood in Pringsheim s sense, i.e., n means that min(n,..., n d ). 9. Restricted convergence of summability means For a given τ we define a cone (see Figure 6) by N d τ := {n N d : τ n i /n j τ, i, j =,..., d}. In this section we investigate the convergence of the summability means over this cone, the multi-dimensional Hardy space Hp (X d ) the restricted maximal operator defined by σ f α := sup σnf. α n N d τ For the Walsh system let p := max { } α +,..., α d +

18 294 FERENC WEISZ Figure 6. The cone for d = 2. for the trigonometric system { } d p := max d +, α +,...,. α d + Theorem 7. If < α j (j =,..., d) p < p <, then σ α f p C p f H p (f H p (X d )). This theorem is due to the author [7, 75, 77, 78, 82] (for Walsh-Kaczmarz system see Simon [55]). For the Fejér means (i.e., α j =, j =,..., d) there are counterexamples for the boundedness of σ α if p p = /2 (Goginava Nagy [25, 27]). Theorem 8. For the Walsh system the operator σ (α j =, j =,..., d) is not bounded from H p (X d ) to L p (X d ) if < p /2. By interpolation we obtain ([7, 75]) Corollary 4. If < α j (j =,..., d) f L (X d ) then sup ρλ(σ f α > ρ) C f. ρ> The usual density argument Corollary 4 imply the generalization of the Marcinkiewicz-Zygmund result. Corollary 5. If < α j (j =,..., d) f L (X d ), then lim σnf α = f n,n N d τ Note that this corollary is due to the author [7, 68, 75, 78]. For Fejér means of two-dimensional Walsh-Fourier series it can also be found in Gát [9] (see also Móricz, Schipp Wade [4], Simon [56] for Walsh-Kaczmarz system Blahota Gát [5] for a general orthonormal system). The following results are known ([7, 75]) for the norm convergence of σ n f. a.e.

19 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 295 Theorem 9. If < α j (j =,..., d), p < p < n N d τ, then σ α nf H p C p f H p (f H p (X d )). Corollary 6. If < α j (j =,..., d), p < p < f Hp (X d ), then lim σnf α = f in the H n,n N d p -norm. τ. Convergence of summability means over a cone-like set Here we extend the results of the preceding section. First we introduce the cone-like sets, which are generalizations of the cones investigated before. Suppose that for all j = 2,..., d, γ j : R + R + are strictly increasing continuous functions such that lim j γ j = lim j + γ j =. Moreover, suppose that there exist c j,, c j,2, ξ > such that c j, γ j (x) γ j (ξx) c j,2 γ j (x) (x > ). For a fixed τ we define the cone-like set (see Figure 7) by Figure 7. Cone-like set for d = 2. N d τ,γ := {n N d : τ γ j (n ) n j τγ j (n ), j = 2,..., d}. To investigate the convergence of the summability means over these conelike sets, we have to introduce another maximal operator other Hardy spaces. Now we introduce the maximal operator σγ α f := sup σnf. α n N d τ,γ The Hardy spaces Hp γ (T d ) Hp γ [, ) d are given with the norms f H γ p := sup f (Pt P γ2 (t) P γd (t)) t> p

20 296 FERENC WEISZ respectively, where f H γ p := sup s 2 n,...,2 n d f n N 2 n j γ j (2 n ) < 2 n j+ p <, (j = 2,..., d). For the Walsh system let p =. For the trigonometric system we can define a number p < depending only on the functions γ j (see Weisz [79]). The results of the preceding section can be generalized as follows (see Weisz [79, 8, 82], for two-dimensional Walsh-Kaczmarz-Fejér means Nagy [4]). Theorem 2. If < α j (j =,..., d) p p < p <, then σ α γ f p C p f H γ p (f H γ p (X d )). Corollary 7. If < α j (j =,..., d) f L (X d ), then sup ρλ(σγ α f > ρ) C f. ρ> Corollary 8. If < α j (j =,..., d) f L (X d ), then lim σnf α = f n,n N d τ,γ In the two-dimensional case, Corollaries 7 8 were proved by Gát Nagy [22, 24] for Fejér summability, for a general orthonormal system by Nagy [42]. For two-dimensional Fejér means the border point p is essential [4]. a.e.. Unrestricted convergence of summability means In this section we deal with the Hardy spaces H p (X d ) the non-restricted maximal operator introduced by σ α f := sup n N d σ α nf. Now we investigate the convergence of σnf α in Pringsheim s sense, that is, min(n,..., n d ). The next result is due to the author ([69, 74, 7, 73, 72]) (for Walsh-Kaczmarz systems see Simon [55]). Let p 2 := max { α +,..., α d + Theorem 2. If < α j (j =,..., d) p 2 < p <, then }. σ α f p C p f Hp (f H p (X d )). The following unboundedness result was proved by Goginava [25, 26]. Theorem 22. For the Walsh system the operator σ (α j =, j =,..., d) is not bounded from H p (X d ) to L p (X d ) if < p p 2.

21 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 297 By interpolation we get here almost everywhere convergence for functions from the spaces H i (X d ) instead of L (X d ). Corollary 9. If < α j f H i (X d ) (i, j =,..., d) then sup ρλ(σ α f > ρ) C f H i. ρ> Recall that H i (X d ) L(log L) d (X d ) for all i =,..., d. Corollary. If < α j f H i (X d ) (i, j =,..., d) then lim n σα nf = f For the L(log L)[, ) 2 space Walsh system see also Móricz, Schipp Wade [4]. Gát [2, 2] proved for the Fejér means that this corollary does not hold for all integrable functions. Theorem 23. The almost everywhere convergence is not true for all f L (X d ). a.e. Theorem 24. If < α j (j =,..., d) p 2 < p <, then σ α nf Hp C p f Hp (f H p (X d ), n N d ). Corollary. If < α j (j =,..., d), p 2 < p < f H p (X d ) then lim n σα nf = f in the H p -norm. References [] N. K. Bary. A treatise on trigonometric series. Vols. I, II. A Pergamon Press Book. The Macmillan Co., New York, 964. [2] C. Bennett R. Sharpley. Interpolation of operators, volume 29 of Pure Applied Mathematics. Academic Press, Inc., Boston, MA, 988. [3] J. Bergh J. Löfström. Interpolation spaces. An introduction. Springer-Verlag, Berlin-New York, 976. Grundlehren der Mathematischen Wissenschaften, No [4] P. Billard. Sur la convergence presque partout des séries de Fourier-Walsh des fonctions de l espace L 2 (, ). Studia Math., 28: , 966/967. [5] I. Blahota G. Gát. Pointwise convergence of double Vilenkin-Fejér means. Studia Sci. Math. Hungar., 36(-2):49 63, 2. [6] M. Bownik. Boundedness of operators on Hardy spaces via atomic decompositions. Proc. Amer. Math. Soc., 33(2): , 25. [7] P. L. Butzer R. J. Nessel. Fourier analysis approximation. Academic Press, New York-London, 97. Volume : One-dimensional theory, Pure Applied Mathematics, Vol. 4. [8] L. Carleson. On convergence growth of partial sums of Fourier series. Acta Math., 6:35 57, 966. [9] S.-Y. A. Chang R. Fefferman. Some recent developments in Fourier analysis H p -theory on product domains. Bull. Amer. Math. Soc. (N.S.), 2(): 43, 985. [] R. R. Coifman G. Weiss. Extensions of Hardy spaces their use in analysis. Bull. Amer. Math. Soc., 83(4): , 977.

22 298 FERENC WEISZ [] C. Fefferman. On the convergence of multiple Fourier series. Bull. Amer. Math. Soc., 77: , 97. [2] C. Fefferman. On the divergence of multiple Fourier series. Bull. Amer. Math. Soc., 77:9 95, 97. [3] R. Fefferman. Calderón-Zygmund theory for product domains: H p spaces. Proc. Nat. Acad. Sci. U.S.A., 83(4):84 843, 986. [4] L. Fejér. Untersuchungen über Fouriersche Reihen. Math. Ann., 58(-2):5 69, 93. [5] N. J. Fine. Cesàro summability of Walsh-Fourier series. Proc. Nat. Acad. Sci. U.S.A., 4:588 59, 955. [6] S. Fridli. On the rate of convergence of Cesàro means of Walsh-Fourier series. J. Approx. Theory, 76():3 53, 994. [7] S. Fridli. Coefficient condition for L -convergence of Walsh-Fourier series. J. Math. Anal. Appl., 2(2):73 74, 997. [8] N. Fujii. A maximal inequality for H -functions on a generalized Walsh-Paley group. Proc. Amer. Math. Soc., 77(): 6, 979. [9] G. Gát. Pointwise convergence of the Cesàro means of double Walsh series. Ann. Univ. Sci. Budapest. Sect. Comput., 6:73 84, 996. [2] G. Gát. On the divergence of the (C, ) means of double Walsh-Fourier series. Proc. Amer. Math. Soc., 28(6):7 72, 2. [2] G. Gát. Divergence of the (C, ) means of d-dimensional Walsh-Fourier series. Anal. Math., 27(3):5 7, 2. [22] G. Gát. Pointwise convergence of cone-like restricted two-dimensional (C, ) means of trigonometric Fourier series. J. Approx. Theory, 49():74 2, 27. [23] G. Gát U. Goginava. A weak type inequality for the maximal operator of (C, α)- means of Fourier series with respect to the Walsh-Kaczmarz system. Acta Math. Hungar., 25(-2):65 83, 29. [24] G. Gát K. Nagy. Pointwise convergence of cone-like restricted two-dimensional Fejér means of Walsh-Fourier series. Acta Math. Sin. (Engl. Ser.), 26(2): , 2. [25] U. Goginava. Maximal operators of Fejér means of double Walsh-Fourier series. Acta Math. Hungar., 5(4):333 34, 27. [26] U. Goginava. Maximal (C, α, β) operators of two-dimensional Walsh-Fourier series. Acta Math. Acad. Paed. Nyiregyh., 24:29 24, 28. [27] U. Goginava K. Nagy. On the Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system. Period. Math. Hungar., 55(): 8, 27. [28] L. Grafakos. Classical modern Fourier analysis. Pearson Education, Inc., Upper Saddle River, NJ, 24. [29] C. Herz. Bounded mean oscillation regulated martingales. Trans. Amer. Math. Soc., 93:99 25, 974. [3] R. A. Hunt. On the convergence of Fourier series. In Orthogonal Expansions their Continuous Analogues (Proc. Conf., Edwardsville, Ill., 967), pages Southern Illinois Univ. Press, Carbondale, Ill., 968. [3] J.-L. Journé. Two problems of Calderón-Zygmund theory on product-spaces. Ann. Inst. Fourier (Grenoble), 38(): 32, 988. [32] A. Kolmogoroff. Une série de Fourier-Lebesgue divergente presque partout. Fundamenta Mathematicae, 4(): , 923. [33] A. Kolmogoroff. Un serie de fourier-lebesgue divergente partout. C. R. Acad. Sci. Paris, 4(83): , 926. [34] R. H. Latter. A characterization of H p (R n ) in terms of atoms. Studia Math., 62():93, 978.

23 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 299 [35] H. Lebesgue. Recherches sur la convergence des séries de fourier. Math. Ann., 6(2):25 28, 95. [36] S. Z. Lu. Four lectures on real H p spaces. World Scientific Publishing Co., Inc., River Edge, NJ, 995. [37] Z. A. Marcinkiewicz, J. On the summability of double fourier series. Fundamenta Mathematicae, 32():22 32, 939. [38] F. Móricz. On the convergence of double orthogonal series. Anal. Math., 2(4):287 34, 976. [39] F. Móricz. The maximal Fejér operator on the spaces H L. In Approximation theory function series (Budapest, 995), volume 5 of Bolyai Soc. Math. Stud., pages János Bolyai Math. Soc., Budapest, 996. [4] F. Móricz, F. Schipp, W. R. Wade. Cesàro summability of double Walsh-Fourier series. Trans. Amer. Math. Soc., 329():3 4, 992. [4] K. Nagy. On the restricted summability of Walsh-Kaczmarz-Fejér means. Georgian Math. J., 22():3 4, 25. [42] K. Nagy. Almost everywhere convergence of cone-like restricted two-dimensional Fejér means with respect to Vilenkin-like systems. St. Petersburg Mathematical Journal, 25:65 64, 24, Originally published in Algebra i Analiz 25(4):25 38, 23. [43] R. E. A. C. Paley. A Remarkable Series of Orthogonal Functions (I). Proc. London Math. Soc., S2-34(): [44] M. Riesz. Sur la sommation des séries de Fourier. Acta Szeged, Sect. Math., :4 3. [45] F. Schipp. Über die Divergenz der Walsh-Fourierreihen. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 2:49 62, 969. [46] F. Schipp. Über gewisse Maximaloperatoren. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 8:89 95, 975. [47] F. Schipp. Pointwise convergence of expansions with respect to certain product systems. Anal. Math., 2():65 76, 976. [48] F. Schipp. Multiple Walsh analysis. In Theory applications of Gibbs derivatives (Kupari-Dubrovnik, 989), pages Mat. Inst., Belgrade, 989. [49] F. Schipp P. Simon. On some (L, H)-type maximal inequalities with respect to the Walsh-Paley system. In Functions, series, operators, Vol. I, II (Budapest, 98), volume 35 of Colloq. Math. Soc. János Bolyai, pages North-Holl, Amsterdam, 983. [5] F. Schipp, W. R. Wade, P. Simon. Walsh series. Adam Hilger, Ltd., Bristol, 99. An introduction to dyadic harmonic analysis, With the collaboration of J. Pál. [5] F. Schipp F. Weisz. Tree martingales a.e. convergence of Vilenkin-Fourier series. Math. Pannon., 8():7 35, 997. [52] P. Simon. Verallgemeinerten Walsh-Fourier Reihen. I. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 6:3 3 (974), 973. [53] P. Simon. On the divergence of Vilenkin-Fourier series. Acta Math. Hungar., 4(3-4):359 37, 983. [54] P. Simon. Investigations with respect to the Vilenkin system. Ann. Univ. Sci. Budapest. Eötvös Sect. Math., 27:87, 985. [55] P. Simon. Cesaro summability with respect to two-parameter Walsh systems. Monatsh. Math., 3(4):32 334, 2. [56] P. Simon. On the Cesaro summability with respect to the Walsh-Kaczmarz system. J. Approx. Theory, 6(2):249 26, 2. [57] P. Simon F. Weisz. Weak inequalities for Cesàro Riesz summability of Walsh- Fourier series. J. Approx. Theory, 5(): 9, 28. [58] P. Sjölin. An inequality of Paley convergence a.e. of Walsh-Fourier series. Ark. Mat., 7:55 57, 969.

24 3 FERENC WEISZ [59] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, oscillatory integrals, volume 43 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. [6] E. M. Stein, M. H. Taibleson, G. Weiss. Weak type estimates for maximal operators on certain H p classes. In Proceedings of the Seminar on Harmonic Analysis (Pisa, 98), number suppl., pages 8 97, 98. [6] E. M. Stein G. Weiss. Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 97. Princeton Mathematical Series, No. 32. [62] A. Torchinsky. Real-variable methods in harmonic analysis, volume 23 of Pure Applied Mathematics. Academic Press, Inc., Orlo, FL, 986. [63] R. M. Trigub E. S. Bellinsky. Fourier analysis approximation of functions. Kluwer Academic Publishers, Dordrecht, 24. [Belinsky on front back cover]. [64] W. R. Wade. Dyadic harmonic analysis. In Harmonic analysis nonlinear differential equations (Riverside, CA, 995), volume 28 of Contemp. Math., pages [65] W. R. Wade. Summability estimates of double Vilenkin-Fourier series. Math. Pannon., ():67 75, 999. [66] F. Weisz. Martingale Hardy spaces their applications in Fourier analysis, volume 568 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 994. [67] F. Weisz. Cesàro summability of one- two-dimensional Walsh-Fourier series. Anal. Math., 22(3): , 996. [68] F. Weisz. Cesàro summability of two-dimensional Walsh-Fourier series. Trans. Amer. Math. Soc., 348(6):269 28, 996. [69] F. Weisz. Cesàro summability of two-parameter trigonometric-fourier series. J. Approx. Theory, 9():3 45, 997. [7] F. Weisz. Cesàro summability of two-parameter Walsh-Fourier series. J. Approx. Theory, 88(2):68 92, 997. [7] F. Weisz. (C, α) means of d-dimensional trigonometric-fourier series. Publ. Math. Debrecen, 52(3-4):75 72, 998. Dedicated to Professors Zoltán Daróczy Imre Kátai. [72] F. Weisz. (C, α) means of several-parameter Walsh- trigonometric-fourier series. East J. Approx., 6(2):29 56, 2. [73] F. Weisz. The maximal (C, α, β) operator of two-parameter Walsh-Fourier series. J. Fourier Anal. Appl., 6(4):389 4, 2. [74] F. Weisz. The maximal (C, α, β) operator on H p (T T ). Approx. Theory Appl. (N.S.), 6():52 65, 2. [75] F. Weisz. Maximal estimates for the (C, α) means of d-dimensional Walsh-Fourier series. Proc. Amer. Math. Soc., 28(8): , 2. [76] F. Weisz. (C, α) summability of Walsh-Fourier series. Anal. Math., 27(2):4 55, 2. [77] F. Weisz. Summability of multi-dimensional Fourier series Hardy spaces, volume 54 of Mathematics its Applications. Kluwer Academic Publishers, Dordrecht, 22. [78] F. Weisz. Summability results of Walsh- Vilenkin-Fourier series. In Functions, series, operators (Budapest, 999), pages János Bolyai Math. Soc., Budapest, 22. [79] F. Weisz. Restricted summability of Fourier series Hardy spaces. Acta Sci. Math. (Szeged), 75(-2):97 27, 29. [8] F. Weisz. Restricted summability of multi-dimensional Vilenkin-Fourier series. Ann. Univ. Sci. Budapest. Sect. Comput., 35:35 37, 2. [8] F. Weisz. Boundedness of operators on Hardy spaces. Acta Sci. Math. (Szeged), 78(3-4):54 557, 22. [82] F. Weisz. Summability of multi-dimensional trigonometric Fourier series. Surv. Approx. Theory, 7: 79, 22.

25 CONVERGENCE OF TRIGONOMETRIC AND WALSH-FOURIER SERIES 3 [83] J. M. Wilson. A simple proof of the atomic decomposition for H p (R n ), < p. Studia Math., 74():25 33, 982. [84] J. M. Wilson. On the atomic decomposition for Hardy spaces. Pacific J. Math., 6():2 27, 985. [85] A. Zygmund. Trigonometric series. Vol. I, II. Cambridge Mathematical Library. Cambridge University Press, Cambridge, third edition, 22. With a foreword by Robert A. Fefferman. Received January 7, 25 Department of Numerical Analysis, Eötvös L. University, H-7 Budapest, Pázmány P. sétány /C., Hungary address: weisz@inf.elte.hu