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

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.

1. Introduction

Consider the following singular operator with Cauchy kernel:𝑡)=1𝑆𝑓(2𝜋𝑖^𝜋_−𝜋𝑓(𝑠)𝑑𝑠1−𝑒^{−𝑖(𝑠−𝑡)}, (1.1) where 𝑓∈𝐿_𝑝,𝜌(−𝜋,𝜋), 1<𝑝<+∞, is an appropriate density, 𝜌(𝑡) is a weight function of the form 𝜌(𝑡)=^𝑟_𝑘=1𝑡−𝑡_𝑘^𝑝𝛽𝑘, (1.2){𝑡_𝑘}^𝑟₁⊂[−𝜋,𝜋] (𝑡_𝑖≠𝑡_𝑗 for 𝑖≠𝑗), {𝛽_𝑘}^𝑟₁⊂𝑅 are real numbers.

Under 𝐿_𝑝,𝜌 we understand a Lebesgue weight space with the norm𝑓_𝑝,𝜌≡^𝜋_−𝜋𝑓(𝑡)^𝑝𝜌(𝑡)𝑑𝑡^1/𝑝. (1.3)

Bounded action of the operator 𝑆 in the spaces 𝐿_𝑝,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 operator 𝑆 is bounded in 𝐿_𝑝,𝜌 if and only if the inequalities −1𝑝<𝛽_𝑘<1𝑞,𝑘=11,𝑟,𝑝+1𝑞=1, (1.4) 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 𝑝𝛽_𝑘. It is known that the classic system of exponents {𝑒^𝑖𝑛𝑡}_𝑛∈𝑍 (𝑍 are integers) forms a basis in 𝐿_𝑝,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

Let 𝐿⁰_𝑝,𝜔≡𝐿_𝑝,𝜔(0,𝜋), 𝜔(𝑡)—a weight function of the form𝜔(𝑡)≡^𝑟_𝑘=0𝑡−𝜏_𝑘^𝑝𝛼𝑘, (2.1) where 0=𝜏₀<𝜏₁<⋯<𝜏_𝑟=𝜋, {𝛼_𝑘}^𝑟₀⊂𝑅.

Denote the space of even (odd) functions in 𝐿_𝑝,𝜌 by 𝐿⁺_𝑝,𝜌 (𝐿⁻_𝑝,𝜌), that is,𝐿^±_𝑝,𝜌≡𝑓∈𝐿_𝑝,𝜌∶𝑓(−𝑡)=±𝑓(𝑡),𝑡∈[−𝜋,𝜋]. (2.2) We'll need the following identity:11−𝑒^{𝑖(𝜃−𝜑)}−11−𝑒^{𝑖(𝜃+𝜑)}=𝑖2sin𝜑.sin𝜑−𝜃/2sin𝜃+𝜑/2 (2.3) Indeed, we have11−𝑒^{𝑖(𝜃−𝜑)}−11−𝑒^{𝑖(𝜃+𝜑)}=1−𝑒^{−𝑖(𝜃−𝜑)}1−𝑒^{𝑖(𝜃−𝜑)}1−𝑒^{−𝑖(𝜃−𝜑)}−1−𝑒^{−𝑖(𝜃+𝜑)}1−𝑒^{𝑖(𝜃+𝜑)}1−𝑒^{−𝑖(𝜃+𝜑)}=1−cos𝜃−𝜑+𝑖sin𝜃−𝜑1−cos𝜃−𝜑²+sin²−𝜃−𝜑1−cos𝜃+𝜑+𝑖sin𝜃+𝜑1−cos𝜃+𝜑²+sin²𝜃+𝜑=𝑖sin𝜃−𝜑−2−2cos𝜃−𝜑sin𝜃+𝜑2−2cos𝜃+𝜑=𝑖cos𝜃−𝜑/2−2sin𝜃−𝜑/2cos𝜃+𝜑/2=𝑖2sin𝜃+𝜑/22cos𝜃−𝜑/2sin𝜃+𝜑/2−cos𝜃+𝜑/2sin𝜃−𝜑/2=𝑖sin𝜃−𝜑/2sin𝜃+𝜑/22sin𝜑.sin𝜃−𝜑/2sin𝜃+𝜑/2 (2.4) For compactness of the notation, we assume 𝑒(𝑥)≡1/(1−𝑒^𝑖𝑥). Thus,𝑒=𝑖𝜃−𝜑−𝑒𝜃+𝜑2sin𝜑.sin𝜃−𝜑/2sin𝜃+𝜑/2 (2.5) From this identity, we can easily get the following relations:𝑒𝑖𝜃−𝜑−𝑒𝜃+𝜑=−2cos𝜑/2()1cos𝜃/2+1sin𝜑−𝜃/2,𝑒𝑖sin𝜑+𝜃/2𝜃−𝜑−𝑒𝜃+𝜑=−2sin𝜑/2()1sin𝜃/2−1sin𝜑−𝜃/2.sin𝜑+𝜃/2 (2.6) 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 𝐿_𝑝. Thus, the following is valid.

Lemma 2.1. The following identities are true: 11−𝑒^{𝑖(𝜃−𝜑)}−11−𝑒^{𝑖(𝜃+𝜑)}𝑖=−2cos𝜑/2()1cos𝜃/2+1sin𝜑−𝜃/2,1sin𝜑+𝜃/21−𝑒^{𝑖(𝜃−𝜑)}−11−𝑒^{𝑖(𝜃+𝜑)}𝑖=−2sin𝜑/2()1sin𝜃/2−1sin𝜑−𝜃/2.sin𝜑+𝜃/2 (2.7)

In the similar way, we obtain11−𝑒^{𝑖(𝜃−𝜑)}+11−𝑒^{𝑖(𝜃+𝜑)}=𝑒^{−𝑖((𝜃−𝜑)/2)}𝑒^{−𝑖((𝜃−𝜑)/2)}−𝑒^{𝑖((𝜃−𝜑)/2)}+𝑒^{−𝑖((𝜃+𝜑)/2)}𝑒^{−𝑖((𝜃+𝜑)/2)}−𝑒^{𝑖((𝜃+𝜑)/2)}1=−𝑒2𝑖^{−𝑖((𝜃−𝜑)/2)}−1sin𝜃−𝜑/2𝑒2𝑖^{−𝑖((𝜃+𝜑)/2)}1sin𝜃+𝜑/2=−2𝑖cos𝜃−𝜑/2−𝑖sin𝜃−𝜑/2+sin𝜃−𝜑/2cos𝜃+𝜑/2−𝑖sin𝜃+𝜑/21sin𝜃+𝜑/2=−2𝑖cos𝜃−𝜑/2+sin𝜃−𝜑/2cos𝜃+𝜑/21sin𝜃+𝜑/2+1=1−2𝑖sin𝜃.sin𝜃−𝜑/2sin𝜃+𝜑/2 (2.8) Further, we must take into account the following relation:1=1sin𝜃−𝜑/2sin𝜃+𝜑/2()12sin𝜃/2cos𝜑/2+1sin𝜃−𝜑/2,1sin𝜃+𝜑/2=1sin𝜃−𝜑/2sin𝜃+𝜑/2()12cos𝜃/2sin𝜑/2−1sin𝜃−𝜑/2.sin𝜃+𝜑/2 (2.9) As a result, we have𝑒1𝜃−𝜑+𝑒𝜃+𝜑=1−()2𝑖sin𝜃/21sin𝜑/2−1sin𝜃−𝜑/2𝑒1sin𝜃+𝜑/2,𝜃−𝜑+𝑒𝜃+𝜑=1−()2𝑖cos𝜃/21cos𝜑/2+1sin𝜃−𝜑/2sin𝜃+𝜑/2. (2.10)(2.11) Assume𝐾_𝜑(𝜃)=𝑒𝜃−𝜑+𝑒𝜃+𝜑−1. (2.12) Thus,𝐾_𝜑()1−𝜃=−()2𝑖cos𝜃/21cos𝜑/2−1−sin𝜃+𝜑/2=1sin𝜃−𝜑/2()2𝑖cos𝜃/2×1cos𝜑/2+1sin𝜃+𝜑/2sin𝜃−𝜑/2=−𝐾_𝜑(𝜃). (2.13) In sequel the following lemma is valid.

Lemma 2.2. The following identities are true: 11−𝑒^{𝑖(𝜃−𝜑)}+11−𝑒^{𝑖(𝜃+𝜑)}1=1−()2𝑖sin𝜃/21sin𝜑/2−1sin𝜃−𝜑/2,1sin𝜃+𝜑/21−𝑒^{𝑖(𝜃−𝜑)}+11−𝑒^{𝑖(𝜃+𝜑)}1=1−()2𝑖cos𝜃/21cos𝜑/2−1sin𝜃−𝜑/2.sin𝜃+𝜑/2 (2.14)

3. Boundedness of Singular Operators on Subspace of Even Functions

Let 𝑓∈𝐿⁻_𝑝,𝜌. We have𝑡)=1𝑆𝑓(2𝜋𝑖^𝜋_−𝜋𝑓(𝑠)𝑑𝑠1−𝑒^{−𝑖(𝑠−𝑡)}=12𝜋𝑖^𝜋₀𝑓(𝑠)𝑑𝑠1−𝑒^{𝑖(𝑡−𝑠)}+⁰_−𝜋𝑓(𝑠)𝑑𝑠1−𝑒^{𝑖(𝑡−𝑠)}=12𝜋𝑖^𝜋₀11−𝑒^{𝑖(𝑡−𝑠)}−11−𝑒^{𝑖(𝑡+𝑠)}𝑓(𝑠)1𝑑𝑠=−4𝜋^𝜋₀𝐾₁()𝑓(𝑠)1𝑡,𝑠𝑑𝑠=−4𝜋^𝜋₀𝐾₂(𝑡,𝑠)𝑓(𝑠)𝑑𝑠, (3.1) where the kernels 𝐾_𝑖(𝑡,𝑠), 𝑖=1,2, are determined by the expressions𝐾₁()=()𝑡,𝑠cos𝑠/2()1cos𝑡/2))+1sin((𝑠−𝑡/2)),𝐾sin((𝑠+𝑡/2₂()=()𝑡,𝑠sin𝑠/2()1sin𝑡/2))−1sin((𝑠−𝑡/2)).sin((𝑠+𝑡/2 (3.2) Continue the weight 𝜔(𝑡) to the interval (−𝜋,0) by parity and denote by 𝜇:𝜇(𝑡)=𝜔(𝑡)𝜔(),0≤𝑡≤𝜋,−𝑡,−𝜋≤𝑡<0. (3.3) It is obvious that {±𝜏_𝑘}^𝑟₀⊂[−𝜋,𝜋] are the degeneration points of 𝜇(𝑡). Thus, 𝜇(𝑡)=𝜔(|𝑡|), 𝑡∈[−𝜋,𝜋], that is,𝜇(𝑡)=^𝑟_𝑘=0|𝑡|−𝜏_𝑘^𝑝𝛼𝑘. (3.4) Accept the denotation𝑓𝑓∼𝑔as𝑡⟶𝑎,if∃𝛿>0∶𝛿≤𝑔≤𝛿⁻¹,as𝑡⟶𝑎. (3.5) It is easy to see that the following holds|𝑡|−𝜏_𝑘∼𝑡−𝜏_𝑘𝑡+𝜏_𝑘,𝑡∈[−𝜋,𝜋],for𝑘>1. (3.6) As a result, for 𝜇, we get the representation𝜇(𝑡)∼𝜌₀(𝑡)≡|𝑡|^{𝑝𝛼0𝑟}_𝑘=1𝑡−𝜏_𝑘^𝑝𝛼𝑘𝑡+𝜏_𝑘^𝑝𝛼𝑘,𝑡∈[−𝜋,𝜋]. (3.7) It is obvious that the singular integral 𝑆 boundedly acts in 𝐿_𝑝,𝜇(−𝜋,𝜋) if and only if it boundedly acts in 𝐿_𝑝,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<𝛼_𝑘<1/𝑞, 𝑘=0,𝑟, are fulfilled. Then, from Statement 1, we directly get that 𝑆 boundedly acts from 𝐿⁻_𝑝,𝜌0 to 𝐿_𝑝,𝜌0 and so from 𝐿⁻_𝑝,𝜇 to 𝐿_𝑝,𝜇. Assume𝑘^±()=()𝑠,𝑡sin𝑠/2()1sin𝑡/2)),()∈[]×[],sin((𝑠±𝑡/2𝑠,𝑡−𝜋,𝜋−𝜋,𝜋 (3.8) and consider the integral operator 𝐼^±:𝐼^±𝑓=^𝜋_−𝜋𝑘^±(𝑠,𝑡)𝑓(𝑠)𝑑𝑠, (3.9) with the kernel 𝑘^±(𝑠,𝑡). We have (𝑓∈𝐿⁻_𝑝,𝜇(−𝜋,𝜋)):𝑆𝑓_𝑝,𝜇=1𝐼4𝜋⁻𝑓−𝐼⁺𝑓_𝑝,𝜇≤1𝐼4𝜋⁻𝑓_𝑝,𝜇+𝐼⁺𝑓_𝑝,𝜇, (3.10) where𝑓(𝑡)=𝑓(𝑡)[],[).,𝑡∈0,𝜋0,𝑡∈−𝜋,0 (3.11) Consequently,𝐼⁻𝑓^𝑝_𝑝,𝜇=^𝜋_−𝜋𝐼⁻𝑓(𝑡)^𝑝𝜇(𝑡)𝑑𝑡∼^𝜋_−𝜋𝐼⁻𝑓^𝑝𝜌₀(𝑡)=𝑑𝑡_{𝜋−𝜋𝜋−𝜋}𝑓(𝑠)sin(𝑠/2)𝑑𝑠sin((𝑠−𝑡)/2)^𝑝𝜌₀(𝑡𝑡)sin2^−𝑝𝑑𝑡. (3.12) It is obvious that𝜌₀(𝑡𝑡)sin2^−𝑝∼𝜃(𝑡), (3.13) where 𝜃(𝑡)=|𝑠|^{𝑝(𝛼0−1)}∏^𝑟_𝑘=1|𝑡−𝜏_𝑘|^𝑝𝛼𝑘|𝑡+𝜏_𝑘|^𝑝𝛼𝑘. So, if the inequalities−1𝑝<𝛼₀1−1<𝑞1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟, (3.14) hold, then from Statement 1 we obtain𝐼⁻𝑓^𝑝_{𝐿𝑝,𝜇}≤𝑐^𝜋_−𝜋^𝜋_−𝜋𝑓(𝑠)sin(𝑠/2)sin((𝑠−𝑡)/2)𝑑𝑠^𝑝𝜃(𝑡)𝑓(𝑠)(𝑠)𝑑𝑡≤𝑐sin𝑠𝜃2^𝑝_{𝐿𝑝,𝜃}≤𝑐^𝜋₀𝑓(𝑠)𝑠^𝑝𝜃(𝑠)𝑑𝑠=𝑐𝑓^𝑝_{𝐿0𝑝,𝜔}, (3.15) 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 𝐿⁰_𝑝,𝜔 to 𝐿_𝑝,𝜇. The same conclusion is true for the operator 𝐼⁺ as well. As a result, we get that while fulfilling the conditions𝛼₀∈−1𝑝,1𝑞1𝑞1,1+𝑞,𝛼_𝑘∈−1𝑝,1𝑞,𝑘=1,𝑟, (3.16) the operator 𝑆 boundedly acts from 𝐿⁻_𝑝,𝜇 to 𝐿_𝑝,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 𝐿⁻_𝑝,𝜇 to 𝐿⁺_𝑝,𝜇.

Now, consider the case when 𝛼₀=1/𝑞 and 𝛼_𝑘, 𝑘=1,𝑟, satisfy conditions (3.16). Let 𝑝_±𝜀=𝑝±𝜀 and 𝑞_±𝜀 be a number conjugated to 𝑝_±𝜀. It is obvious that the relations𝛼₀∈−1𝑝_±𝜀,1𝑞_±𝜀1𝑞_±𝜀1,1+𝑞_±𝜀,𝛼_𝑘∈−1𝑝_±𝜀,1𝑞_±𝜀,𝑘=1,𝑟, (3.17) are fulfilled for sufficiently small 𝜀>0. Then, from the previous reasonings we get that the operator 𝑆 boundedly acts from 𝐿⁻_{𝑝±𝜀,𝜇} to 𝐿⁺_{𝑝±𝜀,𝜇}. As a result, it follows from the Riesz-Torin theorem (see, e.g., [7, page 144]) that the operator 𝑆 boundedly acts from 𝐿⁻_𝑝,𝜇 to 𝐿⁺_𝑝,𝜇. We get the following.

Statement 2. Let the inequalities −1𝑝<𝛼₀1<2−𝑝1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟, (3.18) be fulfilled. Then, the operator 𝑆 boundedly acts from 𝐿⁻_𝑝,𝜇 to 𝐿⁺_𝑝,𝜇.

Now, consider the representation of the operator 𝑆 by the kernel 𝐾₁(𝑡,𝑠). Having paid attention to the expressioncos(𝑠/2)()=))cos𝑡/2sin((𝜋−𝑠/2)),sin((𝜋−𝑡/2 (3.19) 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 𝛼_𝑟 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 function 𝜔 be defined by the expression (2.1) and assume that the inequalities −1𝑝<𝛼₀1<2−𝑝1,−𝑝<𝛼_𝑚1<2−𝑝1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟−1, (3.20) are fulfilled. Then the singular operator 𝑆: 𝑆𝑓=^𝜋_−𝜋𝑘(𝑡,𝑠)𝑓(𝑠)𝑑𝑠, (3.21) with Cauchy kernel 𝑘(𝑡,𝑠)=1/(1−𝑒^{−𝑖(𝑠−𝑡)}), boundedly acts from 𝐿⁻_𝑝,𝜇 to 𝐿⁺_𝑝,𝜇, where 𝜇(𝑡)=𝜔(|𝑡|), 𝑡∈(−𝜋,𝜋).

4. Boundedness of Singular Operators on Subspace of Odd Functions

Let the weight function 𝜔(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 𝐾()=1𝑠;𝑡1−𝑒^{𝑖(𝑡−𝑠)}−12,(𝑠;𝑡)∈(−𝜋,𝜋)×(−𝜋,𝜋). (4.1) Appropriate integral operator denote by 𝒦:𝑡)=1𝒦𝑓(2𝜋^𝜋_−𝜋𝐾(𝑠;𝑡)𝑓(𝑠)𝑑𝑠. (4.2) Let the following inequalities−1𝑝<𝛼_𝑘<1𝑞,𝑘=0,𝑟, (4.3) be fulfilled. We have^𝜋_−𝜋𝑓(𝑡)𝑑𝑡=^𝜋_−𝜋𝑓(𝑡)𝜇^1/𝑝𝜇^−1/𝑝𝑑𝑡≤𝑓_𝑝,𝜇^𝜋_−𝜋𝜇^−𝑞/𝑝𝑑𝑡^1/𝑞. (4.4)

From (4.3) it follows that −𝑞𝛼_𝑘>−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 𝐿_𝑝,𝜇, if the inequalities (4.3) hold. In particular, it follows that the operator 𝒦 boundedly acts from 𝐿⁺_𝑝,𝜇 to 𝐿_𝑝,𝜇, if the inequalities (4.3) are fulfilled, that is,𝒦𝑓_{𝐿𝑝,𝜇}≤𝑐𝑓_{𝐿+𝑝,𝜇},∀𝑓∈𝐿⁺_𝑝,𝜇, (4.5) where 𝑐 is a constant independent from 𝑓. On the other hand for 𝑓∈𝐿⁺_𝑝,𝜇 we have𝑡)=1𝒦𝑓(2𝜋^𝜋₀𝑓(𝑠)[𝐾(𝑠;𝑡)+𝐾(−𝑠;𝑡)−1]𝑑𝑠. (4.6) Pay an attention to the relation (2.13), we obtain that [𝒦𝑓](−𝑡)=−[𝒦𝑓](𝑡), 𝑡∈(−𝜋,𝜋). Then from (4.5) yields𝒦𝑓_{𝐿0𝑝,𝜔}≤𝑐𝑓_{𝐿0𝑝,𝜔},∀𝑓∈𝐿⁰_𝑝,𝜔. (4.7) Now, let1−1−𝑝<𝛼₀1<−𝑝1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟 (4.8) be fulfilled. Assume𝐾^±()1𝑠;𝑡=−()2𝑖sin𝑡/2())),()∈()×().sin𝑠/2sin((𝑡∓𝑠/2𝑠;𝑡0,𝜋0,𝜋 (4.9) Denote the integral operator with kernel 𝐾^±(𝑠;𝑡) by 𝑆^±, that is,𝑆^±𝑓𝑡)=1(2𝜋^𝜋₀𝐾^±(𝑠;𝑡)𝑓(𝑠)𝑑𝑠,𝑡∈(0,𝜋). (4.10) Taking into account the relation (2.10), for 𝑓∈𝐿⁺_𝑝,𝜇 we have𝑡)=1𝒦𝑓(2𝜋^𝜋₀𝑓(𝑠𝐾()()]=1)[𝑠;𝑡+𝐾−𝑠;𝑡−1𝑑𝑠2𝜋^𝜋₀𝐾⁺(𝑠;𝑡)−𝐾⁻(𝑠;𝑡)𝑓(𝑠)𝑑𝑠=𝑆⁺−𝑆⁻𝑓, (4.11) that is,𝒦𝑓(𝑡)=𝑆⁺−𝑆⁻𝑓(𝑡),𝑡∈(−𝜋,𝜋). (4.12) Take into account the nonparity [𝒦𝑓](𝑡) on (−𝜋,𝜋) we obtain2𝒦𝑓^𝑝_{𝐿0𝑝,𝜔}=𝒦𝑓^𝑝_{𝐿𝑝,𝜇}≤𝑆⁺𝑓_{𝐿𝑝,𝜇}+𝑆⁻𝑓_{𝐿𝑝,𝜇}^𝑝. (4.13) Let 𝑓(𝑡)=𝑓(𝑡)(),().,𝑡∈0,𝜋0,𝑡∈−𝜋,0 (4.14) Consider the operator 𝑆⁺. We have𝑆⁺𝑓𝑡)=()(sin𝑡/22𝜋^𝜋₀𝑓(𝑠)sin⁻¹(𝑠/2)))()sin((𝑡−𝑠/2𝑑𝑠=sin𝑡/22𝜋^𝜋_−𝜋𝑓(𝑠)sin⁻¹(𝑠/2)sin((𝑡−𝑠)/2)𝑑𝑠. (4.15) Thus𝑆⁺𝑓^𝑝_{𝐿𝑝,𝜇}=𝑐^𝜋_−𝜋𝑆⁺𝑓(𝑡)^𝑝𝜇(𝑡)𝑑𝑡=𝑐^𝜋_−𝜋^𝜋_−𝜋𝑓(𝑠)sin⁻¹(𝑠/2)sin((𝑡−𝑠)/2)𝑑𝑠^𝑝𝑡sin2^𝑝𝜇(𝑡)𝑑𝑡. (4.16)

Further, we must take into account the expression sin(𝑡/2)∼𝑡, 𝑡∈(−𝜋,𝜋). As a result, from the previous relation we have𝑆⁺𝑓^𝑝_{𝐿𝑝,𝜇}≤𝑐^𝜋_−𝜋^𝜋_−𝜋𝑓(𝑠)sin⁻¹(𝑠/2)sin((𝑡−𝑠)/2)𝑑𝑠^𝑝𝜇(𝑡)𝑑𝑡, (4.17) where𝜇(𝑡)=|𝑡|^{𝑝(𝛼0𝑟+1)}_𝑘=1|𝑡|−𝑡_𝑘^𝑝𝛼𝑘. (4.18) It is clear that for weight function 𝜇(𝑡) Muckenhoupt condition is fulfilled and applying Statement 1 to the expression (4.17) we obtain𝑆⁺𝑓^𝑝_{𝐿𝑝,𝜇}≤𝑐𝑓(𝑠)sin⁻¹𝑠2^𝑝_𝐿𝑝,𝜇=𝑐^𝜋_−𝜋𝑓(𝑠)^𝑝sin⁻¹𝑠2^𝑝𝜇(𝑠)𝑑𝑠≤𝑐^𝜋₀𝑓(𝑠)^𝑝|𝑠|^−𝑝𝜇(𝑠)𝑑𝑠=𝑐^𝜋₀𝑓(𝑠)^𝑝𝜔(𝑠)𝑑𝑠=𝑐𝑓^𝑝_{𝐿+𝑝,𝜇}. (4.19) In the similar way we establish the validity of the inequality𝑆⁻𝑓_{𝐿𝑝,𝜇}≤𝑐𝑓_{𝐿+𝑝,𝜇},∀𝑓∈𝐿⁺_𝑝,𝜇. (4.20) If the inequalities (4.8) hold, as a result, we have𝒦𝑓_{𝐿𝑝,𝜇}≤𝑐𝑓_{𝐿+𝑝,𝜇},∀𝑓∈𝐿⁺_𝑝,𝜇. (4.21) Consider the case𝛼₀1=−𝑝1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟. (4.22) Take sufficiently small 𝜀>0 and determine 𝑝^±_𝜀=𝑝±𝜀. Acting similarly to the case 𝐿⁻_𝑝,𝜇 (par.3) and accept the Riesz-Torin theorem we obtain boundedly acting of the operator 𝒦 from 𝐿⁺_𝑝,𝜇 to 𝐿⁻_𝑝,𝜇 (since [𝒦𝑓](𝑡)∈𝐿⁻_𝑝,𝜇). Thus, if the following inequalities1−1−𝑝<𝛼₀<1𝑞1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟, (4.23) are fulfilled, then the operator 𝒦 boundedly acts from 𝐿⁺_𝑝,𝜇 to 𝐿⁻_𝑝,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 𝛼_𝑟 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 function 𝜔 be defined by the expression (2.1) and 𝜇(𝑡)≡𝜔(|𝑡|), 𝑡∈(−𝜋,𝜋). Assume that the inequalities 1−1−𝑝<𝛼₀<1𝑞1,−1−𝑝<𝛼_𝑟<1𝑞1,−𝑝<𝛼_𝑘<1𝑞,𝑘=1,𝑟−1, (4.24) are fulfilled. Then the singular operator 𝒦: 𝒦𝑓=^𝜋_−𝜋𝐾(𝑠;𝑡)𝑓(𝑠)𝑑𝑠, (4.25) with Cauchy-type kernel 𝐾(𝑠;𝑡)=(1/(1−𝑒^{−𝑖(𝑠−𝑡)}))−1/2, boundedly acts from 𝐿⁺_𝑝,𝜇 to 𝐿⁻_𝑝,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 𝐿⁰_𝑝,𝜔. 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 𝐿⁰_𝑞,𝜔 is a space conjugated to 𝐿⁰_𝑝,𝜔, and an arbitrary continuous functional 𝑙_𝑔 on 𝐿⁰_𝑝,𝜔, generated by 𝑔∈𝐿⁰_𝑞,𝜔, realized by the formula𝑙_𝑔𝑓=^𝜋₀𝑓𝑔𝜔𝑑𝑡,∀𝑓∈𝐿⁰_𝑝,𝜔, (5.1) where (.) is a complex conjugation.

Take 𝑔(𝑡)=𝜔⁻¹(𝑡)sin𝑛𝑡, 𝑛∈𝑁. We have𝜔⁻¹(𝑡)sin𝑛𝑡^𝑞_{𝐿0𝑞,𝜔}=^𝜋₀𝜔⁻¹(𝑡)sin𝑛𝑡^𝑞𝜔(𝑡)𝑑𝑡=^𝜋₀𝜔^1−𝑞(𝑡)|sin𝑛𝑡|^𝑞𝑑𝑡. (5.2)

Since sin𝑛𝑡∼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⊂𝐿⁰_𝑞,𝜔, if the inequalities𝛼₀1<2−𝑝,𝛼_𝑚1<2−𝑝,𝛼_𝑘<1𝑞,𝑘=1,𝑟−1, (5.3) are fulfilled.

Take 𝑔(𝑡)=(2/𝜋)𝜔⁻¹(𝑡)sin𝑛𝑡 and denote by 𝜗_𝑛 generated by its functional, that is,𝜗_𝑛𝑓=2𝜋^𝜋₀𝑓(𝑡)sin𝑛𝑡𝑑𝑡,∀𝑓∈𝐿⁰_𝑝,𝜔. (5.4) It is clear that 𝜗_𝑛(sin𝑘𝑡)=𝛿_𝑛𝑘, ∀𝑛,𝑘∈𝑁, where 𝛿_𝑛𝑘 is a Kronecker's symbol. Consider‖sin𝑛𝑡‖^𝑝_{𝐿0𝑝,𝜔}=^𝜋₀|sin𝑛𝑡|^𝑝𝜔(𝑡)𝑑𝑡. (5.5) Thus, {sin𝑛𝑡}_𝑛∈𝑁⊂𝐿_𝑝,𝜔, if 𝛼₀>−1/𝑝−1, 𝛼_𝑟>−1/𝑝−1, 𝛼_𝑘>−1/𝑝, 𝑘=0,𝑟−1. As a result, we obtain that if the inequalities1−1−𝑝<𝛼₀1<2−𝑝1,−1−𝑝<𝛼_𝑟1<2−𝑝1,−𝑝<𝛼_𝑘<1𝑞,𝑘=0,𝑟−1, (5.6) hold, then the system {sin𝑛𝑡}_𝑛∈𝑁 is minimal in 𝐿⁰_𝑝,𝜔.

Now consider the completeness of the system {sin𝑛𝑡}_𝑛∈𝑁 in 𝐿⁰_𝑝,𝜔. Suppose that for some 𝑔∈𝐿⁰_𝑞,𝜔,^𝜋₀sin𝑛𝑡𝑔𝜔𝑑𝑡=0,∀𝑛∈𝑁, (5.7) holds. We have^𝜋₀𝑔𝜔𝑑𝑡=^𝜋₀𝑔𝜔^1/𝑞𝜔^1/𝑝𝑑𝑡≤𝑔_{𝐿0𝑞,𝜔}^𝜋₀𝜔𝑑𝑡^1/𝑝. (5.8) It is easy to see that if the inequalities𝛼_𝑘1>−𝑝,𝑘=0,𝑟, (5.9) are fulfilled, then 𝜔∈𝐿₁. Then from the previous relation we get 𝑔𝜔∈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 𝐿⁰_𝑝,𝜔. So, the following statement is true.

Statement 3. Let the weight function 𝜔(𝑡) be defined by expression (2.1). The system of sines {sin𝑛𝑡}_𝑛∈𝑁 is minimal in 𝐿⁰_𝑝,𝜔, if the inequalities (5.6) are fulfilled. It is complete in 𝐿⁰_𝑝,𝜔, if the inequalities (5.9) are fulfilled. Moreover, it forms a basis in 𝐿⁰_𝑝,𝜔, if the inequalities −1𝑝<𝛼_𝑘<1𝑞,𝑘=0,𝑚 (5.10) hold.

The basicity of system of sines in 𝐿⁰_𝑝,𝜔, when the inequalities (5.10) hold, follows from the basicity of system of exponent {𝑒^𝑖𝑛𝑡}_𝑛∈𝑍 in 𝐿_𝑝,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 function 𝜔(𝑡) defined by expression (2.1). The system of cosines 1∪{cos𝑛𝑡}_𝑛∈𝑁 is minimal (forms a basis) in 𝐿⁰_𝑝,𝜔, if the inequalities (5.10) are fulfilled. It is complete in 𝐿⁰_𝑝,𝜔, if the inequalities (5.9) holds.