# On Singular Integrals with Cauchy Kernel on   Weight Subspaces: The Basicity Property of Sines   and Cosines Systems in Weight Spaces

Authors: Tofig Isa Najafov; Saeed Farahani
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

A singular operator with Cauchy kernel on the subspaces of weight Lebesgue space is
considered. A sufficient condition for a bounded action of this operator from a subspace
to another subspace of weight Lebesgue space of functions is found. These conditions are
not identical with Muckenhoupt conditions. Moreover, the completeness, minimality, and
basicity of sines and cosines systems are considered.


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## Body

## 1. Introduction

Consider the following singular operator with Cauchy kernel:(1.1)[Sf](t)=12πi∫-ππf(s)ds1-e-i(s-t),
where f∈Lp,ρ(-π,π), 1<p<+∞, is an appropriate density, ρ(t) is a weight function of the form
(1.2)ρ(t)=∏k=1r|t-tk|pβk,{tk}1r⊂[-π,π] (ti≠tj for i≠j), {βk}1r⊂R are real numbers.UnderLp,ρ we understand a Lebesgue weight space with the norm(1.3)‖f‖p,ρ≡(∫-ππ|f(t)|pρ(t)dt)1/p.Bounded action of the operatorS in the spaces Lp,ρ plays an important role in many problems of mathematics including the theory of bases. This direction has been well developed and treated in the known monographs. We will need the following.Statement 1.
The operatorS is bounded in Lp,ρ if and only if the inequalities
(1.4)-1p<βk<1q,k=1,r¯,1p+1q=1,
are fulfilled.Concerning this fact a one can see the monograph [1] and papers [2–4]. Inequalities (1.4) are Muckenhoupt condition with respect to the weight function ρ(t) with degrees pβk. It is known that the classic system of exponents {eint}n∈Z (Z are integers) forms a basis in Lp,ρ if and only if inequalities (1.4) hold (see, e.g., [3, 4]).It turns that if you consider the singular operator acting on the subspace of the weighted Lebesgue space, then inequality (1.4) is not necessary for the bounded action. At different points of degeneration the change interval of the corresponding exponent is expanded. This paper is devoted to studying these issues.


## 2. Some Necessary Facts

LetLp,ω0≡Lp,ω(0,π), ω(t)—a weight function of the form(2.1)ω(t)≡∏k=0r|t-τk|pαk,
where 0=τ0<τ1<⋯<τr=π, {αk}0r⊂R.Denote the space of even (odd) functions inLp,ρ by Lp,ρ+ (Lp,ρ-), that is,(2.2)Lp,ρ±≡{f∈Lp,ρ:f(-t)=±f(t),t∈[-π,π]}.
We'll need the following identity:(2.3)11-ei(θ-φ)-11-ei(θ+φ)=i2sinφsin((φ-θ)/2)sin((θ+φ)/2).
Indeed, we have(2.4)11-ei(θ-φ)-11-ei(θ+φ)=1-e-i(θ-φ)(1-ei(θ-φ))(1-e-i(θ-φ))-1-e-i(θ+φ)(1-ei(θ+φ))(1-e-i(θ+φ))=1-cos⁡(θ-φ)+isin(θ-φ)(1-cos⁡(θ-φ))2+sin2(θ-φ)-1-cos⁡(θ+φ)+isin(θ+φ)(1-cos⁡(θ+φ))2+sin2(θ+φ)=i[sin(θ-φ)2-2cos⁡(θ-φ)-sin(θ+φ)2-2cos⁡(θ+φ)]=i[cos⁡((θ-φ)/2)2sin((θ-φ)/2)-cos⁡((θ+φ)/2)2sin((θ+φ)/2)]=i2cos⁡((θ-φ)/2)sin((θ+φ)/2)-cos⁡((θ+φ)/2)sin((θ-φ)/2)sin((θ-φ)/2)sin((θ+φ)/2)=i2sinφsin((θ-φ)/2)sin((θ+φ)/2).
For compactness of the notation, we assume e(x)≡1/(1-eix). Thus,(2.5)e(θ-φ)-e(θ+φ)=i2sinφsin((θ-φ)/2)sin((θ+φ)/2).
From this identity, we can easily get the following relations:(2.6)e(θ-φ)-e(θ+φ)=-i2cos⁡(φ/2)cos⁡(θ/2)[1sin((φ-θ)/2)+1sin((φ+θ)/2)],e(θ-φ)-e(θ+φ)=-i2sin(φ/2)sin(θ/2)[1sin((φ-θ)/2)-1sin((φ+θ)/2)].
The authors of the papers [4–7] used these relations earlier while establishing the basicity criterion of the system of sines and cosines with linear phases in Lp. Thus, the following is valid.Lemma 2.1.
The following identities are true:(2.7)11-ei(θ-φ)-11-ei(θ+φ)=-i2cos⁡(φ/2)cos⁡(θ/2)[1sin((φ-θ)/2)+1sin((φ+θ)/2)],11-ei(θ-φ)-11-ei(θ+φ)=-i2sin(φ/2)sin(θ/2)[1sin((φ-θ)/2)-1sin((φ+θ)/2)].In the similar way, we obtain(2.8)11-ei(θ-φ)+11-ei(θ+φ)=e-i((θ-φ)/2)e-i((θ-φ)/2)-ei((θ-φ)/2)+e-i((θ+φ)/2)e-i((θ+φ)/2)-ei((θ+φ)/2)=-12ie-i((θ-φ)/2)sin((θ-φ)/2)-12ie-i((θ+φ)/2)sin((θ+φ)/2)=-12i[cos⁡((θ-φ)/2)-isin((θ-φ)/2)sin((θ-φ)/2)+cos⁡((θ+φ)/2)-isin((θ+φ)/2)sin((θ+φ)/2)]=-12i[cos⁡((θ-φ)/2)sin((θ-φ)/2)+cos⁡((θ+φ)/2)sin((θ+φ)/2)]+1=1-12isinθsin((θ-φ)/2)sin((θ+φ)/2).
Further, we must take into account the following relation:(2.9)1sin((θ-φ)/2)sin((θ+φ)/2)=12sin(θ/2)cos⁡(φ/2)[1sin((θ-φ)/2)+1sin((θ+φ)/2)],1sin((θ-φ)/2)sin((θ+φ)/2)=12cos⁡(θ/2)sin(φ/2)[1sin((θ-φ)/2)-1sin((θ+φ)/2)].
As a result, we have(2.10)e(θ-φ)+e(θ+φ)=1-12isin(θ/2)sin(φ/2)[1sin((θ-φ)/2)-1sin((θ+φ)/2)],(2.11)e(θ-φ)+e(θ+φ)=1-12icos⁡(θ/2)cos⁡(φ/2)[1sin((θ-φ)/2)+1sin((θ+φ)/2)].
Assume(2.12)Kφ(θ)=e(θ-φ)+e(θ+φ)-1.
Thus,(2.13)Kφ(-θ)=-12icos⁡(θ/2)cos⁡(φ/2)[1-sin((θ+φ)/2)-1sin((θ-φ)/2)]=12icos⁡(θ/2)cos⁡(φ/2)×[1sin((θ+φ)/2)+1sin((θ-φ)/2)]=-Kφ(θ).
In sequel the following lemma is valid.Lemma 2.2.
The following identities are true:(2.14)11-ei(θ-φ)+11-ei(θ+φ)=1-12isin(θ/2)sin(φ/2)[1sin((θ-φ)/2)-1sin((θ+φ)/2)],11-ei(θ-φ)+11-ei(θ+φ)=1-12icos⁡(θ/2)cos⁡(φ/2)[1sin((θ-φ)/2)-1sin((θ+φ)/2)].


## 3. Boundedness of Singular Operators on Subspace of Even Functions

Letf∈Lp,ρ-. We have(3.1)[Sf](t)=12πi∫-ππf(s)ds1-e-i(s-t)=12πi[∫0πf(s)ds1-ei(t-s)+∫-π0f(s)ds1-ei(t-s)]=12πi∫0π[11-ei(t-s)-11-ei(t+s)]f(s)ds=-14π∫0πK1(t,s)f(s)ds=-14π∫0πK2(t,s)f(s)ds,
where the kernels Ki(t,s), i=1,2, are determined by the expressions(3.2)K1(t,s)=cos⁡(s/2)cos⁡(t/2)[1sin((s-t)/2)+1sin((s+t)/2)],K2(t,s)=sin(s/2)sin(t/2)[1sin((s-t)/2)-1sin((s+t)/2)].
Continue the weight ω(t) to the interval (-π,0) by parity and denote by μ:(3.3)μ(t)={ω(t),0≤t≤π,ω(-t),-π≤t<0.
It is obvious that {±τk}0r⊂[-π,π] are the degeneration points of μ(t). Thus, μ(t)=ω(|t|), t∈[-π,π], that is,(3.4)μ(t)=∏k=0r||t|-τk|pαk.
Accept the denotation(3.5)f~gast⟶a,if∃δ>0:δ≤|fg|≤δ-1,ast⟶a.
It is easy to see that the following holds(3.6)||t|-τk|~|t-τk||t+τk|,t∈[-π,π],fork>1.
As a result, for μ, we get the representation(3.7)μ(t)~ρ0(t)≡|t|pα0∏k=1r|t-τk|pαk|t+τk|pαk,t∈[-π,π].
It is obvious that the singular integral S boundedly acts in Lp,μ(-π,π) if and only if it boundedly acts in Lp,ρ0(-π,π). Statement 1 is valid also in the case if the Cauchy kernel is replaced by the Hilbert kernel (1/sin((φ-t)/2)). So, assume that the inequalities  -1/p<αk<1/q, k=0,r¯, are fulfilled. Then, from Statement 1, we directly get that S boundedly acts from Lp,ρ0- to Lp,ρ0 and so from Lp,μ- to Lp,μ. Assume(3.8)k±(s,t)=sin(s/2)sin(t/2)1sin((s±t)/2),(s,t)∈[-π,π]×[-π,π],
and consider the integral operator I±:(3.9)I±f=∫-ππk±(s,t)f(s)ds,
with the kernel k±(s,t). We have (f∈Lp,μ-(-π,π)):(3.10)‖Sf‖p,μ=14π‖I-f̃-I+f̃‖p,μ≤14π(‖I-f̃‖p,μ+‖I+f̃‖p,μ),
where(3.11)f̃(t)={f(t),t∈[0,π],0,t∈[-π,0).
Consequently,(3.12)‖I-f̃‖p,μp=∫-ππ|(I-f̃)(t)|pμ(t)dt~∫-ππ|I-f̃|pρ0(t)dt=∫-ππ|∫-ππf̃(s)sin(s/2)dssin((s-t)/2)|pρ0(t)|sint2|-pdt.
It is obvious that(3.13)ρ0(t)|sint2|-p~θ(t),
where θ(t)=|s|p(α0-1)∏k=1r|t-τk|pαk|t+τk|pαk. So, if the inequalities(3.14)-1p<α0-1<1q,-1p<αk<1q,k=1,r¯,
hold, then from Statement 1 we obtain(3.15)‖I-f̃‖Lp,μp≤c∫-ππ|∫-ππf̃(s)sin(s/2)sin((s-t)/2)ds|pθ(t)dt≤c‖f̃(s)sinsθ(s)2‖Lp,θp≤c∫0π|f(s)s|pθ(s)ds=c‖f‖Lp,ω0p,
where c is a constant independent from f (different in different places). As a result, we get that if the inequalities (3.14) hold, the operator I- boundedly acts from Lp,ω0 to Lp,μ. The same conclusion is true for the operator I+ as well. As a result, we get that while fulfilling the conditions(3.16)α0∈(-1p,1q)⋃(1q,1+1q),αk∈(-1p,1q),k=1,r¯,
the operator S boundedly acts from Lp,μ- to Lp,μ. On the other hand, it is easily seen that [Sf](-t)=[Sf](t). As a result, we get that the operator S boundedly acts from Lp,μ- to Lp,μ+.Now, consider the case whenα0=1/q and αk, k=1,r¯, satisfy conditions (3.16). Let p±ɛ=p±ɛ and q±ɛ be a number conjugated to p±ɛ. It is obvious that the relations(3.17)α0∈(-1p±ɛ,1q±ɛ)⋃(1q±ɛ,1+1q±ɛ),αk∈(-1p±ɛ,1q±ɛ),k=1,r¯,
are fulfilled for sufficiently small ɛ>0. Then, from the previous reasonings we get that the operator S boundedly acts from Lp±ɛ,μ- to Lp±ɛ,μ+. As a result, it follows from the Riesz-Torin theorem (see, e.g., [7, page 144]) that the operator S boundedly acts from Lp,μ- to Lp,μ+. We get the following.Statement 2.
Let the inequalities(3.18)-1p<α0<2-1p,-1p<αk<1q,k=1,r¯,
be fulfilled. Then, the operator S boundedly acts from Lp,μ- to Lp,μ+.Now, consider the representation of the operatorS by the kernel K1(t,s). Having paid attention to the expression(3.19)cos⁡(s/2)cos⁡(t/2)=sin((π-s)/2)sin((π-t)/2),
similar to the previous case we establish that the boundedness of the operator S holds also in the case when the change interval of the exponent αr extends by (-1/p,2-1/p). In the conclusion we get that the following main theorem is valid.Theorem 3.1.
Let the weight functionω be defined by the expression  (2.1) and  assume that the inequalities
(3.20)-1p<α0<2-1p,-1p<αm<2-1p,-1p<αk<1q,k=1,r-1¯,
are fulfilled. Then the singular operator S:
(3.21)Sf=∫-ππk(t,s)f(s)ds,
with Cauchy kernel k(t,s)=1/(1-e-i(s-t)), boundedly acts from Lp,μ- to Lp,μ+, where  μ(t)=ω(|t|), t∈(-π,π).


## 4. Boundedness of Singular Operators on Subspace of Odd Functions

Let the weight functionω(t) be defined by expression  (2.1)  and assume μ(t)≡ω(|t|), t∈(-π,π). Denote by K(s;t) the following Cauchy-type kernel
(4.1)K(s;t)=11-ei(t-s)-12,(s;t)∈(-π,π)×(-π,π).
Appropriate integral operator denote by 𝒦:(4.2)[Kf](t)=12π∫-ππK(s;t)f(s)ds.
Let the following inequalities(4.3)-1p<αk<1q,k=0,r¯,
be fulfilled. We have(4.4)∫-ππ|f(t)|dt=∫-ππ|f(t)|μ1/pμ-1/pdt≤‖f‖p,μ(∫-ππμ-q/pdt)1/q.From (4.3) it follows that -qαk>-1, k=0,r¯, and as a result μ-q/p∈L1(-π,π). As a result, from Statement 1 we obtain that the integral operator 𝒦  boundedly acts in Lp,μ, if the inequalities (4.3) hold. In particular, it follows that the operator 𝒦 boundedly acts from Lp,μ+ to Lp,μ, if the inequalities (4.3) are fulfilled, that is,(4.5)‖Kf‖Lp,μ≤c‖f‖Lp,μ+,∀f∈Lp,μ+,
where c is a constant independent from f. On the other hand for f∈Lp,μ+ we have(4.6)[Kf](t)=12π∫0πf(s)[K(s;t)+K(-s;t)-1]ds.
Pay an attention to the relation (2.13), we obtain that [𝒦f](-t)=-[𝒦f](t), t∈(-π,π). Then from (4.5) yields(4.7)‖Kf‖Lp,ω0≤c‖f‖Lp,ω0,∀f∈Lp,ω0.
Now, let(4.8)-1-1p<α0<-1p,-1p<αk<1q,k=1,r¯
be fulfilled. Assume(4.9)K±(s;t)=-12isin(t/2)sin(s/2)sin((t∓s)/2),(s;t)∈(0,π)×(0,π).
Denote the integral operator with kernel K±(s;t) by S±, that is,(4.10)[S±f](t)=12π∫0πK±(s;t)f(s)ds,t∈(0,π).
Taking into account the relation (2.10), for f∈Lp,μ+ we have(4.11)[Kf](t)=12π∫0πf(s)[K(s;t)+K(-s;t)-1]ds=12π∫0π[K+(s;t)-K-(s;t)]f(s)ds=[S+-S-]f,
that is,(4.12)[Kf](t)=[(S+-S-)f](t),t∈(-π,π).
Take into account the nonparity [𝒦f](t) on (-π,π) we obtain(4.13)2‖Kf‖Lp,ω0p=‖Kf‖Lp,μp≤(‖S+f‖Lp,μ+‖S-f‖Lp,μ)p.
Let (4.14)f̃(t)={f(t),t∈(0,π),0,t∈(-π,0).
Consider the operator S+. We have(4.15)[S+f](t)=sin(t/2)2π∫0πf(s)sin-1(s/2)sin((t-s)/2)ds=sin(t/2)2π∫-ππf̃(s)sin-1(s/2)sin((t-s)/2)ds.
Thus(4.16)‖S+f‖Lp,μp=c∫-ππ|[S+f](t)|pμ(t)dt=c∫-ππ|∫-ππf̃(s)sin-1(s/2)sin((t-s)/2)ds|p|sint2|pμ(t)dt.Further, we must take into account the  expressionsin(t/2)~t, t∈(-π,π). As a result, from the previous relation we have(4.17)‖S+f‖Lp,μp≤c∫-ππ|∫-ππf̃(s)sin-1(s/2)sin((t-s)/2)ds|pμ̃(t)dt,
where(4.18)μ̃(t)=|t|p(α0+1)∏k=1r||t|-tk|pαk.
It is clear that for weight function μ̃(t)  Muckenhoupt condition is fulfilled and applying Statement 1 to the expression (4.17)  we obtain(4.19)‖S+f‖Lp,μp≤c‖f̃(s)sin-1s2‖Lp,μ̃p=c∫-ππ|f̃(s)|p|sin-1s2|pμ̃(s)ds≤c∫0π|f(s)|p|s|-pμ̃(s)ds=c∫0π|f(s)|pω(s)ds=c‖f‖Lp,μ+p.
In the similar way we establish the validity of the inequality(4.20)‖S-f‖Lp,μ≤c‖f‖Lp,μ+,∀f∈Lp,μ+.
If the inequalities (4.8) hold, as a result, we have(4.21)‖Kf‖Lp,μ≤c‖f‖Lp,μ+,∀f∈Lp,μ+.
Consider the case(4.22)α0=-1p,-1p<αk<1q,k=1,r¯.
Take sufficiently small ɛ>0 and determine pɛ±=p±ɛ. Acting similarly to the case Lp,μ- (par.3) and accept the Riesz-Torin  theorem we obtain boundedly acting of the operator 𝒦  from Lp,μ+ to Lp,μ- (since [𝒦f](t)∈Lp,μ-). Thus, if the following inequalities(4.23)-1-1p<α0<1q,-1p<αk<1q,k=1,r¯,
are fulfilled, then the operator 𝒦 boundedly acts from Lp,μ+ to Lp,μ-.Using the identity (2.11) in the similar way we establish that the same conclusion with respect to the operator 𝒦 is true in the case when the change interval of the exponent αr is expanded on (-1-1/p,1/q). As a result, we obtain the validity of the following theorem.Theorem 4.1.
Let the weight functionω be defined by the expression  (2.1)  and μ(t)≡ω(|t|), t∈(-π,π). Assume that the inequalities
(4.24)-1-1p<α0<1q,-1-1p<αr<1q,-1p<αk<1q,k=1,r-1¯,
are fulfilled. Then the singular operator 𝒦:
(4.25)Kf=∫-ππK(s;t)f(s)ds,
with Cauchy-type kernel K(s;t)=(1/(1-e-i(s-t)))-1/2, boundedly acts from Lp,μ+ to Lp,μ-.


## 5. Completeness, Minimality,  and Basicity of the System of Sines in Weight Space

Consider the system of sines{sinnt}n∈N. Let conditions (3.20) be fulfilled. It is easy to see that then the system {sinnt}n∈N  is minimal in Lp,ω0. The system {(2/π)ω-1(t)sinnt}n∈N, 1/p+1/q=1, is a biorthogonal system to it. Indeed, it is obvious that Lq,ω0 is a space conjugated to Lp,ω0, and an arbitrary continuous functional lg on Lp,ω0, generated by g∈Lq,ω0, realized by the formula(5.1)lg(f)=∫0πfg¯ωdt,∀f∈Lp,ω0,
where (.-) is a complex conjugation.Takeg(t)=ω-1(t)sinnt, n∈N. We have(5.2)‖ω-1(t)sinnt‖Lq,ω0q=∫0π|ω-1(t)sinnt|qω(t)dt=∫0πω1-q(t)|sinnt|qdt.Sincesinnt~t as t→0 and sinnt~π-t as t→π  for every fixed n∈N, then from relation (5.2) follows that {ω-1sinnt}n∈N⊂Lq,ω0, if the inequalities(5.3)α0<2-1p,αm<2-1p,αk<1q,k=1,r-1¯,
are fulfilled.Takeg(t)=(2/π)ω-1(t)sinnt and denote by ϑn generated by its functional, that is,(5.4)ϑn(f)=2π∫0πf(t)sinntdt,∀f∈Lp,ω0.
It is clear that ϑn(sinkt)=δnk, ∀n,k∈N, where  δnk is a Kronecker's symbol. Consider(5.5)‖sinnt‖Lp,ω0p=∫0π|sinnt|pω(t)dt.
Thus, {sinnt}n∈N⊂Lp,ω, if α0>-1/p-1, αr>-1/p-1, αk>-1/p, k=0,r-1¯. As a result, we obtain that if the inequalities(5.6)-1-1p<α0<2-1p,-1-1p<αr<2-1p,-1p<αk<1q,k=0,r-1¯,
hold, then the system {sinnt}n∈N  is minimal in Lp,ω0.Now consider the completeness of the system{sinnt}n∈N in Lp,ω0. Suppose that for some g∈Lq,ω0,(5.7)∫0πsinntg¯ωdt=0,∀n∈N,
holds. We have(5.8)∫0π|g¯ω|dt=∫0π|g¯|ω1/qω1/pdt≤‖g‖Lq,ω0(∫0πωdt)1/p.
It is easy to see that if the inequalities(5.9)αk>-1p,k=0,r¯,
are fulfilled, then ω∈L1. Then from the previous relation we get g¯ω∈L1. Since, the system {sinnt}n∈N is complete in space of continuous on [0,π] functions with sup-norm, which vanishes at the ends of the segment [0,π], then from (5.7) it follows that g(t)=0 a.e. on (0,π). Consequently, the system {sinnt}n∈N is complete in Lp,ω0. So, the following statement is true.Statement 3.
Let the weight functionω(t) be defined by expression (2.1). The system of  sines  {sinnt}n∈N is minimal in Lp,ω0, if the inequalities (5.6) are fulfilled. It is complete in Lp,ω0, if the inequalities (5.9) are fulfilled. Moreover, it forms  a basis in Lp,ω0, if the inequalities
(5.10)-1p<αk<1q,k=0,m¯
hold.The basicity of system of sines inLp,ω0, when the inequalities (5.10) hold, follows from the basicity of system of exponent {eint}n∈Z in Lp,μ, where μ(t)=ω(|t|), t∈(-π,π). The basicity of these systems earlier considered in papers [3, 4, 8, 9].In the similar way we prove the following statement.Statement 4.
Let the weight functionω(t) defined by expression  (2.1). The system of  cosines 1∪{cosnt}n∈N is minimal (forms a basis)  in Lp,ω0, if the inequalities (5.10) are fulfilled. It is complete in Lp,ω0, if the inequalities (5.9) holds.



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