On almost everywhere exponential summability of rectangular partial sums of double trigonometric Fourier series

0.18 MB·

by  Ushangi Goginava

ON ALMOST EVERYWHERE EXPONENTIAL
SUMMABILITY OF RECTANGULAR PARTIAL SUMS OF
DOUBLE TRIGONOMETRIC FOURIER SERIES
7
1
0
USHANGIGOGINAVAAND GRIGORIKARAGULYAN
2

n
a
J

0 Abstract. Inthispaperwestudythea.e. exponentialstrongsumma-
3
bility problem for the rectangular partial sums of double trigonometric
Fourier series of thefunctions from LlogL .
]
P
A
1. Introduction
.
h
t We denote the set of all non-negative integers by N. Let T := [ π,π) =
a
m R/2π and R := ( , ). Denote by L1(T) the class of all me−asurable
functions f on R t−ha∞t a∞re 2π-periodic and satisfy
[

1
f := f < .
v k k1 | | ∞
0 ZT
1
7 The Fourier series of a function f L1(T) with respect to the trigonometric
8 ∈
system is
0
1. ∞
0 (1) c einx,
n
7
n=−∞
1 X
: where
v
1
Xi cn := f(x)e−inxdx
2π
r ZT
a
are the Fourier coefficients of f. Denote by S (x,f) the partial sums of the
n
Fourier series of f and let
n
1
σ (x,f) = S (x,f)
n k
n+1
k=0
X
be the (C,1) means of (1). Fejér [1] proved that σ (f) converges to f uni-
n
formly for any 2π-periodic continuous function. Lebesgue in [18] established
almost everywhere convergence of (C,1) means if f L1(T). The strong
∈
02010 Mathematics Subject Classification 42C10 .
Keywordsandphrases: DoubleFourierseries,strongsummability,exponentialmeans.
1
2 USHANGIGOGINAVAANDGRIGORIKARAGULYAN
summability problem, i.e. the convergence of the strong means
n−1
1
(2) S (x,f) f(x)p, x T, p > 0,
k
n | − | ∈
k=0
X
was first considered by Hardy and Littlewood in [9]. They showed that for
any f Lr(T) (1< r < ) the strong means tend to 0 a.e. as n . The
trigono∈metric Fourier ser∞ies of f L1(T) is said to be (H,p)-sum→m∞able at
x T if the values (2) converge ∈to 0 as n . The (H,p)-summability
pro∈blem in L1(T) has been investigated by M→arc∞inkiewicz [19] for p =2, and
later by Zygmund [34] for the general case 1 p < .
≤ ∞
Let Φ : [0, ) [0, ), Φ(0) = 0, be a continuous increasing function.
∞ → ∞
We say a series with the partial sums s strong Φ-summable to a limit s if
n
n−1
1
lim Φ(s s ) = 0.
k
n→∞n | − |
k=0
X
In [20] Oskolkov first considered the a.e strong Φ-summability problem of
Fourier series with exponentially growing Φ. Namely, he proved a.e strong
Φ-summability of Fourier series if lnΦ(t)= O(t/lnlnt) as t .
→ ∞
In [21] Rodin proved
Theorem R (Rodin). If a continuous function Φ: [0, ) [0, ), Φ(0) =
∞ → ∞
0, satisfies the condition
lnΦ(t)
limsup < ,
t ∞
t→+∞
then for any f L1(T) the relation
∈
n−1
1
(3) lim Φ(S (x,f) f(x) ) = 0
k
n→∞n | − |
k=0
X
holds for a. e. x T.
∈
Karagulyan [11, 12] proved that the exponential growth in Rodin’s theo-
rem is optimal. Moreover, it was proved
TheoremK(Karagulyan). Ifacontinuous increasing functionΦ : [0, )
∞ →
[0, ),Φ(0) = 0, satisfies the condition
∞
lnΦ(t)
limsup = ,
t ∞
t→+∞
then there exists a function f L1(T), for which the relation
∈
n−1
1
limsup Φ(S (x,f) )=
k
n | | ∞
n→∞
k=0
X
holds everywhere on T.
ALMOST EVERYWHERE EXPONENTIAL SUMMABILITY 3
In this paper we study the exponential summability problem for the rect-
angular partial sums of double Fourier series. Let f L1(T2) be a function
∈
with Fourier series
∞
(4) c ei(mx+ny),
nm
m,n=−∞
X
where
1
c = f(x ,x )e−i(mx1+nx2)dx dx
nm 4π2 1 2 1 2
ZT2Z
are the Fourier coefficients of the function f. The rectangular partial sums
of (4) are defined by
M N
S (f)= S (x ,x ,f)= c ei(mx1+nx2).
MN MN 1 2 nm
m=−Mn=−N
X X
We denote by LlogL T2 the class of measurable functions f, with
(cid:0) (cid:1) f log+ f < ,
| | | | ∞
ZT2Z
where log+u := I logu, u > 0. For the rectangular partial sums of
(1,∞)
two-dimensional trigonometric FourierseriesJessen,MarcinkiewiczandZyg-
mund [10] has proved for any f LlogL T2 that
∈
1 n−1m−1 (cid:0) (cid:1)
lim (S (x ,x ,f) f(x ,x )) = 0
ij 1 2 1 2
n,m→∞nm −
i=0 j=0
X X
for a. e. (x ,x ) T2. They also showed that for every non-negative
1 2
∈
function ω : [0, ) [0, ) satisfying ω(t) , ω(t) log+t −1 0 as
∞ → ∞ ↑ ∞ →
t , there exists a function f such that f ω(f ) L1 T2 and the
→ ∞ | | | | ∈(cid:0) (cid:1)
(C,1,1) means of double Fourier series of f diverge a.e..
(cid:0) (cid:1)
The two dimensional a.e. strong rectangular (H,p)-summability, i.e. the
relation
n−1m−1
1
lim S (x ,x ,f) f(x ,x ) p =0 a.e.
ij 1 2 1 2
n,m→∞nm | − |
i=0 j=0
X X
was proved by Gogoladze [8]for f LlogL T2 . These results show that in
∈
two dimensional case the optimal class of functions for (C,1,1) summability
and strong summability coincide. That is th(cid:0)e cl(cid:1)ass of functions LlogL T2 .
We prove the following
(cid:0) (cid:1)
Theorem. If a continuous increasing function Φ: [0, ) [0, ), Φ(0) =
∞ → ∞
0, satisfies the condition
lnΦ(t)
(5) limsup < ,
t→+∞ t/lnlnt ∞
p
4 USHANGIGOGINAVAANDGRIGORIKARAGULYAN
then for any f LlogL T2 the relation
∈
n−(cid:0)1m−(cid:1)1
1
(6) lim Φ(S (x ,x ,f) f(x ,x ) )= 0
ij 1 2 1 2
n,m→∞nm | − |
i=0 j=0
X X
holds for a. e. (x ,x ) T2.
1 2
∈
As a corollary of this result we get the Gogoladze [8] theorem on a.e.
Hp-summability of double Fourier series. From Jessen, Marcinkiewicz and
Zygmund [10] theorem it follows that the class LlogL T2 in our theorem
is necessary in the context of strong summability question. That is, it is not
possible to give a larger convergence space than LlogL(cid:0) T(cid:1)2 . Our method
of proof do not allow to get (6) under the weaker condition
(cid:0) (cid:1)
lnΦ(t)
(7) limsup < .
t→+∞ √t ∞
There is a conjecture that (7) is the optimal bound of Φ ensuring a.e. rect-
angular strong summability (6) for every function f LlogL T2 .
∈
The results on strong summability and approximation by trigonometric
(cid:0) (cid:1)
Fourier series have been extended for several other orthogonal systems, see
Schipp [23, 24, 25], Leindler [14, 15, 16, 17], Totik [26, 27, 28, 29], Gogi-
nava,Gogoladze[5,6],Goginava,Gogoladze, Karagulyan[7],Gat,Goginava,
Karagulyan [3, 4], Weisz [30]-[33].
2. Auxiliary lemmas
The notation a . b will stand for a < c b, where c > 0 is an absolute
·
constant. We shall write a b if the relations a . b and b . a hold at the
∼
same time. Everywhere below q > 1 will be used as the conjugate of p > 1,
that is 1/p+1/q = 1. [a] denotes the integer part of a R.
The maximal function of a function f L1(T) is defi∈ned by
∈
1
Mf (x) := sup f(y) dy,
I:x∈I⊂T|I| Z | |
I
where I is an open interval. The following one dimensional operators intro-
duced by Gabisonia [2] are significant tools in the investigations of strong
summability problems:
q 1/q
k
[nπ] n
n
G(n)f(x):=   f (x+t) + f(x t) dt  ,
p k | | | − |
Xk=1 kZ−1  
  n  
   
G f(x):= supG(n)f(x).
p p
n∈N
Oskolkov’s following lemma plays key role in the proof of the basic lemma.
ALMOST EVERYWHERE EXPONENTIAL SUMMABILITY 5
Lemma 1 (Oskolkov, [20]). For any family of pairwise disjoint intervals
∆ T with centers c it holds the inequality
k k
⊂
q
|∆j|
(8) x T : sup j |x−cj|+|∆j| > λ . exp( cλ), λ > 0,
(cid:12)(cid:12) ∈ p>1 Pp(cid:16)lnln(p+2)(cid:17) (cid:12)(cid:12) −
 
(cid:12) (cid:12)
(cid:12) (cid:12)
where c(cid:12)>0 is an absolute constant. (cid:12)
(cid:12) (cid:12)
One can easily check that
q 1/q q
|∆j| |∆j|
sup j |x−cj|+|∆j| . 1,sup j |x−cj|+|∆j| .
p>1 (cid:16)P (cid:16)plnln(p+2(cid:17)) (cid:17)  p>1 Pp(cid:16)lnln(p+2)(cid:17) 
 
Combining this with (8), we get  
q 1/q
|∆j|
(9) sup j |x−cj|+|∆j| . 1.
Tp>1 (cid:16)P (cid:16)plnln(p+2(cid:17)) (cid:17)
Z
Lemma 2. If f L1(T), then
∈
G f(x) 1 1/2
(10) x T :sup p > λ . f , λ > 0.
∈ plnln(p+2) λ k k1
(cid:12)(cid:26) p>1 (cid:27)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. It i(cid:12)s enough to prove the same estim(cid:12)ate for the modified operators
q 1/q
k
[nπ] n
n
(11) G′f(x):= sup  f (x+t) dt  .
p n∈NXk=1kkZ−1 | |  
  n  
   
Using the Calderon-Zygmund lemma, for the maximal function we get the
relation
∞
(12) R := x T :Mf (x) > √λ = ∆ , λ > 0,
λ k
∈
n o k[=0
where ∆ T are disjoint open intervals such that
k
⊂
1
(13) √λ f(t) dt 2√λ,
≤ ∆ | | ≤
k
| | Z
∆k
1
(14) R f .
| λ|≤ √λk k1
6 USHANGIGOGINAVAANDGRIGORIKARAGULYAN
Denote δn := [(k 1)/n,k/n] and δn(x) := x+δn. Separating the terms in
k − k k
the sum (11) with k satisfying δn(x) R , we get
k ⊂ λ
q 1/q
k
n
n
(15) G′f(x) sup  f(x+t) dt 
p ≤n∈Nk:δknX(x)⊂RλkkZ−1 | |  
  n  
    q 1/q
k
n
n
+sup  f(x+t) dt 
n∈Nk:δknX(x)6⊂RλkkZ−1 | |  
  n  
:= I +II.   
From the definition of R in the case δn(x) R it follows that
λ k 6⊂ λ
k
n
n f (x+t) dt √λ.
| | ≤
kZ−1
n
Thus we conclude
∞ 1/q 1/q
1 1
(16) II √λ . √λ . p√λ.
≤ kq q 1
!
k=1 (cid:18) − (cid:19)
X
Given x T set
∈
min k :δn(x) ∆ if k :δn(x) ∆ =∅,
ki(x)= { k ⊂ i} if {k :δkn(x) ⊂ ∆i} 6=∅.
(cid:26) ∞ { k ⊂ i}
∞
DenoteR := 3∆ andtakeanarbitrary point x T R . Onecaneasily
λ k λ
∈ \
k=1
check theat if kiS(x) 6= ∞, then e
k (x)
i
∆ x c ,
i i
∋ n ∼ | − |
ALMOST EVERYWHERE EXPONENTIAL SUMMABILITY 7
where c is the center of the interval ∆ . Thus for any x / R we obtain
i i λ
∈
e
q 1/q
∞
n
(17) I = sup f (t) dt
   
n∈N Xi=1k:δknX(x)⊂∆i kδnZ(x) | |
  k  
   
q 1/q
∞
n
sup f (t) dt
   
≤ n∈N Xi=1 k:δknX(x)⊂∆i kδnZ(x) | |
  k  
   
q 1/q
∞
n ∆ 1
i
sup | | f (t) dt
≤ n∈Ni=1ki(x)|∆i|Z | |  
X ∆i
   
∞ q 1/q
n ∆
. √λsup | i|
k (x)
n i=1(cid:18) i (cid:19) !
X
∞ q 1/q
∆
. √λ | i| , x / R .
λ
x c + ∆ ∈
i=1(cid:18)| − i| | i|(cid:19) !
X
e
Using Chebyshev’s inequality, from (9), (16) and (17) it follows that
G′f(x)
x T R : sup p > λ
λ
∈ \ plnln(p+2)
(cid:12)(cid:26) p>1 (cid:27)(cid:12)
(cid:12) (cid:12)
(cid:12)(cid:12) e (cid:12)(cid:12) |∆j| q 1/q
. x T R :√λ 1+sup j |x−cj|+|∆j| cλ
(cid:12)(cid:12) ∈ \ λ  p>1 (cid:16)P (cid:16)plnln(p+2(cid:17)) (cid:17)  ≥ (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12)   (cid:12)
e
(cid:12)  q 1/q  (cid:12)
(cid:12) |∆j| (cid:12)
. (cid:12)1 sup j |x−cj|+|∆j| dx (cid:12)
√λ p>1 (cid:16)P (cid:16)plnln(p+2(cid:17)) (cid:17)
ZT
1
. ,
√λ
for an appropriate absolute constant c > 0. Applying homogeneity, one can
get
G′f(x) f 1/2
(18) x T R : sup p > λ . k k1 , λ > 0.
λ
∈ \ plnln(p+2) λ
(cid:12)(cid:26) p>1 (cid:27)(cid:12) (cid:18) (cid:19)
(cid:12) (cid:12)
(cid:12) e (cid:12)
(cid:12) (cid:12)
8 USHANGIGOGINAVAANDGRIGORIKARAGULYAN
Consequently, from (14)-(18) we get
G′f(x)
x T : sup p > λ
∈ plnln(p+2)
(cid:12)(cid:26) p>1 (cid:27)(cid:12)
(cid:12)(cid:12)(cid:12) x T R :sup G′pf((cid:12)(cid:12)(cid:12)x) > λ + R˜
λ λ
≤ ∈ \ plnln(p+2) | |
(cid:12)(cid:26) p>1 (cid:27)(cid:12)
(cid:12) (cid:12)
(cid:12) f 1/e2 f (cid:12)
. (cid:12) k k1 + k k1. (cid:12)
λ √λ
(cid:18) (cid:19)
Again using homogeneity, we obtain (10). (cid:3)
We will need the following estimations.
Lemma 3 (Gabisonia, [2]). If p > 1 and f L1(T2), then
∈
1/p
n−1
1
(19) S (x,f)p . G(n)f(x).
n | j |  p
j=0
X
 
Lemma 4 (Schipp, [22]). If f L1(T2), then
∈
1/p
n−1
1
(20) S (x,f) p . pG f(x).
j 2
n | | 
j=0
X
 
Rodin [21] proved the weak (1,1)-type estimate for the operators G f(x)
p
with a fixed p > 1. From this fact, applying a standard argument, one can
derive
Lemma 5 (Rodin, [21]). Let f LlogL(T). Then
∈
G (f) . 1+ f log f .
k 2 k1 | | | |
ZT
For any function f L1(T2) define
∈
G (x ,x ;f) = G f (x ), G (x ,x ;f)= G f (x ),
p,1 1 2 p x2 1 p,2 1 2 p x1 2
G(n)(x ,x ;f) = G(n)f (x ), G(n)(x ,x ;f) = G(n)f (x ),
p,1 1 2 p x2 1 p,2 1 2 p x1 2
where f () = f(,x ) and f () = f(x , ) are considered as functions on
x2 · · 2 x1 · 1 ·
x and x respectively. Similarly one dimensional partial sums of f(x ,x )
1 2 1 2
with respect to each variables will be denoted by
S (x ,x ,f)= S (x ,f ), S (x ,x ,f)= S (x ,f ).
n,1 1 2 n 1 x2 n,2 1 2 n 2 x1
ALMOST EVERYWHERE EXPONENTIAL SUMMABILITY 9
Lemma 6. If f LlogL(T2), then
∈
1/p
n−1m−1
1 S (x ,x ,f)p
(cid:12) nm i=0 j=0 | i,j 1 2 | ! (cid:12)
(cid:12)(cid:12)(cid:12)spu>p1ns,mu∈pN P Pp2lnln(p+2) > λ(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
 
(cid:12) 1/2 (cid:12)
(cid:12) (cid:12)
(cid:12) 1 (cid:12)
. 1+ f log+ f , λ >0.
λ  | | | |
ZT2Z
  
Proof. Using (19), (20) and generalized Minkowsi’s inequality, we get
n−1m−1 n−1m−1
1 1
S (x ,x ,f) p = S (x ,x ,S (f)) p
i,j 1 2 i,1 1 2 j,2
nm | | nm | |
i=0 j=0 i=0 j=0
X X X X
m−1
1 p
(n)
G (x ,x , S (f))
≤ m p,1 1 2 | j,2 |
Xj=0 (cid:16) (cid:17)
p
1/p
m−1
1
G(n) x ,x , S (f)p
≤  p,1  1 2 m | j,2 |  
j=0
X
  
   1/pp
m−1
1
G x ,x , S (f)p
 p,1 1 2 j,2 
≤ m | | 
j=0
X
  
. pp(G (x ,x ,G (f)))p.  
p,1 1 2 2,2
Hence we obtain
1/p
n−1m−1
1 S (x ,x ,f)p
 nm i=0 j=0 | i,j 1 2 | ! 
Ω = (x1,x2)∈ T2 :spu>p1ns,mu∈pN P Pp2lnln(p+2) > λ
 
 
 
 G (x ,x ,G (f)) 
(x ,x ) T2 : sup p,1 1 2 2,2 > λ , 
1 2
⊂ ∈ plnln(p+2)
(cid:26) p>1 (cid:27)
10 USHANGIGOGINAVAANDGRIGORIKARAGULYAN
then, applying Lemma 2 and 5, we conclude
Ω = I (x ,x )dx dx = dx I (x ,x )dx
Ω 1 2 1 2 2 Ω 1 2 1
| |
TZ2 ZT ZT
1/2
1
. G (x ,x ,f)dx dx
2,2 1 2 1 2
λ 
ZT ZT
 
1/2
1
. 1+ f (x ,x ) log+ f(x ,x ) dx dx
1 2 1 2 1 2
λ  | | | | 
ZT ZT
   1/2
1
. 1+ f (x ,x ) log+ f(x ,x ) dx dx .
1 2 1 2 1 2
λ  | | | | 
TZ2
  
Lemma is proved. (cid:3)
3. Proof of Theorem 1
Let L := L T2 be Orlicz space of functions on T2 generated by the
M M
Young function M(t) = tlog+t. Itis known that L is aBanach space with
(cid:0) (cid:1) M
respect to the Luxemburg norm
f
f := inf λ : λ > 0, M | | 1 < .
k kM  λ ≤  ∞
 XZ (cid:18) (cid:19) 
According to a theorem from ([13], Chap. 2, theorem 9.5) we have
0,5 1+ M( f ) f 1+ M( f )
 | |  ≤ k kM ≤ | |
TZ2 ZT2
 
provided f = 1. Hence from Lemma 6 we conclude
k kM
1/p
n−1m−1
1 S (f)p
(cid:12) nm i=0 j=0 | i,j | ! (cid:12) f 1/2
(21) (cid:12)(cid:12)(cid:12)(cid:12)spu>p1ns,mu∈pN pP2loPglog(p+2) > λ(cid:12)(cid:12)(cid:12)(cid:12). (cid:18)k λkM(cid:19) .
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
 
(cid:12) (cid:12)
 
Indeed, (cid:12)at first we deduce the case of f = 1, then(cid:12)using a homogeneity
(cid:12) k kM (cid:12)
argument, we get the inequality in the general case.