Acta Math. Hungar., 138 (3) (2013), 259–266 DOI: 10.1007/s10474-012-0235-2 First published online May 8, 2012

STRONG SUMMABILITY THEOREMS FOR Hp (Δd ) U. GOGINAVA∗ and L. GOGOLADZE Department of Mathematics, Faculty of Exact and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi 0128, Georgia e-mails: [email protected], [email protected] (Received October 13, 2011; revised November 21, 2011; accepted November 23, 2011)

Abstract. We prove that certain means of the (C, α, . . . , α)-means (α = 1/p − 1) of the d-dimensional trigonometric Fourier series are uniformly bounded operators from the Hardy space Hp to Hp (1  p  2). As a consequence we obtain strong summability theorems concerning (C, α, . . . , α)-means.

Let C d be a d-dimensional space of complex numbers. Denote by Z := (z1 , . . . , zd ),

zj := rj eixj



0  rj < ∞, xj ∈ [−π, π], j = 1, . . . , d

the points from C d . The set 

Δd := Z : |zj | < 1, xj ∈ [−π, π], j = 1, . . . , d



is said to be a d-dimensional polydisc, and the set 

Γd := Z : |zj | = 1, xj ∈ [−π, π], j = 1, . . . , d



will be called the distinguished boundary of the polydisc Δd . Denote, as usual, by Hp (Δd ), p  1 the set of functions 

f (Z) := f r1 eix1 , . . . , rd eixd



which are analytic in Δd , i.e. (1)

f (Z) =

∞  n1 =0

···

∞  nd =0

cn1 ,...,nd (f )

d  n r j einj xj j

j=1

∗ Corresponding

author. Key words and phrases: Hardy space, strong summability. Mathematics Subject Classification: 40F05, 42B08, 42B30.

c 2012 Akad´ 0236-5294/$ 20.00  emiai Kiad´ o, Budapest, Hungary



260

U. GOGINAVA and L. GOGOLADZE

and 



f Hp =

  p |f eix1 , . . . , eixd | dx1 · · · dxd

1/p

.

[−π,π]d

The trigonometric Fourier series of the function f ∈ H1 (Δd ) is the power series +∞ 

(2)

cn1 ,...,nd (f )ei(n1 x+···+nd xd ) ,

n1 ,...,nd =0

where cn1 ,...,nd (f ) =



1 d

(2π)

d

[−π,π]

f (x1 , . . . , xd )e−i(n1 x1 +···+nd xd ) dx1 · · · dxd

are the Fourier coefficients of f . The rectangular partial sums are defined as follows: N1 

SN1 ,...,Nd (f ; x1 , . . . , xd ) :=

···

n1 =0

Nd 

cn1 ,...,nd (f )ei(n1 x1 +···+nd xd ) .

nd =0

We recall that the Ces` aro means of order (α1 , . . . , αd ) of f ∈ Hp (Δd ) at   the point eix1 , . . . , eixd are defined as 

1 ,...,αd ) f, eix1 , . . . , eixd σn(α1 ,...,n d

=

n1 

1 d 

α k =0 1

···

Anjj

nd  d 



α

Anjj −kj ck1 ,...,kd (f )ei(k1 x1 +···+kd xd ) .

kd =0 j=1

j=1

It is known [6] that if f ∈ Hp (Δd ), p  1 then it can be represented as the Poisson integral, i.e. (3)

f (Z) =

1 (2π)d



f (ei(x1 +s1 ) , . . . , ei(xd +sd ) )

d 

P (rj , sj ) ds1 · · · dsd

j=1

[−π,π]d

where P (rj , tj ) :=

∞  1 k + r j cos (kj tj ), 2 k =1 j j

Acta Mathematica Hungarica 138, 2013

j = 1, . . . , d

STRONG SUMMABILITY THEOREMS FOR Hp (Δd )

261

is the Poisson kernel. From (1) and (3) we have f (r1 ei(x1 +t1 ) , . . . , rd ei(xd +td ) )

(4)

=

∞ 

···

n1 =0

=

∞ 

cn1 ,...,nd (f )

nd =0

d  n r j einj xj einj tj j

j=1



1

i(x1 +t1 +s1 )

(2π)d

f (e

,...,e

i(xd +td +sd )

)

d 

P (rj , sj ) ds1 · · · dsd .

j=1

d

[−π,π]

In order to prove the main result we need the d-dimensional analogue of Hardy–Littlewood’s theorem. Theorem A. If f ∈ Hp (Δd ), 0 < p  2, then ∞ cn ,...,n (f ) p  1 d ···  c(p, d)f pHp . d  n1 =0 nd =0 (nj + 1)2−p ∞ 

j=1

The validity of Theorem A is obtained by iteration of the Hardy– Littlewood’s theorem. We note that Theorem A for real Hardy spaces was proved in [3,9] The series (2) is said to be strong summable with exponent k ((H, k)summability) if (5)

1

lim

d min(mj : j=1,...,d)→∞ 

(mj + 1)

j=1

×

m1  n1 =0

···

md  Sn ,...,n (f, x1 , . . . , xd ) − f (x1 , . . . , xd ) k = 0, 1 d nd =0

For the one-dimensional case Hardy and Littlewood [2] proved that if f ∈ Lp (T ), p > 1, then (5) holds almost everywhere. They posed the following problem: is (5) valid almost everywhere if f ∈ L(T )? Marcinkiewicz [5] solved this problem for k = 2, and Zygmund [10] for any k > 0. For the d-dimensional case in [4] the second author proved that if f ∈ d−1 L(ln+ L) (T d ) then its d-multiple Fourier series is (H, k) summable almost Acta Mathematica Hungarica 138, 2013

262

U. GOGINAVA and L. GOGOLADZE

everywhere. This gave rise to the question of strong summability in the norm. Smith [7] and Belinskii [1] proved that if f ∈ H1 (T ), then



m S (f ) − f 1  k L1 = 0. lim m→∞ log m k k=0

Weisz [8] improved this result in two directions: first he proved that if f ∈ Hp (T ), 1/2 < p  1 then 

(6)

1 log n

[p]  m S (f ) p k Hp

k 2−p

k=1

 cp f pHp

where cp is a positive constant, depending only on p, Hp (T ) is a real Hardy space and second, he proved the validity of (6) for the multi-dimensional case. The present paper deals with the problem of strong summability of Ces`aro means of nonnegative order for functions from the space Hp (Δd ), 1  p < 2. Theorem 1. Let f ∈ Hp (Δd ), 1  p  2, α = 1/p − 1. Then m1 md σn1 ,...,nd (f )   1 Hp ···  c(p, d)f pHp , d  ln (m + 1) j n1 =0 nd =0 j=1 (nj + 1) p

(α,...,α)

d 

j=1

where c(p, d) is a positive constant, depending only on p and d.

Proof. From (4) it follows that (see [11], p. 80) (7)





Fx1 ,...,xd ,r1 ,...,rd eit1 , . . . , eitd :=

f (r1 ei(x1 +t1 ) , . . . , rd ei(xd +td ) ) d   j=1

=

m1  n1 =0

···

md  d 

1 − rj eitj



Aαnl σn(α,...,α) f, eix1 , . . . , eixd 1 ,...,nd

nd =0 l=1

=

 α+1

d   n r j einj tj j

j=1

1

1 d

(2π)

d   j=1

Acta Mathematica Hungarica 138, 2013

1 − rj eitj

 α+1

263

STRONG SUMMABILITY THEOREMS FOR Hp (Δd )



×

d 

f (ei(x1 +t1 +s1 ) , . . . , ei(xd +td +sd ) )

P (rj , sj ) ds1 · · · dsd

j=1

d

[−π,π]





It is clear that for every eix1 , . . . , eixd and fixed rj ∈ [0, 1) the function  it 1 Fx1 ,...,xd ,r1 ,...,rd e , . . . , eitd is bounded and (7) is its power series on the boundary Γd . Therefore in view of Theorem A we will have

m1 

(8)

···

n1 =0

  p d d α (α,...,α)  ix1 , . . . , eixd  r pnj A σ f, e nl n1 ,...,nd md j  j=1 l=1 d 

nd =0

(nj + 1)2−p

j=1



 c(p, d)

1

[−π,π]d



d  1 − rj eitj (α+1)p

j=1

i(x1 +t1 +s1 )

f (e

i(xd +td +sd )

,...,e

)

d  j=1

d

[−π,π]

p d  P (rj , sj ) dsj dtk . k=1

Since

1 πd

(9)

d  d

[−π,π]

P (rj , sj ) dsj = 1

j=1

by virtue of Jensen’s inequality, we get

(10)



i(x1 +t1 +s1 )

f (e

i(xd +td +sd )

,...,e

)

j=1

d

[−π,π]





|f (ei(x +t +s ) , . . . , ei(x 1

d

[−π,π]

d 

1

1

d

+td +sd )

)|

p

p P (rj , sj ) dsj

d 

P (rj , sj ) dsj .

j=1

Acta Mathematica Hungarica 138, 2013

264

U. GOGINAVA and L. GOGOLADZE

Since α = 1/p − 1, we have (α + 1)p = 1 and Aαn ∼ nα . Using these facts, (10) and integrating (8) by the variables xj , j = 1, . . . , d, we get md 

σn(α,...,α) ,...,n (f )H

nd =0

d 

p

m1 

(11)

···

n1 =0

1

d

p

d  pnj j=1

rj

(nj + 1)

j=1







[−π,π]d



[−π,π]d



1 d  1 − rj eitj

[−π,π]d

j=1



|f ( e

i(x1 +t1 +s1 )

i(xd +td +sd )

,...,e

)|

d 

p

P (rj , sj ) dsj

d 

j=1

[−π,π]d

dtk

k=1

d 

dxl .

l=1

From (9) and changing the order of integration in (11), more exactly, integrating first by the variables xj , j = 1, . . . , d and then by the variables sj , j = 1, . . . , d and at last by the variables tj , j = 1, . . . , d, from (10) we have md 

σn(α,...,α) ,...,n (f )H

nd =0

d 

p

m1 

(12)

···

n1 =0

1

d

p

d  pnj j=1

rj

(nj + 1)

j=1



 c(p, d)f pHp

d 

[−π,π]d

j=1

dtj

. 1 − rj eitj

It is easy to show that [−π,π]



dt = |1 − reit |



=

 [−π,π]

⎝

1−r|t|π



|t|<1−r

Acta Mathematica Hungarica 138, 2013

(1 − r)2 + 4r sin2 (t/2) ⎞

⎛



+ |t|<1−r

dt

dt (1 − r)2 + 4r sin2 (t/2)

dt +c 1−r



1−r|t|π

dt t

⎠

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STRONG SUMMABILITY THEOREMS FOR Hp (Δd )









 1 + c ln 1/|1 − r|  c1 ln 1/|1 − r| ,

0 < r0  r < 1.

Consequently,

d 

(13) [−π,π]d

j=1

dt

j c 1 − rj eitj

d 





ln 1/|1 − rj | .

j=1

Assuming rj = 1 − 1/(mj + 1) in (12) and taking into account 

1 1− mj + 1

nj



1  1− mj + 1

mj

> e−1 ,

j = 1, . . . , d

when 0 < nj  mj from (12) and (13) we get m1  n1 =0

···

md  nd =0

σn(α,...,α) ,...,n (f )H p

1

d 

d

p

 c(p, d)f pHp

(nj + 1)

d 

ln (mj + 1).



j=1

j=1

This theorem for a real Hardy space H1 (T × · · · × T ) and p = 1 had been proved earlier in a different way by Weisz [8]. From Theorem 1 we get the following Corollary 1. Let f ∈ Hp (Δd ), 1  p  2, α = 1/p − 1. Then m1 md σn1 ,...,nd (f ) − f    1 Hp = 0. lim ··· d  ln (m + 1) min (mj : j=1,...,d)→0 j n1 =0 nd =0 j=1 (nj + 1) (α,...,α)

d 

p

j=1

We note that Theorem 1 was not known even in the one-dimensional case. Acknowledgement. The authors would like to thank Professor Ferenc Weisz for helpful suggestions. References [1] E. S. Belinskii, Strong summability of Fourier series of the periodic functions from Hp (0 < p < 1), Constr. Approx., 12 (1996), 187–195. [2] G. H. Hardy and J. E. Littlewood, On the strong summability of Fourier series, Proc. London Math. Soc., 26 (1926/27), 273–286. [3] B. Jawerth and A. Torchinsky, A note on real interpolation of Hardy spaces in the polydisk, Proc. Amer. Math. Soc., 96 (1986), 227–232. Acta Mathematica Hungarica 138, 2013

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U. GOGINAVA and L. GOGOLADZE: STRONG SUMMABILITY THEOREMS . . .

[4] L. D. Gogoladze, The (H, k)-summability of multiple trigonometric Fourier series (in Russian), Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 937–958. [5] J. Marcinkiewicz, Sur la sommabilit´e forte de s´eries de Fourier, J. London Math. Soc., 14 (1939), 162–168. [6] W. Rudin, Function Theory in Polydiscs, Mathematics Lecture Note Series (New York–Amsterdam, 1969). [7] R. Smith, A strong convergence theorem for H1 (T ), Lecture Notes in Math. 995, Springer (Berlin, Heidelberg, New York, 1984), pp. 169–173. [8] F. Weisz, Strong convergence theorems for Hp (T × · · · × T ), Publ. Math. Debrecen, 58 (2001), 667–678. [9] F. Weisz, Two-parameter Hardy–Littlewood inequalities, Studia Math., 118 (1996), 175–184. [10] A. Zygmund, On the convergence and summability of power series on the circle of convergence. II, Proc. London Math. Soc., 47 (1942), 326–350. [11] A. Zygmund, Trigonometric Series, 3rd ed., Cambridge Univ. Press (London, 2002)

Acta Mathematica Hungarica 138, 2013