Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

385 https://internationalpubls.com 

𝑳𝟏-Convergence of Double Fourier Transform in 𝑳𝒑(𝑹) Spaces, 𝒑 ≥ 𝟏 

 

Sakshi1, Karanvir Singh2 

sakshi.bfcmt@gmail.com1, karanvir@mrsptu.ac.in2 

1 Department of Mathematics, Maharaja Ranjit Singh , Punjab Technical University, Bathinda, Punjab , India. 

 

Article History: 

Received: 18-07-2024 

Revised: 31-08-2024 

Accepted: 14-09-2024 

Abstract  

In this research article, we have given a method which restrict the double fourier transform 

of fϵL^p (R) spaces, 1≤p≤∞ . Further, we have discussed the convergence by using the 

approximate identities. The aim of this paper is to extend the results of  K. Devendra and 

S. Dimple[3] from one dimensional to two-dimensional trigonometric series. 

Keywords: Schwartz space, convolution operator, approximate identities, L^p-

Convergence. 

2020 Mathematics Subject Classification. 42A20, 42A32, 42A38. 

 

1. Introduction: 

Let 𝑓𝜖𝐿1(𝑅). The fourier transform of 𝑓(𝑥, 𝑦) is denoted by 𝑔(휁, 𝜙) and is defined by: 

𝑔(휁, 𝜙) =
1

√2𝜋
∫  

 

𝑅

𝑓(𝑥, 𝑦)𝑒𝑥𝑝−𝜄( 𝑥+𝜙𝑦)𝑑𝑥𝑑𝑦, 휁𝜖𝑅 

If 𝑓, 𝑔𝜖𝐿1(𝑅), then the inverse fourier transform of g is defined as:: 

𝑓(𝑥, 𝑦) =
1

√2𝜋
∫  

 

𝑅

𝑔(휁, 𝜙)𝑒𝑥𝑝𝜄( 𝑥+𝜙𝑦)𝑑휁𝑑𝜙 

for 𝑥𝜖𝑅. 

“As we know that several functions such as elementary constant functions 𝑠𝑖𝑛𝑤𝑡, 𝑐𝑜𝑠𝑤𝑡 do not 

converge in 𝐿1(𝑅) and thus they do not have fourier transforms. But when these functions are 

multiplied by characteristic functions , then the resultiong functions converge in 𝐿1(𝑅) and have fourier 

transforms". 

As we know, Lebesgue lemma states that if 𝑓𝜖𝐿1(𝑅) then 𝑙𝑖𝑚
| |→∞

|𝑔(휁)| = 0. From which it follows that 

“Fourier transform is a continuous linear operator from 𝐿1(𝑅) into 𝐶0(𝑅), the space of all continuous 

functions on R which decay at infinity , i.e. 𝑓(𝑥) → 0 as |𝑥| → ∞. We say that if 𝑓𝜖𝐿1(𝑅) , it is not 

necessary that g also belongs to 𝐿1(𝑅). 

In the present article, we provide a method for restricting Fourier transform of 𝑓𝜖𝐿𝑝(𝑅) spaces using 

the pointwise convergence of convolution operators for approximate identities." 

Definition 1.1. Let 𝜓휀𝐿1(𝑅) suct that 𝜉(0) = 1. Then 𝜓𝜖(𝑥, 𝑦) = 𝜖−1𝜓 (
𝑥

𝜖
,

𝑦

𝜖
) is called an 

approximate identity if 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

386 https://internationalpubls.com 

(i) ∫  
 

𝑅
𝜓𝜖(𝑥, 𝑦)𝑑𝑥𝑑𝑦 = 1 

(ii) 𝑠𝑢𝑝𝜖>0 ∫  
 

𝑅
|𝜓𝜖(𝑥, 𝑦)|𝑑𝑥𝑑𝑦 < ∞, 

(iii) 𝑙𝑖𝑚
𝜖→0

∫  
 

|𝑥|>𝛿,|𝑦|>𝛿
|𝜓𝜖(𝑥, 𝑦)|𝑑𝑥𝑑𝑦 = 0, for all 𝛿 > 0 " 

Proof. We can prove properties (i) and (ii) by following: 

∫  
 

𝑅

𝜓𝜖(𝑥, 𝑦)𝑑𝑥𝑑𝑦 = ∫  
 

𝑅

𝜖−1𝜓 (
𝑥

𝜖
,
𝑦

𝜖
) 𝑑𝑥𝑑𝑦 = ∫  

 

𝑅

𝜓 (
𝑥

𝜖
,
𝑦

𝜖
) 𝑑 (

𝑥

𝜖
,
𝑦

𝜖
) = 1 

For (iii), it follows that: 

∫  
 

|𝑥|>𝛿,|𝑦|>𝛿

𝜓𝜖(𝑥, 𝑦)𝑑𝑥𝑑𝑦 = ∫  
 

|𝑥|>𝛿,|𝑦|>𝛿

1

𝜖
𝜓 (

𝑥

𝜖
,
𝑦

𝜖
) 𝑑𝑥𝑑𝑦

= ∫  
∞

𝛿

1

𝜖
𝜓 (

𝑥

𝜖
,
𝑦

𝜖
) 𝑑𝑥𝑑𝑦 + ∫  

−𝛿

−∞

1

𝜖
𝜓 (

𝑥

𝜖
,
𝑦

𝜖
) 𝑑𝑥𝑑𝑦𝑆𝑢𝑏𝑠𝑡𝑖𝑡𝑢𝑡𝑖𝑛𝑔 𝑧 =

𝑥

𝜖
, 𝑡 =

𝑦

𝜖
 

, we get 

𝑙𝑖𝑚
𝜖→0

∫  
∞

𝛿
𝜖

𝜓(𝑧, 𝑡)𝑑𝑧𝑑𝑡 + ∫  
−

𝛿
𝜖

−∞

𝜓(𝑧, 𝑡)𝑑𝑧𝑑𝑡 = 0. 

Definition 1.2. “A sequence of functions ℎ𝑛𝑛 𝑁
 such that ℎ𝑛(𝑥, 𝑦) = 𝑛ℎ(𝑛𝑥, 𝑛𝑦) where 

𝑛 =
1

휀
, 𝑛 → ∞, 휀 → 0 

is called an approximate identity if 

(i) ∫  
 

𝑅
ℎ𝑛(𝑥, 𝑦)𝑑𝑥𝑑𝑦 = 1 for all n, 

(ii) 𝑠𝑢𝑝𝑛 ∫  
 

𝑅
ℎ𝑛(𝑥, 𝑦)𝑑𝑥𝑑𝑦 < +∞, 

(iii) 𝑙𝑖𝑚
𝑛→∞

∫  
 

|𝑥|>𝛿
ℎ𝑛(𝑥, 𝑦)𝑑𝑥𝑑𝑦 = 0 for every 𝛿 > 0." 

By following the above definition , the following proposition can easily prove: 

Proposition 1.1. “A sequence of functions ℎ𝑛𝑛 𝑁
 with ℎ𝑛 ≥ 0, ℎ𝑛(0,0) = 1 is an approximate identity 

if for every 휀 > 0 there exists 𝑛0휀𝑁 so that for all 𝑛 ≥ 𝑛0 we have ∫  
−

ℎ𝑛 > 1 − 휀. 

Let us consider the class 𝑆∗(𝑅) of 𝐶∞-functions on R which are rapidly decreasing i.e. Schwartz class 

such that 

𝑆∗(𝑅) = 𝑓: 𝑅 → 𝑅, 𝑠𝑢𝑝𝑥→𝑅(𝑥, 𝑦)
𝑑𝑚

𝑑𝑥𝑚

𝑑𝑚

𝑑𝑦𝑚 𝑓(𝑥, 𝑦) < ∞; 𝑛, 𝑚휀𝑁⋃(0)". 

We know that if 𝑓휀𝑆∗(𝑅), then 𝑔휀𝑆∗ and “𝑆∗(𝑅) ⊂ 𝐿𝑝(𝑅)". To prove the denseness of 𝑆∗(𝑅) ⊂

𝐿𝑝(𝑅), we have 휂휀𝑆∗(𝑅) ⇒ |휂(𝑥, 𝑦)| ≤
𝑐

1+|𝑥𝑦|𝑛. 

For 1 ≤ 𝑝 < ∞, 

∫  
 

𝑅

|휂(𝑥, 𝑦)|𝑝𝑑𝑥𝑑𝑦 ≤ ∫  
 

𝑅

𝑐𝑝

(1 + |𝑥𝑦|𝑛)𝑝
< ∞𝑤ℎ𝑖𝑐ℎ 𝑔𝑖𝑣𝑒𝑠 휂휀𝐿𝑝(𝑅) 



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. Define a sequence 휂𝑁 such that 

휂𝑁(𝑥, 𝑦) = 𝑓(𝑥, 𝑦) , if −𝑁 ≤ 𝑥, 𝑦 ≤ 𝑁; and otherwise it will become 0. 

⇒ ∃휂𝑁휀𝑆(𝑅), 𝑓휀𝐿𝑝(𝑅) such that 

∫  
 

𝑅
|휂𝑁 − 𝑓|𝑝𝑑𝑥 → 0. 

as 𝑁 → ∞. 

“Hence 𝑆∗(𝑅) is dense in 𝐿𝑝(𝑅)." 

Proposition 1.2. Let ℎ𝑛 = 𝛼𝑛𝜓𝑛 + (1 − 𝛼𝑛)𝜎𝑛, where {𝜓𝑛}𝑛 𝑁, {𝜎𝑛}𝑛 𝑁 are approximate identities 

and 0 ≤ 𝛼𝑛 ≤ 1. 

“(a) For 1 ≤ 𝑝 ≤ +∞ and every 𝑓휀𝐿𝑝(𝑅), 𝑙𝑖𝑚
𝑛→∞

(ℎ𝑛 − 𝜓𝑛) ⋆ 𝑓 → 0 and 𝑙𝑖𝑚
𝑛→∞

(ℎ𝑛 − 𝜎𝑛) ⋆ 𝑓 → 0. 

(b) For every 𝑓휀𝐿∞(𝑅), 𝑙𝑖𝑚
𝑛→∞

(ℎ𝑛 − 𝜓𝑛) ⋆ 𝑓 → 0 a.e.. 

(c) For 1 ≤ 𝑝 ≤ +∞ , if ∑   
𝑛 (1 − 𝛼𝑛)𝑝 < +∞, then for every 𝑓휀𝐿𝑝(𝑅), 𝑙𝑖𝑚

𝑛→∞
(ℎ𝑛 − 𝜓𝑛) ⋆ 𝑓 → 0" a.e. 

Proof. (a) “If 1 ≤ 𝑝 ≤ +∞ and every 𝑓휀𝐿𝑝(𝑅). 

Using Minkowski’s inequality, 

||(ℎ𝑛 − 𝜓𝑛) ⋆ 𝑓||
𝑝

≤ (1 − 𝛼𝑛) (||𝜎𝑛 ⋆ 𝑓 − 𝑓||
𝑝

+ ||𝜓𝑛 ⋆ 𝑓 − 𝑓||
𝑝

) 

Now, As proved by Singh D. and Singh D. [2] , we have 

If ℎ𝑛(𝑥) is an approximate identity and 𝑓휀𝐿𝑝(𝑅) , then ℎ𝑛 ⋆ 𝑓 → 𝑓휀𝐿𝑝(𝑅). 

So, by using the above, we obtain ||(ℎ𝑛 − 𝜎𝑛) ⋆ 𝑓||
𝑝

→ 0." 

(b) “For 𝑓휀𝐿∞(𝑅), |(ℎ𝑛 − 𝜓𝑛) ⋆ 𝑓| ≤ ||(ℎ𝑛 − 𝜓𝑛) ⋆ 𝑓|| → 0 by part (a)." 

(c) “For 𝑓휀𝐿𝑝(𝑅), 

∫  
 

𝑅
∑   

𝑛 (1 − 𝛼𝑛)𝑝|𝜎𝑛 ⋆ 𝑓(𝑥, 𝑦)|𝑝𝑑𝑥𝑑𝑦 = ∑   
𝑛 || (1 − 𝛼𝑛)𝜎𝑛 ⋆ 𝑓(𝑥, 𝑦)||𝑝

𝑝 ≤ ∑   
𝑛 (1 − 𝜎𝑛)𝑝||𝑓||𝑝

𝑝 <

+∞." 

Then (1 − 𝛼𝑛)𝜎𝑛 ⋆ 𝑓 → 0 a.e. . Similarly (𝛼𝑛 − 1)𝜓𝑛 ⋆ 𝑓 → 0 a.e. 

Definition 1.3. “An approximate identity {ℎ𝑛} is called 𝐿𝑝-good if ℎ𝑛 ⋆ 𝑓 → 𝑓 a.e. for all 𝑓휀𝐿𝑝(𝑅), 

and it is called good if it is 𝐿𝑝-good for every 1 ≤ 𝑝 ≤ +∞. An approximate identity {ℎ𝑛} is called 𝐿𝑝 

-bad if therte exists 𝑓휀𝐿𝑝(𝑅) such that ℎ𝑛 ⋆ 𝑓 not approachable to 𝑓 on a set of positive measure. 

Definition 1.4. Let {𝜓𝑛}𝑛 𝑁 and {𝜎}𝑛 𝑁 be approximate identities , 𝛼𝑛 be a sequence of real numbers 

with 0 ≤ 𝛼𝑛 ≤ 1 and 𝛼𝑛 → 1. We call preturbed approximate identities any approximate identity 

{ℎ𝑛}𝑛 𝑁 of the form ℎ𝑛𝜓𝑛 + (1 − 𝛼𝑛)𝜎𝑛." 

2. Main Results. 

Theorem 2.1. 

(i)“ Given any good approximate identity {𝜓𝑛}𝑛 𝑁 there exists a perturbed approximate identity 

{ℎ𝑛}𝑛휀𝑁 such that 𝑓휀𝐿𝑞(𝑅) 

(ℎ𝑛 ⋆ 𝑓)(휁, 𝜙) = ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙) (ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)) → 𝑓(𝑥, 𝑦)1 ≤ 𝑞 < 𝑝 

(ii) (ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)) → 𝑓(𝑥, 𝑦) for 𝑞 > 𝑝 and (ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)) not approaches to𝑓(𝑥, 𝑦) for 1 ≤

𝑞 ≤ 𝑝. 



Communications on Applied Nonlinear Analysis 

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(iii) (ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)) → 𝑓(𝑥, 𝑦) for 𝑞 = ∞ 

(ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙) not approachable to 𝑓(𝑥, 𝑦) for 1 ≤ 𝑞 < ∞." 

Proof. (i) Let 

𝑔𝑛(𝑥) =
1

√2𝜋
∫  

 

𝑅

𝑒𝜄(𝑥 +𝑦𝜙)ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)𝑑휁𝑑𝜙 =
1

√2𝜋
∫  

 

𝑅

𝑒𝜄(𝑥 +𝑦𝜙)ℎ�̂�(휁, 𝜙) ∫  
 

𝑅

𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦

=
1

√2𝜋
∫  

 

𝑅

ℎ𝑛(𝑥 + 𝑦)𝑓(𝑥, 𝑦)𝑑𝑥𝑑𝑦𝑜𝑟 = (ℎ𝑛 ⋆ 𝑓)(𝑥, 𝑦)ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)

=
1

√2𝜋
∫  

 

𝑅

𝑒𝜄(𝑥 +𝑦𝜙)ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙)𝑑휁𝑑𝜙 = (ℎ𝑛 ⋆ 𝑓)(𝑥, 𝑦)“𝐹𝑖𝑥 𝑞 ≥ 𝑝 

and taking 1 − 𝛼𝑛 =
1

(𝑛𝑙𝑜𝑔2𝑛)1/𝑝
. Since ∑   

𝑛 (1 − 𝛼𝑛)𝑞 < +∞ and 𝜓𝑛 is an 𝐿𝑞-good approximate 

identity , using Proposition 1.4. we obtain that ℎ𝑛 is also an 𝐿𝑞-good approximate identity." 

Hence for 𝑞 ≥ 𝑝,(ℎ𝑛 ⋆ 𝑓)(𝑥, 𝑦) → 𝑓(𝑥, 𝑦) 

Now, we have to prove that for each 1 ≤ 𝑞 < 𝑝. there exists 𝑓𝑞휀𝐿𝑞(𝑅) so that “𝑙𝑖𝑚𝑠𝑢𝑝𝑘|𝑥|𝑘 𝑑𝑘

𝑑𝑥𝑘 (ℎ𝑘 ⋆

𝑓𝑞 → ∞)" on a set of positive measure. 

Set, 

𝑓𝑞(𝑥, 𝑦) =
1

(𝑥𝑦𝑙𝑜𝑔2(𝑥/2,𝑦/2))
1/𝑞 𝜒[0,1](𝑥)휀𝐿𝑞(𝑅). 

Take 

𝑟𝑛 =
1

𝑛1+1/𝑝(𝑙𝑜𝑔𝑛)2/𝑝 , 𝑎𝑛 = 𝑟𝑛

1

𝑝+1 =
1

𝑛1/𝑝
(𝑙𝑜𝑔𝑛)

2

𝑝(𝑝+1), 

“𝐽𝑛 = [𝑎𝑛 − 𝑟𝑛, 𝑎𝑛 + 𝑟𝑛] and 

𝑈𝑛 = [−𝑎𝑛 + 𝑟𝑛, −𝑎𝑛+1 + 𝑟𝑛+1]," 

for sufficiently large n and for all 𝑘 ≥ 𝑛, 𝑥휀𝑈𝑘. 

ℎ𝑘 ⋆ 𝑓𝑞(𝑥, 𝑦) ≥ (1 − 𝛼𝑘)𝜎𝑘 ⋆ 𝑓𝑞(𝑥) ≥
1

(𝑘𝑙𝑜𝑔2(𝑘))
1/𝑝

∫  
 

−𝐽𝑘

𝜎𝑘(𝑥, 𝑦)𝑓𝑞(𝑥 − 𝑦)𝑑𝑥𝑑𝑦𝑁𝑜𝑤, 𝑤𝑒 ℎ𝑎𝑣𝑒, 

ℎ𝑘 ⋆ 𝑓𝑞(𝑥, 𝑦) ≥
𝑓𝑞(𝐶𝑟𝑘

(𝑙𝑜𝑔𝑘)2/𝑝+1)

(𝑘𝑙𝑜𝑔2𝑘)1/𝑝
∫  

 

−𝐽𝑘

𝜎𝑘(𝑥, 𝑦)𝑑𝑥𝑑𝑦 

or , 𝑓𝑞(𝐶𝑟𝑘
(𝑙𝑜𝑔𝑘)2/𝑝+1) =

𝑘1/𝑞+1/𝑝𝑞(𝑙𝑜𝑔𝑘)
2

𝑝𝑞(𝑝+1)

𝐶1/𝑞(𝑙𝑜𝑔(𝐶/2𝑘(𝑝+1)/𝑝(𝑙𝑜𝑔𝑘)2/𝑝(𝑝+1)))
2/𝑞 

 

Then, ℎ𝑘 ⋆ 𝑓𝑞(𝑥, 𝑦) ≥ 𝐶𝑘

1

𝑞
−

1

𝑝
+

1

𝑝𝑞𝐻𝑞(𝑘) > 𝑘𝛿 ≥ 𝑛𝛿), 

 

where, 𝐻𝑞(𝑘) =
(𝑙𝑜𝑔𝑘)

2
𝑝𝑞(𝑝+1)

−
2
𝑝

𝐶1/𝑞(𝑙𝑜𝑔(𝐶/2𝑘(𝑝+1)/𝑝(𝑙𝑜𝑔𝑘)2/𝑝(𝑝+1))
2/𝑞 



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Vol 32 No. 1 (2025) 

 

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and 0 < 𝛿 < 1/𝑞 − 1/𝑝 + 1/𝑝𝑞. 

So, 
𝑑𝑘

𝑑𝑥𝑘

𝑑𝑘

𝑑𝑦𝑘 (ℎ𝑘 ⋆ 𝑓𝑞(𝑥, 𝑦)) ≥ 𝐶
𝑑𝑘

𝑑𝑥𝑘

𝑑𝑘

𝑑𝑦𝑘 (𝑘1/𝑞−1/𝑝+1/𝑝𝑞𝐻𝑞(𝑘)) 

or, |𝑥𝑦|𝑛 𝑑𝑛

𝑑𝑥𝑛

𝑑𝑛

𝑑𝑦𝑛
(ℎ𝑘 ⋆ 𝑓𝑞(𝑥, 𝑦)) ≥ |𝑥𝑦|𝑛 ∫  

 

−𝐽𝑘
𝑓𝑞(𝑥 − 𝑦)

𝑑𝑛

𝑑𝑥𝑛

𝑑𝑛

𝑑𝑦𝑛
𝜎𝑘(𝑥, 𝑦)𝑑𝑥𝑑𝑦 

for 𝑘 ≥ 𝑛, |𝑥𝑦|𝑘 𝑑𝑘

𝑑𝑥𝑘

𝑑𝑘

𝑑𝑦𝑘 (ℎ𝑘 ⋆ 𝑓𝑞(𝑥, 𝑦)) ≥ |𝑥𝑦|𝑛 𝑑𝑛

𝑑𝑥𝑛 𝑛𝛿 ≥ |𝑥𝑦|𝑛 𝑑𝑛

𝑑𝑥𝑛

𝑑𝑛

𝑑𝑦𝑛 (
1

(𝑥−𝑦)𝑝𝛿) 

                                                                            = 
|𝑥𝑦|𝑛(−1)𝑛(𝑝𝛿+𝑛−1)!

(𝑝𝛿)!(𝑥−𝑦)𝑝𝛿+𝑛  

                                                                          ≥ |𝑥𝑦|𝑛 (−1)𝑛(𝑝𝛿+𝑛−1)!

(𝑝𝛿)!𝐶𝑟𝑛(𝑙𝑜𝑔𝑛)2/𝑝+1(𝑙𝑜𝑔𝑛)2𝛿/𝑝+1 → ∞ as 𝑛 → ∞. 

“In view of Sawyer’s Principle [4] , there exists a functions 𝑓휀𝐿𝑞([0,1)) ⊆ 𝐿𝑞(𝑅) such that 

𝑙𝑖𝑚𝑠𝑢𝑝𝑛|𝑥𝑦|𝑛 𝑑𝑛

𝑑𝑥𝑛

𝑑𝑛

𝑑𝑦𝑛
(ℎ𝑛 ⋆ 𝑓) → ∞ a.e. on a set of positive measure in R. It follows that (ℎ𝑛 ⋆ 𝑓) not 

belongs to 𝑆(𝑅) or ℎ𝑛 ⋆ 𝑓not approachable to 𝑓 or ℎ�̂�(휁, 𝜙)𝑓(휁, 𝜙) not approachable to 𝑓(𝑥, 𝑦) for 

1 ≤ 𝑞 < 𝑝. 

Let 𝑝𝑛 be a decreasing sequence of real numbers such that 𝑝1 > 𝑝2 >. . . . . 𝑝𝑛 >. . . . . . 𝑝. for each 𝑝𝑖 we 

can construct a perturbation {ℎ𝑛
𝑖 }𝑛 of {𝜓𝑛} that is 𝐿𝑞-good for 𝑞 ≥ 𝑝𝑖, and 𝐿𝑞-bad for 1 ≥ 𝑞 < 𝑝𝑖. 

Consider a sequence of blocks {𝑇𝑘}𝑘 𝑁, where 𝑇𝑘 = {ℎ𝑛𝑘−1+1, . . . . . , ℎ𝑛𝑘
𝑘 } and {𝑛𝑘} is a sequence of 

positive integers increasing to infinity. Let 𝑆𝑘 = {𝑛𝑘−1 + 1, . . . . , 𝑛𝑘}. and let {ℎ𝑛}𝑛 = 𝑈𝑘𝑇𝑘. Now, fix 

𝑞 > 𝑝. There exists 𝑛0휀𝑁 so that for all 𝑛 > 𝑛0 we have 𝑝𝑛 < 𝑞". 

∑  ∞
𝑘=𝑛0

∑   
𝑛 𝑆𝑘

(1 − 𝛼𝑛
𝑘)𝑞 ≤ ∑  ∞

𝑘=𝑛0
∑   

𝑛 𝑆𝑘

1

(𝑛𝑙𝑜𝑔2𝑛)

𝑞/𝑝𝑛0
≤ ∑   

𝑛
1

(𝑛𝑙𝑜𝑔2𝑛)𝑞/𝑝𝑛0
< ∞. 

Using Proposition 1.2(c), we get ℎ𝑛 ⋆ 𝑓 → 𝑓 for 𝑓휀𝐿𝑞(𝑅), 𝑞 > 𝑝, or ℎ�̂�(휁, 𝜓)𝑓(휁, 𝜓) → 𝑓(𝑥, 𝑦) for 

𝑞 > 𝑝. 

Now consider a sequence “𝐶𝑖
𝑁 → ∞ as 𝑖 → ∞. Since {ℎ𝑛

𝑖 }𝑛 is 𝐿𝑞-bad for all 𝑞 < 𝑝𝑖, it is also 𝐿𝑝-bad. 

These exists 𝑓𝑖휀𝐿𝑝([0,1)) and 𝜆𝑖
𝑁 > 0 such that 

||𝑠𝑢𝑝𝑛>𝑛𝑖−1
ℎ𝑛

𝑖 ⋆ 𝑓𝑖(𝑥, 𝑦) ∥> ∫  ||ℎ𝑛
𝑖 ⋆ 𝑓𝑖(𝑥, 𝑦) ∥𝑝 𝑑𝑥𝑑𝑦 > 𝐶𝑁||𝑓𝑖(𝑥𝑦 − 𝜆𝑖

𝑁)||𝑝
𝑝
 

                                          =  2𝐶𝑖
𝑁 , [||𝑓𝑖(𝑥𝑦 − 𝜆𝑖

𝑁)||𝑝 = 21−𝑖, 𝐶𝑁 = 2(𝑖−1)𝑝+1𝐶𝑖
𝑁]." 

It follows that there exists 𝑛𝑖 > 𝑛𝑖−1, so that ||𝑠𝑢𝑝𝑛𝑖−1<𝑛≤𝑛𝑖
ℎ𝑛

𝑖 ⋆ 𝑓𝑖 ∥> 𝐶𝑖
𝑁. 

Set      𝑓 = ∑   
𝑖 𝑓𝑖, then ||𝑓||𝑝 ≤ ∑   

𝑖 ||𝑓𝑖||𝑝 ≤ 2. 

“Suppose that {ℎ𝑛} satisfies a weak (𝑝, 𝑝) inequality in 𝐿𝑝([0,1)), We know that if 𝜇 be a finite positive 

Borel measure, then there exists a sequence 𝜇𝑛 of atomic measure that converges to 𝜇 weakly or if f 

has compact support then 

∫  
 

𝑅

𝑑𝜇𝑛𝑓(𝑥, 𝑦) → ∫  
 

𝑅

𝑓(𝑥, 𝑦)𝑑𝜇𝑑𝛾 

where, 𝜇𝑛 → 𝜇 and 𝛾𝑛 → 𝛾, weakly. 

If 𝑓휀𝐿1(𝑅), 𝑑𝜇𝑑𝛾 = |𝑓(𝑥, 𝑦)|𝑑𝑥𝑑𝑦 is a finite Borel measure, so we can find 

𝛾𝑛𝜇𝑛 = ∑  𝑁
𝑖=1 𝐶𝑖

𝑁𝛿𝜆𝑖
𝑁 → 𝜇𝛾 weakly. 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

390 https://internationalpubls.com 

Consider 

|{𝑠𝑢𝑝𝑛(ℎ𝑛
𝑖 ⋆ 𝑓)}| = ∫  

 

−𝐽𝑘

|ℎ𝑛
𝑖 (𝑥, 𝑦)𝑓(𝑥 − 𝑦)|

𝑝
𝑑𝑥𝑑𝑦 ≤ ∫  

 

−𝐽𝑘

|ℎ𝑛
𝑖 (𝑥, 𝑦)𝑑𝜇𝑛(𝑥 − 𝑦)|

𝑝
𝑑𝑥𝑑𝑦

≤ || ∑  

𝑁

𝑖=1

𝑓(𝑥𝑦 − 𝜆𝑖
𝑁)𝐶𝑖

𝑁||𝑝
𝑝 ∑  

𝑁

𝑖=1

𝐶𝑖
𝑁||𝑓(𝑥𝑦 − 𝜆𝑖

𝑁)||𝑝
𝑝 ≤ 𝐶0

𝑁||𝑓||𝑝
𝑝
 

                                             = 2𝑝𝐶0
𝑁." (1) 

On the other hand 

|{𝑠𝑢𝑝𝑛(ℎ𝑛 ⋆ 𝑓)}| ≤ |{𝑠𝑢𝑝𝑛𝑖−1<𝑛≤𝑛𝑖
(ℎ𝑛

𝑖 ⋆ 𝑓(𝑖))}| > 𝐶𝑖
𝑁 (2) 

Combining equations, we get 

𝐶0
𝑁 > 𝐶𝑖

𝑁𝐵𝑢𝑡 𝐶𝑖
𝑁 → ∞ 

 as 𝑖 → +∞. Hence ℎ𝑛 ⋆ 𝑓 not approachable to f in 𝐿𝑝([0,1)). Since the spaces 𝐿𝑞([0,1)) are nested , 

{ℎ𝑛} is 𝐿𝑞([0,1))-bad for all 1 ≤ 𝑞 ≤ 𝑝. Therefore, such a choice of {𝑛𝑘} makes {ℎ𝑛} 𝐿𝑞(𝑅)-bad for 

all 1 ≤ 𝑞 ≤ 𝑝. This implies that ℎ�̂�(휁, 𝜓)𝑓(휁, 𝜓) not approachable to 𝑓(𝑥, 𝑦) for 1 ≤ 𝑞 ≤ 𝑝. 

(iii) “Let {𝜓𝑛}𝑛 𝑁 be a good approximate identity and let {휁𝑛}𝑛 𝑁 be any approximate identity. Let 

{𝑝𝑛} be a sequence of real numbers satisfying 

1 ≤ 𝑝1 < 𝑝2 <. . . . . . < 𝑝𝑛 → ∞ 

Consider the blocks {𝑇𝑘}, where each block 𝑇𝑘 is related to 𝑝𝑖 , for 𝑖휀𝑆𝑛, let 

ℎ𝑖 = 𝛼𝑖
𝑘𝜓𝑖

𝑘 + (1 − 𝛼𝑖
𝑘)𝜎𝑖

𝑘 . 

Choose 𝑛𝑘 such that 𝛼𝑖
𝑘 → 1. Then since {𝜓𝑛} is 𝐿∞ good, 

𝜓𝑛 ⋆ 𝑓 → 𝑓a.e. for all 𝑓휀𝐿∞(𝑅), 

and, 𝛼𝑖
𝑘𝜓𝑖

𝑘 ⋆ 𝑓 → 𝑓 a.e. for all 𝑓휀𝐿∞(𝑅) 

Since, 𝜎𝑖
𝑘 ⋆ 𝑓(𝑥) ≤ ||𝑓||∞. 

(1 − 𝛼𝑖
𝑘)𝜎𝑖

𝑘 ⋆ 𝑓 → 0 a.e. for all 𝑓휀𝐿∞(𝑅). 

it follows that  ℎ𝑛 ⋆ 𝑓 → 0 a.e. for all 𝑓휀𝐿∞(𝑅). 

This implies that (ℎ�̂�(휁, 𝜓)𝑓(휁, 𝜓)) → 𝑓(𝑥, 𝑦) for 𝑞 = ∞. 

The approximate identity {ℎ𝑛
𝑘}𝑛 is 𝐿𝑝𝑚-bad for every 𝑚휀{1, . . . . , 𝑘}, since it is 𝐿𝑞-bad for every 1 ≤

𝑞 ≤ 𝑝𝑘. There exists 𝑓𝑚
𝑘휀𝐿𝑝𝑚([0,1)) with ||𝑓𝑚

𝑘(𝑥𝑦 − 𝜆𝑚
𝑘(𝑁)

|| = 2−𝑘, 𝜆𝑚
𝑘(𝑁)

> 0 and 𝑛𝑚
𝑘 > 𝑚𝑘−1 so that 

|{𝑠𝑢𝑝𝑛𝑘−1
< 𝑛 < 𝑛𝑚

𝑘 (ℎ𝑛
𝑘 ⋆ 𝑓𝑚

𝑘)}| > 𝐶𝑁||𝑓𝑚
𝑘(𝑥𝑦 − 𝜆𝑚

𝑘(𝑁)
||𝑝𝑚

𝑝𝑚 = 
𝐶𝑘

𝑁

2𝑘𝑝𝑚
 

Let 𝑓 = ∑   
𝑘≥𝑘0

𝑓𝑘0

𝑘 , then ||𝑓||𝑝𝑘0
< 2. 

So, |{𝑠𝑢𝑝𝑛(ℎ𝑛 ⋆ 𝑓)}| ≤ 𝐶0||𝑓||𝑝𝑘0

𝑝𝑘0 ≤ 2𝑝𝑘0 𝐶0
𝑁.               (3) 

Hence, |{𝑠𝑢𝑝𝑛(ℎ𝑛 ⋆ 𝑓)}| ≥ |{𝑠𝑢𝑝𝑛𝑘−1
< 𝑛 < 𝑛𝑚

𝑘 (ℎ𝑛
𝑘 ⋆ 𝑓𝑘0

𝑘 )}| >
𝐶𝑘

𝑁

2
𝑘𝑝𝑘0

"                       (4) 

 



Communications on Applied Nonlinear Analysis 

ISSN: 1074-133X 

Vol 32 No. 1 (2025) 

 

391 https://internationalpubls.com 

Using the equations, we get 

𝐶0
𝑁 >

𝐶𝑘
𝑁

2𝑘𝑝𝑘0
(𝑘+1)

→ +∞ 

Thus we conclude that ℎ�̂�(휁, 𝜓)𝑓(휁, 𝜓) not approachable to 𝑓(𝑥, 𝑦) for 1 ≤ 𝑞 ≤ ∞.          

Hence, the proof is completed. 

References: 

[1] A.Bellow, Perturbation of a sequence , Advances in Mathematics, 78(1989), 131-139. 

[2] K. Devendra and S. Dimple, Fourier Transform in 𝐿𝑝(𝑅) Spaces , 𝑝 ≥ 1, 3(2011), 14-25. 

[3] K. Reinhold-Larsson, Discrepancy of behaviour of perturbed sequences in 𝐿𝑝-spaces , Proc. Amer. Math. Soc., 

120(1994), 865-874. 

[4] S. Sawyer , Maximal inequalities of weak type, Ann. of Math, 84(2)(1966), 157-174.