Some Complementary Results on Convergence of Double Singular

Modelling & Application & Theory ISSN: 2548–0596 Vol. 1, No. 1, 2016, pp. 13-22

Some Complementary Results on Convergence of Double Singular Integral Operators Gumrah Uysal1,∗ , Vishnu Narayan Mishra2 1 2

Dep. Computer Technologies, Karabuk University, Karabuk, Turkey. Mathematics, Indira Gandhi National Tribal University, Madhya Pradesh, India

Abstract. The paper is devoted to the study of pointwise approximation of functions f ∈ L1,ϕ (D) by double singular integral operators with radial kernels at µ−generalized Lebesgue points. Here, ϕ : R2 → R+ is a weight function satisfying some sharp conditions including almost everywhere differentiability on its domain, and L1,ϕ (D) is the collection of all measurable and non-integrable functions for which ϕf is integrable on D, where D = ha, bi × hc, di is an arbitrary bounded open, semi open or closed rectangular region or D = R2 . Keywords. µ−generalized Lebesgue point, radial kernel, weighted approximation. 2000 MSC. 41A35, 41A25, 45P05.

1. Introduction As far as we know, Fichtenholz [8] was the first mathematician who introduced the theory of integrals depending on parameters. Later on, many researchers studied and used this theory in different subjects of mathematics including approximation by integral operators. Now, we highlight some of the studies as follows: In [25], Taberski proved some theorems on the pointwise convergence of the two-parameter convolution type linear singular integrals in the following form: Zπ

Lλ ( f ; x) =

f (t) Kλ (t − x) dt, x ∈ h−π, πi , λ ∈ Λ ⊂ R+ 0,

(1)

−π

where Kλ (t) is the kernel fulfilling appropriate conditions. The indicated study also contains the lemma which is a generalization of well-known Natanson’s lemma [18]. The works by Gadjiev [10] and Rydzewska [19], which are continuations of Taberski [25], are devoted to the study of pointwise convergence of the operators of type (1) on some planar sets consisting of the characteristic points of various types. Then Bardaro [2] presented significant results about the rate of pointwise convergence of some classes of the linear singular integral operators. Distinctively, Esen [7] obtained some approximation results concerning the pointwise convergence and the rate of pointwise convergence of nonconvolution type linear singular integral operators at p−Lebesgue points (1 ≤ p < ∞). In the mentioned works and in many others the kernel function satisfies the conditions equivalent to usual approximate identities (see, for example, [5, 23]). In this context, the notion of approximate identity is directly related to characterization of δ −function presented by Dirac [6]. ∗ Corresponding

author (E-mail) [email protected]

14

G. Uysal, V. N. Mishra

In [26], Taberski advanced his previous analysis [25] by studying the pointwise approximation of functions f ∈ L1 (R) which are 2π−periodic with respect to each variable, where R = h−π, πi × h−π, πi by the three-parameter family of convolution type double singular integral operators in the following form: ZZ

Lλ ( f ; x, y) =

f (t, s)Kλ (t − x, s − y) dsdt, (x, y) ∈ R,

(2)

R

where λ ∈ Λ ⊂ R+ 0 . We have to mention here the paper by Gahariya [11], whose main concern was approximating functions of two variables by the operators of type (2) at their Lebesgue points as λ tends to infinity. For further studies concerning pointwise approximation by the operators of type (2), we refer the reader to [16, 20, 21, 22]. Some researchers considered the theory of pointwise approximation in weighted Lebesgue spaces. Therefore, Alexits [1], Mamedov [17] and Esen [7] presented necessary conditions satisfied by kernel functions to obtain a weighted convergence, separately. Also, Taberski [27] studied the weighted pointwise convergence of double singular integral operators of type (2) using two dimensional counterparts of the conditions obtained by Alexits [1]. Esen [7] used the characterization based on almosteverywhere differentiability of the product of weight function and kernel function and eventually, she obtained that the monotonicity property of this product coincided with the monotonicity property of the kernel function. This idea is used in some works, such as [31]. Also, for the analogous characterization based on directly monotonicity of product of weight function and kernel function, we refer the reader to [17]. In [28], Uysal et al. investigated the pointwise convergence of double singular integral operators with radial kernels in the following setting: ZZ

Lλ ( f ; x, y) =

f (t, s)Hλ (t − x, s − y) dsdt, (x, y) ∈ D, λ ∈ Λ,

(3)

D

where D = ha, bi × hc, di is an arbitrary bounded closed, semi-closed or open rectangular region or D = R2 , and f ∈ L p (D) (1 ≤ p < ∞) . Later on, some weighted pointwise approximation results for the operator of type (3) were presented in [29] using two dimensional counterpart of the characterization explained above used by Esen [7]. In this method, the weight function ϕ : R2 → R+ is almost everywhere differentiable ϕ(t+x,s+y) on its domain and possesses some properties, such as φ (t, s) = sup , for every (t, s) ∈ D in ϕ(t,s) (x,y)∈D

order to obtain the well-definiteness of the operators of type (3). The indicated property is used in many major works, such as [9, 24]. On the other hand, in recent paper [30], same authors studied the weighted pointwise convergence of the operators of type (3) in L1,ϕ R2 by using different weight function. In this work, the handled weight function ϕ : R2 → R+ was submultiplicative, i.e., ϕ (t + x, s + y) ≤ ϕ (t, s) ϕ (x, y) , (t, s) , (x, y) ∈ R2 besides other vital properties. This property is used in many major works with different purposes, for example, Beurling [3] used it while defining his algebra. Also, for more information on submultiplicative weight functions, we refer the reader to [13, 14, 15, 17]. This paper may be seen as a continuation and further generalization of [28]. In this paper, our main concern is to prove the pointwise convergence of the operators of type (3) at µ−generalized Lebesgue points as (x, y, λ ) tends to (x0 , y0 , λ0 ). Here, ϕ > 0 is a weight function satisfying some sharp conditions and L1,ϕ (D) f is the collection of all measurable and non-integrable functions for which ϕ is integrable on D, where D = ha, bi × hc, di is bounded open, semi open or closed rectangular region or D = R2 . We will use two dimensional counterpart of the conditions presented by Esen [7] as in [29].

1, 1, 2016, pp. 13-22

Some Complementary Results on Convergence of Double Singular Integral Operators

15

The paper is organized as follows: In Section 2, we give some preliminary concepts. In Section 3, main results are presented. In Section 4, the rate of pointwise convergence of the operators of type (3) is established.

2. Preliminaries 2 is said to be radial, if there exists a function K : R+ → R such that Definition 1. A function H ∈ L R 1 0 √ t 2 + s2 almost everywhere [4]. H (t, s) = K In the following definition we used the class definition given in [28]. Definition 2. Class Aϕ Let Hλ : R2 × Λ → R be a radial i.e., there exists a function Kλ : √function, + 2 2 t + s holds for almost every (t, s) ∈ R2 , R0 × Λ → R such that the equality given as Hλ (t, s) = Kλ where Λ is a given set of non-negative numbers with accumulation point λ0 . In addition, let φ (t, s) = ϕ(t+x,s+y) sup ϕ(x,y) for every (t, s) ∈ D and ϕ : R2 → R+ . (x,y)∈D

Hλ (t, s) belongs to class Aϕ , if the following conditions are satisfied: √ (a) Hλ (t, s) = Kλ t 2 + s2 is non-negative and integrable as a function of (t, s) on R2 for each fixed λ ∈ Λ. q x02 + y20 tends to infinity as λ tends to λ0 . (b) For fixed (x0 , y0 ) ∈ D, Kλ RR √ (c) limλ →λ0 R2 Kλ t 2 + s2 dsdt − 1 = 0. h i √ (d) limλ →λ0 sup √ 2 2 Kλ t 2 + s2 = 0, ∀ξ > 0. hRR ξ < t +s √ i 2 + s2 dsdt = 0, ∀ξ > 0. √ (e) limλ →λ0 t K λ < t 2 +s2

ξ q

2 2 φ (., .) K (f) (.) + (.) ≤ M < ∞, ∀λ ∈ Λ. λ

L1 (R2 ) √ (g) Kλ t 2 + s2 is non-increasing with respect to t on [0, ∞) and non-decreasing on (−∞, 0], similarly, √ Kλ t 2 + s2 is non-increasing with respect to s on [0, ∞) and non-decreasing on (−∞, 0] , for any √ λ ∈ Λ. Kλ t 2 + s2 is bimonotonically increasing with respect to (t, s) on [0, ∞) × [0, ∞) and (−∞, 0] × (−∞, 0] and similarly, bimonotonically decreasing with respect to (t, s) on [0, ∞) × (−∞, 0] and (−∞, 0] × [0, ∞) for any λ ∈ Λ. Throughout this paper we suppose that the kernel function Hλ (t, s) belongs to class Aϕ . Remark 2.1. The studies [9, 26, 27, 29], among others, are used as main reference works in the construction stage of class Aϕ . Therefore, we refer the reader to see the indicated works. For more information about the concept of bimonotonicity, the reader may see also, for example, [12]. Our main results are based on the following theorem. Theorem 1. Let 1 ≤ p < ∞. If f ∈ L p,ϕ (D) , then Lλ ( f ; x, y) defines a continuous transformation acting on L p,ϕ (D) [29].

16


3. Pointwise Convergence Theorem 2. Suppose that D = ha, bi×ha, ci is an arbitrary bounded closed, semi-closed or open rectangular region. If (x0 , y0 ) ∈ D is a common µ−generalized Lebesgue point of the functions f ∈ L1,ϕ (D) and ϕ ∈ L1 (D) , then

Lλ ( f ; x, y) = f (x0 , y0 )

lim

(x,y,λ )→(x0 ,y0 ,λ0 )

under the conditions q 2 2 ∂ Kλ (t − x) + (s − y) ∂t

×

∂ ϕ (t, s) > 0, for each fixed (x, y) ∈ D, ∂t

(4)

×

∂ ϕ (t, s) > 0, for each fixed (x, y) ∈ D, ∂s

(5)

q 2 2 ∂ Kλ (t − x) + (s − y) ∂s and ∂ 2 Kλ

q

(t − x)2 + (s − y)2

∂ 2 ϕ (t, s) > 0, for each fixed (x, y) ∈ D (6) ∂t∂ s ∂t∂ s q provided that first and second order (mixed) partial derivatives of Kλ (t − x)2 + (s − y)2 and ϕ (t, s) ×

with respect to (t, s) exist almost everywhere on R2 , on any set Z on which the function x0Z+δ y0Z+δ q 0 0 2 2 (t − x) + (s − y) {µ1 (|x0 − t|)}t {µ2 (|y0 − s|)}s dsdt ϕ (t, s) Kλ

(7)

x0 −δ y0 −δ

y0Z+δ

+2µ1 (|x0 − x|)

y0 −δ x0Z+δ

+2µ2 (|y0 − y|)

x0 −δ

0 ϕ (x, s) Kλ (|s − y|) {µ2 (|y0 − s|)}t ds 0 ϕ (t, y) Kλ (|t − x|) {µ1 (|x0 − t|)}t dt

+4ϕ (x, y) Kλ (0) µ1 (|x0 − x|)µ2 (|y0 − y|) remains bounded as (x, y, λ ) tends to (x0 , y0 , λ0 ).

Proof. Let (x0 , y0 ) ∈ D be a common µ−generalized Lebesgue point (for the definition and related concepts, see [20, 28, 30] and the references therein) of the functions f ∈ L1,ϕ (D) and ϕ ∈ L1 (D) . Let |x − x0 | < δ2 and |y − y0 | < δ2 for a given δ > 0. The proof will be given for the case 0 < x0 − x < δ2 and 0 < y0 − y < δ2 for all δ > 0 satisfying x0 + δ < b, x0 − δ > a, y0 + δ < d and y0 − δ > c.

1, 1, 2016, pp. 13-22


17

According to the definition of µ−generalized Lebesgue point used in [28], for all given ε > 0, there exists δ > 0 such that for all h, k satisfying 0 < h, k ≤ δ , the inequality Zx0 Zy0 f (t, s) − f (x0 , y0 ) dsdt < µ1 (h) µ2 (k) (8) ϕ (t, s) ϕ (x0 , y0 ) x0 −δ y0 −δ

holds.

Set I(x, y, λ ) := |Lλ ( f ; x, y) − f (x0 , y0 )|. It is easy to see that ZZ f (t, s) Hλ (t − x, s − y) dsdt − f (x0 , y0 ) I(x, y, λ ) = D q ZZ 2 2 f (t, s) Kλ = (t − x) + (t − y) dsdt − f (x , y ) 0 0 D

≤

ZZ D

q 2 2 f (t, s) − f (x0 , y0 ) ϕ (t, s) Kλ (t − x) + (s − y) dsdt ϕ (t, s) ϕ (x0 , y0 )

q f (x0 , y0 ) ZZ 2 2 + ϕ (t, s) Kλ (t − x) + (s − y) dsdt − ϕ (x0 , y0 ) . ϕ (x0 , y0 ) D holds. Further, we may easily write    q ZZ   f (t, s)  ZZ f (x0 , y0 ) 2 2 + ϕ (t, s) K I(x, y, λ ) ≤ − (t − x) + (s − y) dsdt λ ϕ (t, s) ϕ (x0 , y0 )     Bδ D\Bδ q f (x0 , y0 ) ZZ 2 2 ϕ (t, s) Kλ (t − x) + (s − y) dsdt − ϕ (x0 , y0 ) + ϕ (x0 , y0 ) D = I1 + I2 + I3 , n o where Bδ := (t, s) : (t − x0 )2 + (s − y0 )2 ≤ δ 2 , (x0 , y0 ) ∈ D .

From Theorem 1 in [28], we see that I3 → 0 as (x, y, λ ) tends to (x0 , y0 , λ0 ) .

From (4) − (6), we see that the monotonicity properties of Kλ

q

2

2

(t − x) + (s − y)

and ϕ (t, s) coincides

for each fixed (x, y) ∈ D. Therefore, we can write for the integral I1 : f (x0 , y0 ) δ |b − a| |c − d| . I1 ≤ Kλ √ sup ϕ (t, s) k f kL1,ϕ (D) + ϕ (x0 , y0 ) 2 D\Bδ

Hence, I2 → 0 as (x, y, λ ) tends to (x0 , y0 , λ0 ) by condition (d) .

18


The following inequality holds for the integral I21 :   y0  x0Z+δ Z q  Zx0 Zy0 f (t, s) f (x0 , y0 ) 2 2 + ϕ (t, s) K I21 ≤ − (t − x) + (s − y) dsdt λ  ϕ (t, s) ϕ (x0 , y0 )  x0 y0 −δ

x0 −δ y0 −δ

+

  Zx0 y0Z+δ 

x0 −δ y0



+

x0Z+δ y0Z+δ  x0

y0

q f (t, s) f (x0 , y0 ) 2 2 ϕ (t, s) K − (t − x) + (s − y) dsdt λ  ϕ (t, s) ϕ (x0 , y0 )

= I211 + I212 + I213 + I214 .

Since I21 ≤ I211 + I212 + I213 + I214 , it is sufficient to show that the terms on the right hand side of the last inequality tends to zero as (x, y, λ ) → (x0 , y0 , λ0 ) on Z. Let us consider the integral I211 .

For this aim, we define the new function as follows: Zx0 Zy0 f (x0 , y0 ) f (u, v) F (t, s) := ϕ (u, v) − ϕ (x0 , y0 ) dvdu. t s

From (8), for all t and s satisfying 0 < x0 − t ≤ δ and 0 < y0 − s ≤ δ , we have |F (t, s)| ≤ ε µ1 (x0 − t) µ2 (y0 − s) . Now, we can handle the integral I211 . From Theorem 2.5 in [26], we have the following equality: q Zx0 Zy0 2 2 f (t, s) − f (x0 , y0 ) ϕ (t, s) Kλ I211 = (t − x) + (s − y) dsdt ϕ (t, s) ϕ (x0 , y0 ) x0 −δ y0 −δ Zy0

Zx0

= (LS)

ϕ (t, s) Kλ

q

(t − x)2 + (s − y)2 dF (t, s) ,

x0 −δ y0 −δ

where (LS) denotes Lebesgue-Stieltjes integral.

(9)

1, 1, 2016, pp. 13-22


Two-dimensional integration by parts method (see Theorem 2.2, p.100 in [26]) gives us q Zx0 Zy0 2 2 ϕ (t, s) Kλ (t − x) + (s − y) dF (t, s) x0 −δ y0 −δ Zx0 Zy0

=

q 2 2 F (t, s) d ϕ (t, s) Kλ (t − x) + (s − y)

x0 −δ y0 −δ Zx0

q F (t, y0 − δ ) dt ϕ (t, y0 − δ ) Kλ (t − x)2 + (y0 − y − δ )2

+ x0 −δ Zy0

+

q F (x0 − δ , s) ds ϕ (x0 − δ , s) Kλ (x0 − x − δ )2 + (s − y)2

y0 −δ

+F (x0 − δ , y0 − δ ) ϕ (x0 − δ , y0 − δ ) Kλ

q (x0 − x − δ )2 + (y0 − y − δ )2 .

From (9), we can write |I211 | ≤ ε

Zx0

Zy0

x0 −δ y0 −δ

q 2 2 (s − x) + (t − y) µ1 (x0 − t) µ2 (y0 − s) d ϕ (t, s) Kλ Zx0

+ε µ2 (δ ) x0 −δ Zy0

+ε µ2 (δ ) y0 −δ

q (t − x)2 + (y0 − y − δ )2 µ1 (x0 − t) dt ϕ (t, y0 − δ ) Kλ q µ2 (y0 − s) ds ϕ (x0 − δ , s) Kλ (x0 − x − δ )2 + (s − y)2

+ε µ2 (δ ) ϕ (x0 − δ , y0 − δ ) Kλ

q

(x0 − δ − x)2 + (y0 − δ − y)2 .

If we continue as in the proof of Theorem 1 in [29] (for the similar situation, see [26, 20]), then we have q Zx0 Zy0 0 0 |I211 | ≤ ε (t − x)2 + (s − y)2 {µ1 (x0 − t)}t {µ2 (y0 − s)}s dsdt ϕ (t, s) Kλ x0 −δ y0 −δ

+2ε µ1 (x0 − x)

+2ε µ2 (y0 − y)

Zy0

y0 −δ Zx0

x0 −δ

0 ϕ (x, s) Kλ (|s − y|) {µ2 (y0 − s)}t ds 0 ϕ (t, y) Kλ (|t − x|) {µ1 (x0 − t)}t dt

+4εKλ (0) ϕ (x, y) µ1 (x0 − x)µ2 (y0 − y).

19

20


For the integrals I212 , I213, and I214 the proof is similar to the above one. Collecting the estimates for I212 , I213 and I214 , we have x0Z+δ y0Z+δ q 0 0 |I21 | ≤ ε ϕ (t, s) Kλ (t − x)2 + (s − y)2 {µ1 (|x0 − t|)}t {µ2 (|y0 − s|)}s dsdt x0 −δ y0 −δ

y0Z+δ

+2ε µ1 (|x0 − x|)

y0 −δ x0Z+δ

+2ε µ2 (|y0 − y|)

x0 −δ


+4εKλ (0) ϕ (x, y) µ1 (|x0 − x|)µ2 (|y0 − y|). Therefore, by the hypothesis, if the points (x, y, λ ) are sufficiently close to (x0 , y0 , λ0 ) , we have I21 ≤ ε pC. The above inequality is obtained for other cases of initial assumptions |x − x0 | < completes the proof.

δ 2

and |y − y0 | < δ2 . This

2 Theorem 3. Suppose that D = R2 , and the hypotheses (4)-(7) of Theorem 2 are satisfied. If (x0 , y0 ) ∈ R 2 2 is a common µ−generalized Lebesgue point of the functions f ∈ L1,ϕ R and ϕ ∈ L1 R , then

lim

Lλ ( f ; x, y) = f (x0 , y0 )

(x,y,λ )→(x0 ,y0 ,λ0 )

as (x, y, λ ) tends to (x0 , y0 , λ0 ) . Proof. The proof of this theorem is quite similar to that of Theorem 2, and it is omitted.

4. Rate of Convergence Theorem 4. Suppose that the hypotheses of Theorem 2 are satisfied. Let x0Z+δ y0Z+δ

∆ (x, y, λ , δ ) =

ϕ (t, s) Kλ

q

x0 −δ y0 −δ

0 0 (t − x)2 + (s − y)2 {µ1 (|x0 − t|)}t {µ2 (|y0 − s|)}s dsdt

y0Z+δ

+2µ1 (|x0 − x|)

y0 −δ x0Z+δ

+2µ2 (|y0 − y|)

x0 −δ


+4ϕ (x, y) Kλ (0) µ1 (|x0 − x|)µ2 (|y0 − y|), where 0 < δ ≤ δ0 for a fixed (and finite!) positive number δ0 , and the following conditions are satisfied:

1, 1, 2016, pp. 13-22


21

(i) ∆ (x, y, λ , δ ) tends to 0 as (x, y, λ ) tends to (x0 , y0 , λ0 ) , for some δ > 0. (ii) For we have Kλ (ξ ) =o(∆ every ξ >0, y, λ ) tends to (x0 , y0 , λ0 ) . (x, y, λ , δ )) as (x, q RR 2 2 (iii) ϕ (t, s) Kλ (t − x) + (s − y) dsdt − ϕ (x0 , y0 ) =o(∆ (x, y, λ , δ )) as (x, y, λ ) tends to (x0 , y0 , λ0 ) . D

Then, at each common µ−generalized Lebesgue point of f ∈ L1,ϕ (D) and ϕ ∈ L1 (D) we have |Lλ ( f ; x, y) − f (x0 , y0 )| = o (∆ (x, y, λ , δ )) as (x, y, λ ) tends to (x0 , y0 , λ0 ). Proof. The assertion is obvious by the hypotheses of Theorem 2 and assumptions (i)-(iii). Thus it is omitted. Theorem 5. Suppose that the hypotheses of Theorem 3 are satisfied. Let x0Z+δ y0Z+δ q 0 0 2 2 (t − x) + (s − y) {µ1 (|x0 − t|)}t {µ2 (|y0 − s|)}s dsdt ∆ (x, y, λ , δ ) = ϕ (t, s) Kλ x0 −δ y0 −δ

y0Z+δ

+2µ1 (|x0 − x|)

y0 −δ x0Z+δ

+2µ2 (|y0 − y|)

x0 −δ


+4ϕ (x, y) Kλ (0) µ1 (|x0 − x|)µ2 (|y0 − y|), where 0 < δ ≤ δ0 for a fixed (and finite!) positive number δ0 , and the following conditions are satisfied: (i) ∆ (x, y, λ , δ ) tends to 0 as (x, y, λ ) tends to (x0 , y0 , λ0 ) , for some δ > 0. (ii) For every ξ > 0, we have Kλ (ξ ) =o(∆ (x, √y, λ , δ ))as (x, y, λ ) tends to (x0 , y0 , λ0 ) . RR √ (iii) For every ξ > 0, we have K t 2 + s2 dsdt =o(∆ (x, y, λ , δ ))as (x, y, λ ) tends to (x0 , y0 , λ0 ) . ξ ≤ t 2 +s2 λ q RR 2 2 (iv) ϕ (t, s) Kλ (t − x) + (s − y) dsdt − ϕ (x0 , y0 ) =o(∆ (x, y, λ , δ )) as (x, y, λ ) tends to (x0 , y0 , λ0 ) . R2 Then, at each common µ−generalized Lebesgue point of f ∈ L1,ϕ R2 and ϕ ∈ L1 R2 , we have |Lλ ( f ; x, y) − f (x0 , y0 )| = o (∆ (x, y, λ , δ )) as (x, y, λ ) tends to (x0 , y0 , λ0 ). Proof. The assertion is obvious by the hypotheses of Theorem 2 and assumptions (i)-(iv). Thus it is omitted.

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