The present paper concerns with the Fatou type convergence properties of the \(r-th\) and \((r+1)-th\) derivatives of the nonlinear singular integral operators defined as

\[

\left( I_{\lambda}f\right) (x)=\int\limits_{a}^{b}K_{\lambda}(t-x,f(t))\,{\rm d}t,\,\,\,\,\,\,\,x\in\left( a,b\right) ,

\]

acting on functions defined on an arbitrary interval \(\left( a,b\right)

,\) where the kernel \(K_{\lambda}\) satisfies some suitable assumptions. The present study is a continuation and extension of the results established in the paper [7]

H. Erhan Altin

Harun Karsli

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REVUE D’ANALYSE NUMÉRIQUE ET DE THÉORIE DE L’APPROXIMATIONRev. Anal. Numér. Théor. Approx., vol. 42 (2013) no. 1, pp. 37–48ictp.acad.ro/jnaatON FATOU TYPE CONVERGENCE OF HIGHER DERIVATIVES OFCERTAIN NONLINEAR SINGULAR INTEGRAL OPERATORSHARUN KARSLI∗ and H. ERHAN ALTIN†Abstract. The present paper concerns with the Fatou type convergence prop-erties of the r− th and (r + 1)− th derivatives of the nonlinear singular integraloperators defined as(Iλf) (x) =b∫aKλ(t− x, f(t)) dt, x ∈ (a, b) ,acting on functions defined on an arbitrary interval (a, b) , where the kernel Kλsatisfies some suitable assumptions. The present study is a continuation andextension of the results established in the paper [7].MSC 2000. 41A35, 47G10, 47H30.Keywords. nonlinear singular integral operator, pointwise convergence, Fatoutype convergence.1. INTRODUCTIONLet Λ be a nonempty set of indices with a topology and λ0 be an accu-mulation point of Λ in this topology. By U(θ) we denote the family of allneighborhoods of the neutral element θ of R, and x0 is a fixed accumulationpoint of R. We take a family K of functions Kλ : R xR→R, where Kλ(t, 0) = 0for all t ∈ R and λ ∈ Λ, such that Kλ(t, u) is integrable over R with respect tot, in the sense of Lebesgue measure, for all values of the index λ and secondvariable u. The family K will be called a kernel. In addition, if the kernelfunction Kλ(t, u) is continuous in R for every t ∈ R, then the kernel functionis called Carathéodory kernel function.In this paper we are concerned with the Fatou type pointwise convergenceof r − th and (r + 1) − th derivatives of certain family of nonlinear singular∗Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics,14280, Golkoy Bolu, Turkey, e-mail: karsli h@ibu.edu.tr.†Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics,14280, Golkoy Bolu, Turkey, e-mail: erhanaltin@ibu.edu.tr.38 Harun Karsli and H. Erhan Altin 2integral operators (Iλf) of the form(1) (Iλf) (x) =b∫aKλ(t− x, f(t)) dt, x ∈ (a, b) ,acting on functions defined on an arbitrary interval (a, b) , where the kernelfunction Kλ satisfies some suitable assumptions. In these theorems the conver-gence is restricted to some subsets of the plane, i.e. the Fatou type convergenceis discussed, whenever the first parameter tends to an accumulation point x0,at which the function f has finite r− th and (r+ 1)− th derivatives, whereasthe second one tends to an accumulation point λ0 of a given index set Λ.Further results on convergence of the operators (1) and its linear cases canbe found in [1]-[12].In particular, we obtain the rate of the Fatou type pointwise convergence forthe nonlinear family of singular integral operators (1) to the point x0, at whichthe function f has finite r−th and (r+1)−th derivatives, as (x, λ)→ (x0, λ0) .The results presented in this paper are the continuation and extension of thoseestablished in [7] in which the kernel function Kλ satisfies[∂∂xKλ(t− x, u)−∂∂xKλ(t− x, v)]= ∂∂xLλ(t− x) [u− v]for every t, u, v ∈ R and for any λ ∈ Λ.Throughout this paper we assume that the function Kλ : R xR→R satisfiesthe following conditions;a ) Let Lλ(t) be an integrable function such that for any fixed r ∈ N(2)[∂r∂xrKλ(t− x, u)−∂r∂xrKλ(t− x, v)]= ∂r∂xrLλ(t− x) [u− v] ,holds for every t u, v ∈ R and for any λ ∈ Λ.b) limλ→λ0∫R \ULλ(t)dt = 0, for every U ∈ U(0).c ) limλ→λ0[sup|t|≥δLλ(t)]= 0, for every δ > 0.d ) limλ→λ0∫RLλ(t) dt = 1.According to a) it is easy to see that Kλ : R xR→R is a kernel function.We introduce a function∼f∈ L1(R) as(3)∼f(t) :={f(t), t ∈ (a, b) ,0, t /∈ (a, b) ,(See [6]-[9]).3 Fatou type convergence of nonlinear singular integral operators 39Theorem 1. [3] Let 1 ≤ p <∞ and assume that Kλ(t, u) is a Carathéodorykernel function. If f ∈ Lp (a, b), then (Iλf) ∈ Lp (a, b) and‖Iλf‖Lp(a,b) ≤ H(λ) ‖f‖Lp(a,b)for every λ ∈ Λ.This kind of existence theorem is also valid in general functional spaces (seee.g. [3]).2. CONVERGENCE OF THE DERIVATIVESLet us define, for any constants Cν > 0 (ν = 1, 2, ..., r) , the set(4) Dr :={(x, λ) ∈ I x Λ : |x− x0|ν∫ x0−x+δx0−x−δ|t|r−ν∣∣ ∂r∂trLλ(t)∣∣ dt < Cν} ,where I = (a, b) is an arbitrary interval in R.Now, we are ready to investigate the approximation for finite r-th derivativesof the operator (Iλf) in L1(I).Theorem 2. Let the function Lλ(t) and its derivatives∂ν∂tνLλ(t), (ν =1, 2, ..., r) be continuous with respect to t on (−∞,∞) and Lλ(t) be integrablewith respect to t for each fixed λ ∈ Λ. Suppose that the conditions c) and d)together with(5) supλ∈Λ∫ x0−x+δx0−x−δ|t|r∣∣ ∂r∂trLλ(t)∣∣ dt <∞ and limλ→λ0sup0<δ≤|t|∣∣ ∂r∂trLλ(t)∣∣ = 0hold for every δ > 0, are satisfied. Suppose that the function f ∈ L1(I) has atx0 a finite r − th derivative.Then(6) lim ∂r∂xr (Iλf) (x) = f(r)(x0)as (x, λ)→ (x0, λ0) and (x, λ) ∈ Dr.Proof. Suppose that a < x0 < b and 0 < |x0 − x| < δ (δ > 0).We construct a function(7) g(t) = f(x0) + (t− x0)f ′(x0) + ...+ (t−x0)rr! f(r)(x0)so that g(r)(t) = f (r)(x0).Firstly we shall prove this theorem for the function g(t). For this purpose weintroduce a function∼g∈ L1(R) as follows;(8)∼g(t) :={g(t), t ∈ (a, b) ,0 , t /∈ (a, b) .40 Harun Karsli and H. Erhan Altin 4Applying the operator Iλ to the function g(t), we have(Iλg) (x) =b∫aKλ(t− x, g(t))dtand according to (8) we can rewrite the last equality as follows;(9) (Iλg) (x) =∫RKλ(t− x,∼g(t))dt =(Iλ∼g)(x)and hence∂r∂xr (Iλg) (x) =∂r∂xr∫RKλ(t− x,∼g(t))dt =∫R∼g(t) ∂r∂xrLλ(t− x)dt= (−1)r∫R∼g(t) ∂r∂trLλ(t− x)dt =∫R∼g(r)(t)Lλ(t− x)dt.In the above, substituting (7), we get∣∣∣ ∂r∂xr (Iλg) (x)− f (r)(x0)∣∣∣ =∣∣∣∣∣∣b∫af (r)(x0)Lλ(t− x)dt− f (r)(x0)∣∣∣∣∣∣=∣∣∣f (r)(x0)∣∣∣∣∣∣∣∣∣b∫aLλ(t− x)dt− 1∣∣∣∣∣∣ .Hence, from condition d) one has(10) lim(x,λ)→(x0,λ0)∂r∂xr (Iλg) (x) = f(r)(x0).We denote Iλ(x) :=∂r∂xr (Iλg) (x)−∂r∂xr (Iλf) (x).Thanks to (10), for completing the proof of the theorem, it is sufficient toshow thatlim(x,λ)→(x0,λ0)|Iλ(x)| = 0,which gives (6).5 Fatou type convergence of nonlinear singular integral operators 41It is easy to see that|Iλ(x)| =∣∣∣∣∣∣∫R∼f(t) ∂r∂trLλ(t− x)dt−∫R∼g(t) ∂r∂trLλ(t− x)dt∣∣∣∣∣∣=∣∣∣∣∣∣b∫a[f(t)− g(t)] ∂r∂trLλ(t− x)dt∣∣∣∣∣∣ ≤x0−δ∫a|f(t)− g(t)|∣∣ ∂r∂trLλ(t− x)∣∣dt+x0+δ∫x0−δ|f(t)−g(t)|∣∣ ∂r∂trLλ(t− x)∣∣ dt+ b∫x0+δ|f(t)−g(t)|∣∣ ∂r∂trLλ(t− x)∣∣ dt=: I1(x, λ) + I2(x, λ) + I3(x, λ).To estimate I1(x, λ) and I3(x, λ), we use the following method:I1(x, λ) =x0−δ∫a|f(t)− g(t)|∣∣ ∂r∂trLλ(t− x)∣∣ dt≤ sup0<δ≤|t|∣∣ ∂r∂trLλ(t− x)∣∣ b∫a|f(t)− g(t)|dtandI3(x, λ) =b∫x0+δ|f(t)− g(t)|∣∣ ∂r∂trLλ(t− x)∣∣dt≤ sup0<δ≤|t|∣∣ ∂r∂trLλ(t− x)∣∣ b∫a|f(t)− g(t)| dt.Now we shall rewrite I2(x, λ) as follows,I2(x, λ) =x0+δ∫x0−δ∣∣∣f(t)−g(t)(t−x0)r ∣∣∣ |(t− x0)r| ∣∣ ∂r∂trLλ(t− x)∣∣ dt.For every ε > 0 there exists a δ > 0 such that;I2(x, λ) ≤ εx0+δ∫x0−δ|(t− x0)r|∣∣ ∂r∂trLλ(t− x)∣∣dt.LetI2,1(x, λ) :=x0+δ∫x0−δ|(t−x0)r|∣∣ ∂r∂trLλ(t−x)∣∣ dt= x0−x+δ∫x0−x−δ|(t+ x− x0)r|∣∣ ∂r∂trLλ(t)∣∣dt.42 Harun Karsli and H. Erhan Altin 6It can be rewrite I2,1(x, λ) as follows:I2,1(x, λ) =x0−x+δ∫x0−x−δ|(t+ x− x0)r − tr + tr|∣∣ ∂r∂trLλ(t)∣∣ dt≤x0−x+δ∫x0−x−δ|(t+ x− x0)r − tr|∣∣ ∂r∂trLλ(t)∣∣dt+ x0−x+δ∫x0−x−δ|t|r∣∣ ∂r∂trLλ(t)∣∣dt=: I2,1,1(x, λ) + I2,1,2(x, λ).From (5) I2,1,2(x, λ) is finite. So it is sufficient to show that I2,1,1(x, λ) is finite.Using the obvious identity(11) an − bn = (a− b)(an−1 + an−2b+ ...+ abn−2 + bn−1)yieldsI2,1,1(x, λ) = |x− x0|x0−x+δ∫x0−x−δ∣∣∣(t+ x− x0)r−1 + ...+ tr−1∣∣∣ ∣∣ ∂r∂trLλ(t)∣∣dt≤ |x− x0|x0−x+δ∫x0−x−δ∣∣∣(t+ x− x0)r−1∣∣∣ ∣∣ ∂r∂trLλ(t)∣∣dt+ |x− x0|x0−x+δ∫x0−x−δ∣∣∣(t+ x− x0)r−2 t∣∣∣ ∣∣ ∂r∂trLλ(t)∣∣ dt...(12)+ |x− x0|x0−x+δ∫x0−x−δ∣∣(t+ x− x0) tr−2∣∣ ∣∣ ∂r∂trLλ(t)∣∣ dt+ |x− x0|x0−x+δ∫x0−x−δ|t|r−1∣∣ ∂r∂trLλ(t)∣∣ dt.Applying the formula (11) successively to the right-hand side of (12), we cansee that I2,1,1(x, λ) is less than or equal to the linear combinations of|x− x0|νx0−x+δ∫x0−x−δ|t|r−ν∣∣ ∂r∂trLλ(t)∣∣ dt, (ν = 1, 2, ..., r).Taking into account (5) and (4), we show that I2,1(x, λ) is finite on any planarset Dr.7 Fatou type convergence of nonlinear singular integral operators 43Hence|Iλ(x)| ≤ I1(x, λ) + I2(x, λ) + I3(x, λ)≤ εI2(x, λ) + 2 sup0<δ≤|t|∣∣ ∂r∂trLλ(t− x)∣∣ b∫a|f(t)− g(t)|dt.Since f(t) − g(t) belongs to L1(a, b), in view of (5), (4) and the condition c),we obtainlim(x,λ)→(x0,λ0)|Iλ(x)| = 0,i.e.,lim(x,λ)→(x0,λ0)∂r∂xr (Iλf) (x) = lim(x,λ)→(x0,λ0)∂r∂xr (Iλg) (x).This completes the proof of the Theorem. Secondly, we will investigate the approximation for finite (r+1)−th deriva-tives of the operator (Iλf) in L1(I).Theorem 3. Let the function Lλ(t) and its derivatives∂ν∂tνLλ(t), (ν =1, 2, ..., r, r + 1) be continuous with respect to t on (−∞,∞) and Lλ(t) be in-tegrable with respect to t for each fixed λ ∈ Λ. Suppose that conditions c) andd) are satisfied. Also we assume that the relationssupλ∈Λx0−x+δ∫x0−x−δ|t|r+1∣∣∣ ∂r+1∂tr+1Lλ(t)∣∣∣ dt <∞ and limλ→λ0 sup0<δ≤|t|∣∣∣ ∂r+1∂tr+1Lλ(t)∣∣∣ = 0hold for every δ > 0. Suppose that the function f ∈ L1(I) has at x0 finitederivatives f(r+1)+ (x0) and f(r+1)− (x0).Thenlim ∂r+1∂xr+1(Iλf) (x) = Bf(r+1)+ (x0) + (1−B)f(r+1)− (x0)as (x, λ)→ (x0, λ0) and (x, λ) ∈ Dr+1, where(13) lim(x,λ)→(x0,λ0)∞∫x0Lλ(t− x)dt = B, 0 ≤ B ≤ 1.Proof. Let a < x0 < b and 0 < x0 − x < δ2 (δ > 0). Setting(14)g(t) :=f(x0) + ...+(t−x0)rr! f(r)(x0) +(t−x0)r+1(r+1)! f(r+1)− (x0), a < t < x0,f(x0) + ...+(t−x0)rr! f(r)(x0) +(t−x0)r+1(r+1)! f(r+1)+ (x0), x0 ≤ t < b.Note that(Iλg) (x) =b∫aKλ(t− x, g(t))dt =∫RKλ(t− x,∼g(t))dt.44 Harun Karsli and H. Erhan Altin 8Differentiating both sides of the last equality (r + 1) times with respect to x,one has∂r+1∂xr+1(Iλg) (x) =b∫ag(r+1)(t)Lλ(t− x)dt.By (14), the last equality can be rewritten in the form∂r+1∂xr+1(Iλg) (x) = f(r+1)− (x0)x0∫aLλ(t− x)dt+ f(r+1)+ (x0)b∫x0Lλ(t− x)dt.The hypothesis (13) yieldslim(x,λ)→(x0,λ0)∂r+1∂xr+1(Iλg) (x) = B f(r+1)+ (x0) + (1−B)f(r+1)− (x0).Define|Iλ(x)| :=∣∣∣ ∂r+1∂xr+1 (Iλg) (x)− ∂r+1∂xr+1 (Iλf) (x)∣∣∣ .In order to complete the proof of the theorem, it is sufficient to showlim(x,λ)→(x0,λ0)|Iλ(x)| = 0.One has|Iλ(x)| =∣∣∣∣∣∣b∫a[f(t)− g(t)] ∂r+1∂tr+1Lλ(t− x)dt∣∣∣∣∣∣≤x0−δ∫a|f(t)− g(t)|∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt+x0∫x0−δ∣∣∣ f(t)−g(t)(t−x0)r+1∣∣∣ ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣dt+b∫x0+δ|f(t)− g(t)|∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt+x0+δ∫x0∣∣∣ f(t)−g(t)(t−x0)r+1∣∣∣ ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣dt=: I1(x, λ) + I2(x, λ) + I3(x, λ) + I4(x, λ).(15)9 Fatou type convergence of nonlinear singular integral operators 45Since f has at x0 a finite (r+ 1)− th right and left derivatives, then for everyε > 0 there exists a δ > 0 such thatI2(x, λ) =x0∫x0−δ∣∣∣ f(t)−g(t)(t−x0)r+1∣∣∣ ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt≤ εx0∫x0−δ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt(16)andI4(x, λ) =x0+δ∫x0∣∣∣ f(t)−g(t)(t−x0)r+1∣∣∣ ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt≤ εx0+δ∫x0∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt.(17)Thus we getI2(x, λ) + I4(x, λ) ≤ εx0+δ∫x0−δ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt.SetI2,1(x, λ) + I4,1(x, λ) :=x0+δ∫x0−δ∣∣∣(t− x0)r+1∣∣∣ ∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣ dt.Using the same method as in the proof of Theorem 2 we deduce I2,1(x, λ) +I4,1(x, λ) is less than or equal to a linear combination of|x− x0|νx0−x+δ∫x0−x−δ|t|r+1−ν∣∣∣ ∂r+1∂tr+1K(t, λ)∣∣∣ dt, (ν = 1, ..., r + 1).By virtue of (4), the term I2(x, λ) + I4(x, λ) is bounded.Now, we consider the integrals I1(x, λ) and I3(x, λ), respectively.I1(x, λ) =x0−δ∫a|f(t)− g(t)|∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣dt≤M supδ2<|u|∣∣∣ ∂r+1∂ur+1Lλ(u)∣∣∣(18)46 Harun Karsli and H. Erhan Altin 10andI3(x, λ) =b∫x0+δ|f(t)− g(t)|∣∣∣ ∂r+1∂tr+1Lλ(t− x)∣∣∣dt≤M supδ2<|u|∣∣∣ ∂r+1∂ur+1Lλ(u)∣∣∣ .(19)Using (16), (19) in (15) one has|Iλ(x)| ≤ I1(x, λ) + I2(x, λ) + I3(x, λ) + I4(x, λ)≤ ε [I2,1(x, λ) + I4,1(x, λ)] & + 2M supδ2<|u|∣∣∣ ∂r+1∂ur+1Lλ(u)∣∣∣ .(20)Under the hypotheses of the theorem, (20) yieldslim(x,λ)→(x0,λ0)|Iλ(x)| = lim(x,λ)→(x0,λ0)∣∣∣ ∂r+1∂xr+1 (Iλg) (x)− ∂r+1∂xr+1 (Iλf) (x)∣∣∣ = 0.This completes the proof. Remark 4. If B = 12 , then we havelim(x,λ)→(x0,λ0)∂r+1∂xr+1(Iλf) (x) =f(r+1)+ (x0)+f(r+1)− (x0)2 . Remark 5. Let I = (a, b) be an arbitrary bounded interval in R and f ∈L1(a, b) be a (b−a)-periodic function. In this case the proofs of the Theoremsare similar. 3. EXAMPLESExample 6. A special case of the function Kλ(t, u) satisfying the condi-tions, is the linear case with respect to the second variable, i.e.,Kλ(t, u) = Zλ(t)u.This case is widely used in Approximation Theory [4]. Example 7. We introduce the functionKλ(t, u) ={(r + 1)λr+1tr u+ u, t ∈ [0, 1λ ],0, t /∈ [0, 1λ ],where Λ = [1,∞) is a set of indices with natural topology and λ0 = ∞ is anaccumulation point of Λ in this topology.First of all Kλ(t, u) is a kernel, i.e., Kλ(t, 0) = 0.It is seen that for every u ∈ R,∂r∂xrKλ(t− x, u) ={(−1)r(r + 1)!λr+1 u, t− x ∈ [0, 1λ ],0, t− x /∈ [0, 1λ ].11 Fatou type convergence of nonlinear singular integral operators 47According to (2), we obtain∂r∂xrLλ(t− x) ={(−1)r(r + 1)!λr+1, t− x ∈ [0, 1λ ],0, t− x /∈ [0, 1λ ].This impliesLλ(t− x) ={(r + 1)λr+1(t− x)r, t− x ∈ [0, 1λ ],0, t− x /∈ [0, 1λ ].Moreover ∫RLλ(t)dt =∫[0,1λ ](r + 1)λr+1trdt = 1 <∞.It is easy to see thatlimλ→∞∫R \ULλ(t)dt = 0,for every U ∈ U(0) andlimλ→∞[sup|t|≥δLλ(t)]= 0,for every δ > 0. Acknowledgement. The authors would like to express their sincerethanks to the referee and to Professor Octavian Agratini for their very carefuland intensive study of this manuscript.REFERENCES[1] C. Bardaro, H. Karsli and G. Vinti, On pointwise convergence of linear integraloperators with homogeneous kernels, Integral Transforms and Special Functions, 19(6)(2008), pp. 429–439.[2] C. Bardaro, H. Karsli and G. Vinti, Nonlinear integral operators with homogeneouskernels: pointwise approximation theorems, Applicable Analysis, 90 (2011) No. 3–4, pp.463–474.[3] C. Bardaro, J. Musielak and G. Vinti, Nonlinear Integral Operators and Applica-tions, De Gruyter Series in Nonlinear Analysis and Applications, 9 (2003), xii + 201pp.[4] Butzer P.L. and R.J. Nessel, Fourier Analysis and Approximation, V.1, AcademicPress, New York, London, 1971.[5] A.D. Gadjiev, On convergence of integral operators depending on two parameters, Dokl.Acad. Nauk. Azerb. SSR, XIX (1963) No. 12, pp. 3–7.[6] H. Karsli, Convergence and rate of convergence by nonlinear singular integral operatorsdepending on two parameters, Applicable Analysis, 85 (2006) No. 6-7, pp. 781–791.[7] H. Karsli, Convergence of the derivatives of nonlinear singular integral operators, J.Math. Anal. Approx. Theory, 2 (2007) No. 1, pp. 53–61.[8] H. Karsli, On approximation properties of a class of convolution type nonlinear singularintegral operators, Georgian Math. Jour., 15 (2008), No. 1, pp. 77–86.[9] H. Karsli and Gupta V., Rate of convergence by nonlinear integral operators forfunctions of bounded variation, Calcolo, 45, (2) (2008), pp. 87–99.48 Harun Karsli and H. Erhan Altin 12[10] J. Musielak, On some approximation problems in modular spaces. In ConstructiveFunction Theory 1981 ( Proc. Int. Conf. Varna, June 1-5, 1981), pp. 455-461, Publ.House Bulgarian Acad. Sci., Sofia 1983.[11] T. Swiderski and E. Wachnicki, Nonlinear Singular Integrals depending on two pa-rameters, Commentationes Math., XL (2000), pp. 181–189.[12] R. Taberski, Singular integrals depending on two parameters, Rocznicki Polskiego to-warzystwa matematycznego, Seria I. Prace matematyczne, VII (1962), pp. 173–179.Received by the editors: June 28, 2012.

On Fatou type convergence of higher derivatives of certain nonlinear singular integral operators

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On Fatou type convergence of higher derivatives of certain nonlinear singular integral operators

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