A new approach to nonlinear singular integral operators depending on three parameters

Let 1 ≤ p < ∞, and δ0,δ1∈R+it be fixed numbers. A point(x0,y0)∈R2at which the following relations

and

it hold uniformly with respect to almost every} t∈Ris called a μ – p – generalized Lebesgue point of locally p– integrable function (i.e., a function whose p –th power is locally integrable) g: R2→R Here, R→R is increasing and absolutely continuous on 0 < h ≤ δ₀ and μ₁(0) = 0 and also, μ2:R→R is increasing and absolutely continuous on 0 < k ≤ δ₁ and μ₂(0) = 0.

Basically Definition 2.1 is obtained by combining the characterization of the function μ(t) presented by Gadjiev 9 with the definition of d–point given by Siudut 21. Also, some diferent modifications are done according to our problem's needs, such as predispozing the definition to Lpφ space. On the other hand, for some other μ-generalized Lebesgue point definitions, we reer the reader to [8, 25] and 28.

Let λ₀ be an accumulation point of the non-negative set of numbers Λ or λ₀ = ∞, andφ:R2→R+be a locally bounded weightfunction such that the following inequality

holds for every (t,s)∈R2

A family (K_λ)_λ∈Λ consisting of the functions R2×R→R is called class A, if the following conditions hold:

(a) Kλ(t,s,0)=0for every (t,s)∈R2 and for each λ∈Λ, and Kλ(.,.,u)∈L1(R2) for every u∈R and for each λ∈Λ.

(b) There exists a family (L_λ)_λ∈Λ consisting of the (globally) integrable functions L λ: R2→R such that the Lipschitz inequality given by

holds for every (t,s)∈R2,u,v∈R, and for each fixed λ∈Λ.

(c) lim(x,y,λ)→(x0;y0;λ0)∬R2Kλ(t−x,s−y,u(x0;y0),(t,s))dsdt−u=0for everyu∈R and for any (x0,y0)∈R2.

(d) For every ξ>0,limλ→λ0supξ≤t2+s2[φ(t,s)Lλ(t,s)]dsdt=0.

(e)For every ξ>0,limλ→λ0∬ξ⩽t2+s2[φ(t,s)Lλ(t,s)]dsdt=0.

(f) ∥φLλ∥L1(R2)≤M<∞for everyλ∈Λ.

(g) L λ(t,s) is non-increasing on [0,∞) and non-decreasing on (−∞,0] for each fixed λ∈Λ as a function of t. Similarly, L λ(t,s) is non-increasing on [0,∞) and non-decreasing on (−∞,0] for each fixed λ∈Λ as a function of s.

Throughout this paper the kernel function K_λ belongs to class A.

The studies [11, 13, 16, 21] and 18, among others, are used as main reerence works in the construction stage of class A. Therefore, we refer the reader to see the indicated works. On the other hand, we recommend the reader to compare the usage of the inequality (6) used in 11 with the current study Also, for the Lipschitz inequality included in the definition class $\mathcal{A}$, we reer the reader to see the works [14, 18].

Existence of the operators of type (3) is guaranteed by the conditions of class A, that isTλ(f;x,y)∈Lpφ(R2)wheneverf∈Lpφ(R2).

A first example is the linear kernel. Let Λ be a set of non-negative numbers such that Λ = (0,;∞) with accumulation point λ₀ = 0. Now, the definition of the functionR2×R→Ris as follows:

one may easily observe that given function K_λ belongs to class A. For detailed analysis of the function L_λ(t,s), we recommend the reader to see 21.

Define the kernel function such that

where λ∈ N and ∈₀ = \∞. This kernel is the two dimensional analogue of the kernel given in 16.

It is easy to see that the conditions of class A are satisfied. Observe that one may take the desired linear kernel as

The appropriate weight functions, which are defined on R2, may be given by φ1(t,s)=et+s and φ2(t,s)=(1+|t|)(1+|s|).

If(x0,y0)∈R2is a common μ-generalized Lebesgue point of the functionsf∈Lpφ(R2)(1≤p<∞) and φ then

and

where 0 < δ < min {δ₀, δ₁, are bounded as (x,y,λ) tends to (x₀,y₀,λ₀).

Let 0<|x0−x|<δ2. Further, let 0<y0−y<δ2,, and (x₀,y₀) be a common μ-generalized Lebesgue point of the functions f∈Lp(2)(1p<∞)andφ. The proof of theorem will be given for the case 1 < p < ∞. The proof for the case p = 1 is similar. Now, set I(x,y,λ)=|Tλ(f;x,y)−f(x0,y0)|. Using condition (c) , we obtain

Using condition (b), it is easy to see that the following inequality holds:

Since whenever m,n being positive numbers the inequality (m+n)p≤2p(mp+np) holds (see, e.g., 38) we have

Now, applying Hölder's inequality (see, e.g., 38) to the first integral of the resulting inequality, we have

where

Moreover, the following inequality holds:

where (t−x0)2+(s−y0)2<δ2}.

Now, applying the inequality given by (m+n)p<_2p(mp+np) once more to the right hand side of the resulting inequality, we obtain

Recalling the initial assumptions 0<|y0−y|<δ2, we may define the following set as follows:

Comparing geometric representations of the setsBδ gives the inclusion relation such thatR2∖Bδ⊆R2∖Aδ, where

In the light of these relations, we may write

Rearranging and rewriting the last inequality, we obtain

Boundedness of the term (x,y,λ) which follows from condition (f).On the other hand, I4(x,y,λ)→0 by condition(c).Lastly, I2(x,y,λ)→0 by conditions (d) and (e) , respectively.

Now, we may write the following inequality for the integral I3(x,y,λ)

where Qδ=(x0−δ,x0+δ)×(y0−δ,y0+δ).Observe that I₃₁(x,y,λ) may be written in the following form:

It is easy to see that the following inequality holds:

Now, we pass to the integral iλ(x,y,t).

This integral may be written in the following form:

Now, we consider the integraliλ1(x,y,t) From relation (5), for every ϵ>0 there exists a corresponding numberδ>0 such that the expression

holds for every 0<k<_δ<min{δ0,δ1}.

Define the function F(t,s) by

From (10), we have

From (9) and (10), for every ssatisfying0<y0−s<_δ<min{δ0,δ1} we have

(12)for any fixed t∈R By virtue of (10) and (11), we have

where (LS) denotes Lebesgue-Stieltjes integral.

Using integration by parts and applying (12), we have the following inequality:

It is easy to see that (for the similar situation, see [7, 9]) the following inequality holds:

Applying integration by parts to the right hand side of the last inequality, we have the following expression:

Remaining variational operations are evaluated using condition (g). Hence, we obtain

Using preceding method, we can estimate the integral iλ2(x,y,t)as follows:

(14)Combining (13) and (14), we obtain

Similar calculations for the integral jλ(x,y,s)yield

Thus, we have

Since epsilon is arbitrary and L_λ is integrable with respect to eachvariable, the desired result follows from hypotheses (8)and (7), i.e.,(x,y,λ)→(x0,y0,λ0)

Note that the same conclusion is obtained for the case 0<y−y0<δ2 Thus the proof is completed.

it Suppose that the hypotheses of Theorem 3.1 are satisfied.

Let

(i) △(x,y,λ,δ)→0as(x,y,λ)tends to(x0,y0,λ0)for someδ>0.

(ii) For everyξ>0we havesupξ⩽t2+s2varphi(t,s)Lλ(t,s)]=o(△(x,y,λ,δ))as(x,y,λ)tendsto(x0,y0,λ0)

(iii) For everyu∈Rwe have∬R2Kλ(t−x,s−y,uφ(x0,y0)φ(t,s))dsdt−u=o(△(x,y,λ,δ))tends to(x0,y0,λ0)