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

(1)

Volume 2011, Article ID 717501,13pages doi:10.1155/2011/717501

Research Article

On Singular Integrals with Cauchy Kernel on

Weight Subspaces: The Basicity Property of Sines

and Cosines Systems in Weight Spaces

Tofig Isa Najafov

1

_{and Saeed Farahani}

2

1_{Nakhchivan State University, University Campus, AZ 7000 Nakhchivan, Azerbaijan}

2_{Institute of Mathematics and Mechanics, Azerbaijan National Academy of Sciences,}

AZ 1141 Baku, Azerbaijan

Correspondence should be addressed to Tofig Isa Najafov,s [email protected]

Received 3 February 2011; Accepted 28 March 2011

Academic Editor: Shyam Kalla

Copyrightq2011 T. I. Najafov and S. Farahani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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:

Sft 1

2πi

_π

−π

fsds

1−e−is−t, 1.1

wheref ∈Lp,ρ−π, π, 1< p <∞, is an appropriate density,ρtis a weight function of the

form

ρt

r

k1

|t−tk|pβk, 1.2

(2)

UnderLp,ρwe understand a Lebesgue weight space with the norm

_f

p,ρ≡

π

−π

_f_tp

ρtdt

1/p

. 1.3

Bounded action of the operator S 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 operatorSis bounded inLp,ρif and only if the inequalities

−1

p < βk< 1

q, k1, r, 1

p

1

q 1, 1.4

are fulfilled.

Concerning this fact a one can see the monograph1and papers2–4. Inequalities

1.4are Muckenhoupt condition with respect to the weight functionρtwith degreespβk.

It is known that the classic system of exponents{eint_}

n∈ZZare integersforms a basis inLp,ρ

if and only if inequalities1.4holdsee, 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

LetL0

p,ω ≡Lp,ω0, π,ωt—a weight function of the form

ωt≡

r

k0

|t−τk|pαk, 2.1

where 0τ0< τ1<· · ·< τr π,{αk}r0⊂R.

Denote the space of evenoddfunctions inLp,ρbyLp,ρL−p,ρ, that is,

L±_p,ρ ≡f ∈Lp,ρ:f−t ±ft, t∈−π, π

. 2.2

We’ll need the following identity:

1 1−eiθ−ϕ −

1 1−eiθϕ

i 2

sinϕ

(3)

Indeed, we have

1 1−eiθ−ϕ −

1 1−eiθϕ

1−e−iθ−ϕ

1−eiθ−ϕ₁₋_e−iθ−ϕ−

1−e−iθϕ

1−eiθϕ₁₋_e−iθϕ

1−cos

θ−ϕisinθ−ϕ

1−cosθ−ϕ2sin2θ−ϕ−

1−cosθϕisinθϕ

1−cosθϕ2sin2θϕ

i

sinθ−ϕ 2−2 cosθ−ϕ −

sinθϕ 2−2 cosθϕ

i

cosθ−ϕ/2 2 sinθ−ϕ/2−

cosθϕ/2 2 sinθϕ/2

i

2

cosθ−ϕ/2sinθϕ/2−cosθϕ/2sinθ−ϕ/2 sinθ−ϕ/2sinθϕ/2

i

2

sinϕ

sinθ−ϕ/2sinθϕ/2.

2.4

For compactness of the notation, we assumeex≡1/1−eix_{. Thus,}

eθ−ϕ−eθϕ i 2

sinϕ

sinθ−ϕ/2sinθϕ/2. 2.5

From this identity, we can easily get the following relations:

eθ−ϕ−eθϕ−i 2

cosϕ/2 cosθ/2

1

sinϕ−θ/2

1 sinϕθ/2

,

eθ−ϕ−eθϕ−i 2

sinϕ/2 sinθ/2

1

sinϕ−θ/2−

1 sinϕθ/2

.

2.6

The authors of the papers4–7used these relations earlier while establishing the basicity criterion of the system of sines and cosines with linear phases inLp. Thus, the following is

valid.

Lemma 2.1. The following identities are true:

1 1−eiθ−ϕ −

1

1−eiθϕ −

i 2

cosϕ/2 cosθ/2

1

sinϕ−θ/2

1 sinϕθ/2

,

1 1−eiθ−ϕ −

1

1−eiθϕ −

i 2

sinϕ/2 sinθ/2

1

sinϕ−θ/2−

1 sinϕθ/2

.

(4)

In the similar way, we obtain

1 1−eiθ−ϕ

1 1−eiθϕ

e−iθ−ϕ/2

e−iθ−ϕ/2₋_eiθ−ϕ/2

e−iθϕ/2

e−iθϕ/2₋_eiθϕ/2

−1

2i

e−iθ−ϕ/2

sinθ−ϕ/2− 1 2i

e−iθϕ/2

sinθϕ/2

−1

2i

cosθ−ϕ/2−isinθ−ϕ/2

sinθ−ϕ/2

cosθϕ/2−isinθϕ/2 sinθϕ/2

−1

2i

cosθ−ϕ/2 sinθ−ϕ/2

cosθϕ/2 sinθϕ/2

1

1− 1 2i

sinθ

sinθ−ϕ/2sinθϕ/2.

2.8

Further, we must take into account the following relation:

1

sinθ−ϕ/2sinθϕ/2

1

2 sinθ/2cosϕ/2

1

sinθ−ϕ/2

1 sinθϕ/2

,

1

sinθ−ϕ/2sinθϕ/2

1

2 cosθ/2sinϕ/2

1

sinθ−ϕ/2 −

1 sinθϕ/2

.

2.9

As a result, we have

eθ−ϕeθϕ1− 1 2i

sinθ/2 sinϕ/2

1

sinθ−ϕ/2 −

1 sinθϕ/2

, 2.10

eθ−ϕeθϕ1− 1 2i

cosθ/2 cosϕ/2

1

sinθ−ϕ/2

1 sinθϕ/2

. 2.11

Assume

Kϕθ e

θ−ϕeθϕ−1. 2.12

Thus,

Kϕ−θ −

1 2i

cosθ/2 cosϕ/2

1

−sinθϕ/2−

1 sinθ−ϕ/2

1

2i

cosθ/2 cosϕ/2

×

1

sinθϕ/2

1 sinθ−ϕ/2

−Kϕθ.

2.13

(5)

Lemma 2.2. The following identities are true:

1 1−eiθ−ϕ

1

1−eiθϕ 1−

1 2i

sinθ/2 sinϕ/2

1

sinθ−ϕ/2 −

1 sinθϕ/2

,

1 1−eiθ−ϕ

1

1−eiθϕ 1−

1 2i

cosθ/2 cosϕ/2

1

sinθ−ϕ/2−

1 sinθϕ/2

.

2.14

3. Boundedness of Singular Operators on Subspace of Even Functions

Letf∈L−_p,ρ. We have

Sft 1

2πi

_π

−π

fsds 1−e−is−t

1 2πi

π

0

fsds 1−eit−s

₀

−π

fsds 1−eit−s

1

2πi

_π

0

1 1−eit−s −

1 1−eits

fsds− 1 4π

_π

0

K1t, sfsds

− 1

4π

_π

0

K2t, sfsds,

3.1

where the kernelsKit, s,i1,2, are determined by the expressions

K1t, s coss/2

cost/2

1

sins−t/2

1 sinst/2

,

K2t, s

sins/2 sint/2

1

sins−t/2−

1 sinst/2

.

3.2

Continue the weightωtto the interval−π,0by parity and denote byμ:

μt

⎧ ⎨ ⎩

ωt, 0≤t≤π,

ω−t, −π≤t <0.

3.3

It is obvious that{±τk}r0 ⊂ −π, πare the degeneration points ofμt. Thus,μt ω|t|,

t∈−π, π, that is,

μt

r

k0

||t| −τk|pαk. 3.4

Accept the denotation

f∼g ast−→a, if∃δ >0 :δ≤f g

(6)

It is easy to see that the following holds

||t| −τk| ∼ |t−τk||tτk|, t∈−π, π, fork >1. 3.6

As a result, forμ, we get the representation

μt∼ρ0t≡ |t|pα0

r

k1

|t−τk|pαk|tτk|pαk, t∈−π, π. 3.7

It is obvious that the singular integral S boundedly acts in Lp,μ−π, π if and only if it

boundedly acts inLp,ρ0−π, π. Statement1is valid also in the case if the Cauchy kernel is

replaced by the Hilbert kernel1/sinϕ−t/2. So, assume that the inequalities −1/p < αk<1/q,k0, r, are fulfilled. Then, from Statement1, we directly get thatSboundedly acts

fromL−_p,ρ₀toLp,ρ0and so fromL−p,μtoLp,μ. Assume

k±s, t sins/2 sint/2

1

sins±t/2, s, t∈−π, π×−π, π, 3.8

and consider the integral operatorI±:

I±f

_π

−π

k±s, tfsds, 3.9

with the kernelk±s, t. We havef ∈L−_p,μ−π, π:

Sf_p,μ 1 4π

I−f−If

p,μ≤

1 4π

I−f

p,μ

If

p,μ , 3.10

where

ft

⎧ ⎨ ⎩

ft, t∈0, π,

0, t∈−π,0.

3.11

Consequently,

I−fp

p,μ

π

−π

I−ftpμtdt∼

π

−π

I−fpρ0tdt

π

−π

π

−π

fssins/2ds sins−t/2

p

ρ0t

sint

2

−pdt.

3.12

It is obvious that

ρ0t

sin t

2

(7)

whereθt |s|pα0−1r

k1|t−τk|pαk|tτk|pαk. So, if the inequalities

−1

p < α0−1< 1

q, −

1 p < αk<

1

q, k1, r, 3.14

hold, then from Statement1we obtain

I−fp

Lp,μ

≤c

π

−π

π

−π

fssins/2 sins−t/2ds

p

θtdt≤cfssinsθs 2

p

Lp,θ

≤c

π

0

_f_s_sp

θsdscfp_L0

p,ω,

3.15

wherecis a constant independent fromf diﬀerent in diﬀerent places. As a result, we get that if the inequalities3.14hold, the operatorI−boundedly acts fromL0

p,ωtoLp,μ. The same

conclusion is true for the operator I as well. As a result, we get that while fulfilling the conditions

α0∈

−1

p, 1 q

1 q,1

1

q , αk∈

−1

p, 1

q , k1, r, 3.16

the operator S boundedly acts from L−_p,μ to Lp,μ. On the other hand, it is easily seen that

Sf −t Sft. As a result, we get that the operatorSboundedly acts fromL−_p,μtoL_p,μ. Now, consider the case whenα0 1/qandαk,k 1, r, satisfy conditions3.16. Let

p±εp±εandq±εbe a number conjugated top±ε. It is obvious that the relations

α0∈

− 1

p±ε

, 1 q±ε

1 q±ε

,1 1 q±ε

, αk∈

− 1

p±ε

, 1 q±ε

, k1, r, 3.17

are fulfilled for suﬃciently smallε >0. Then, from the previous reasonings we get that the operatorSboundedly acts from L−_p_±ε_,μ toL_p_±ε_,μ. As a result, it follows from the Riesz-Torin theoremsee, e.g.,7, page 144that the operatorSboundedly acts fromL−_p,μtoL_p,μ. We get the following.

Statement 2. Let the inequalities

−1

p < α0<2− 1

p, −

1 p < αk<

1

q, k1, r, 3.18

be fulfilled. Then, the operatorSboundedly acts fromL−_p,μtoL_p,μ.

Now, consider the representation of the operatorSby the kernelK1t, s. Having paid

attention to the expression

coss/2 cost/2

sinπ−s/2

(8)

similar to the previous case we establish that the boundedness of the operatorSholds also in the case when the change interval of the exponentαr 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

−1

p < α0<2− 1

p, −

1

p < αm<2− 1

p, −

1 p < αk<

1

q, k1, r−1, 3.20

are fulfilled. Then the singular operatorS:

Sf

_π

−π

kt, sfsds, 3.21

with Cauchy kernelkt, s 1/1−e−is−t_{, boundedly acts from}_L−

p,μtoLp,μ, whereμt ω|t|,

t∈−π, π.

4. Boundedness of Singular Operators on Subspace of Odd Functions

Let the weight function ωt be defined by expression2.1 and assumeμt ≡ ω|t|,t ∈

−π, π. Denote byKs;tthe following Cauchy-type kernel

Ks;t 1

1−eit−s −

1

2, s;t∈−π, π×−π, π. 4.1

Appropriate integral operator denote byK:

Kft 1

2π

π

−π

Ks;tfsds. 4.2

Let the following inequalities

−1

p < αk< 1

q, k0, r, 4.3

be fulfilled. We have

_π

−π

ftdt

_π

−π

ftμ1/p_μ−1/p_dt_≤_f p,μ

π

−π

μ−q/p_dt

1/q

. 4.4

From4.3it follows that−qαk > −1,k 0, r, and as a resultμ−q/p ∈L1−π, π. As a

(9)

inequalities4.3hold. In particular, it follows that the operatorKboundedly acts fromL_p,μ toLp,μ, if the inequalities4.3are fulfilled, that is,

_K_f

Lp,μ ≤cfLp,μ, ∀f ∈L

p,μ, 4.5

wherecis a constant independent fromf. On the other hand forf∈L_p,μwe have

Kft 1

2π

π

0

fsKs;t K−s;t−1ds. 4.6

Pay an attention to the relation2.13, we obtain thatKf−t −Kft,t∈−π, π. Then from4.5yields

_Kf_L0

p,ω ≤cfL0p,ω, ∀f∈L

0

p,ω. 4.7

Now, let

−1− 1

p < α0 <− 1

p, −

1 p < αk<

1

q, k1, r 4.8

be fulfilled. Assume

K±s;t −1 2i

sint/2

sins/2sint∓s/2, s;t∈0, π×0, π. 4.9

Denote the integral operator with kernelK±s;tbyS±, that is,

S±ft 1 2π

_π

0

K±s;tfsds, t∈0, π. 4.10

Taking into account the relation2.10, forf∈L_p,μwe have

Kft 1

2π

_π

0

fsKs;t K−s;t−1ds

1

2π

π

0

Ks;t−K−s;tfsdsS−S−f,

4.11

that is,

Kft S−S−ft, t∈−π, π. 4.12

Take into account the nonparityKfton−π, πwe obtain

2Kfp_L0

p,ω

_K_fp Lp,μ ≤

Sf_L

p,μS

−_f

Lp,μ

p

(10)

Let

ft

⎧ ⎨ ⎩

ft, t∈0, π,

0, t∈−π,0.

4.14

Consider the operatorS. We have

Sft sint/2 2π

π

0

fssin−1s/2 sint−s/2 ds

sint/2 2π

π

−π

fssin−1s/2

sint−s/2 ds. 4.15

Thus

Sfp_L

p,μ c

_π

−π

Sftpμtdtc

_π

−π

_π

−π

p

sin t 2

pμtdt. 4.16

Further, we must take into account the expression sint/2∼t,t∈−π, π. As a result, from the previous relation we have

Sfp_L

p,μ ≤c

_π

−π

_π

−π

p

μtdt, 4.17

where

μt |t|pα01

r

k1

||t| −tk|pαk. 4.18

It is clear that for weight function μt Muckenhoupt condition is fulfilled and applying Statement1to the expression4.17we obtain

Sfp_L

p,μ ≤c

_f_s_sin−1s

2

p

Lp,μ

c

_π

−π

_f_sp

sin−1s 2

pμsds

≤c

_π

0

fsp|s|−pμsdsc

_π

0

fspωsdscfp_L

p,μ.

4.19

In the similar way we establish the validity of the inequality

_S−_f

Lp,μ ≤cfLp,μ, ∀f ∈L

p,μ. 4.20

If the inequalities4.8hold, as a result, we have

_Kf_L

p,μ ≤cfLp,μ, ∀f ∈L

(11)

Consider the case

α0−

1

p, −

1 p < αk<

1

q, k1, r. 4.22

Take suﬃciently smallε >0 and determinep±_ε p±ε. Acting similarly to the caseL−_p,μpar.3 and accept the Riesz-Torin theorem we obtain boundedly acting of the operatorKfromL_p,μ toL−_p,μsinceKft∈L−_p,μ. Thus, if the following inequalities

−1−1

p < α0 < 1

q, −

1 p < αk<

1

q, k1, r, 4.23

are fulfilled, then the operatorKboundedly acts fromL_p,μtoL−_p,μ.

Using the identity2.11in the similar way we establish that the same conclusion with respect to the operatorKis true in the case when the change interval of the exponentαr 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 functionωbe defined by the expression2.1andμt ≡ ω|t|,t ∈

−π, π. Assume that the inequalities

−1− 1

p < α0< 1

q, −1−

1 p < αr <

1

q, −

1 p < αk<

1

q, k1, r−1, 4.24

are fulfilled. Then the singular operatorK:

Kf

π

−π

Ks;tfsds, 4.25

with Cauchy-type kernelKs;t 1/1−e−is−t₋_{1/2, boundedly acts from}_L

p,μtoL−p,μ.

5. Completeness, Minimality, and Basicity of the

System of Sines in Weight Space

Consider the system of sines{sinnt}_n_∈_N. Let conditions3.20be fulfilled. It is easy to see that then the system{sinnt}_n_∈_Nis minimal inL0

p,ω. The system{2/πω−1tsinnt}n∈N, 1/p

1/q1, is a biorthogonal system to it. Indeed, it is obvious thatL0

q,ωis a space conjugated to

L0p,ω, and an arbitrary continuous functionallgonL0p,ω, generated byg∈L0q,ω, realized by the

formula

lg

f

π

0

fgωdt, ∀f∈L0_p,ω, 5.1

(12)

Takegt ω−1_t_sin_nt,_n_∈_{N. We have}

ω−1tsinntq

L0 q,ω

_π

0

ω−1tsinntqωtdt

_π

0

ω1−qt|sinnt|qdt. 5.2

Since sinnt∼tast → 0 and sinnt∼π−tast → πfor every fixedn∈N, then from relation5.2follows that{ω−1_sin_nt_}

n∈N⊂L0q,ω, if the inequalities

α0<2−

1

p, αm<2− 1

p, αk< 1

q, k1, r−1, 5.3

are fulfilled.

Takegt 2/πω−1_t_sin_nt_{and denote by}_ϑ

ngenerated by its functional, that is,

ϑn

f 2

π

0

ftsinnt dt, ∀f∈L0_p,ω. 5.4

It is clear thatϑnsinkt δnk,∀n, k∈N, whereδnkis a Kronecker’s symbol. Consider

sinntp_L0

p,ω

π

0

|sinnt|pωtdt. 5.5

Thus,{sinnt}_n_∈_N ⊂Lp,ω, ifα0 >−1/p−1,αr >−1/p−1,αk >−1/p,k 0, r−1. As a result,

we obtain that if the inequalities

−1− 1

p < α0<2− 1

p, −1−

1

p < αr <2− 1

p, −

1 p < αk<

1

q, k0, r−1, 5.6

hold, then the system{sinnt}_n_∈_Nis minimal inL0

p,ω.

Now consider the completeness of the system{sinnt}_n_∈_N in L0

p,ω. Suppose that for

someg∈L0

q,ω,

_π

0

sinnt gω dt0, ∀n∈N, 5.7

holds. We have

_π

0

gωdt

_π

0

gω1/q_ω1/p_dt_≤_g L0

q,ω

π

0

ω dt

1/p

. 5.8

It is easy to see that if the inequalities

αk>−

1

(13)

are fulfilled, thenω∈L1. Then from the previous relation we getgω ∈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 segment0, π, then from5.7it follows thatgt 0 a.e. on

0, π. Consequently, the system{sinnt}_n_∈_Nis complete inL0

p,ω. So, the following statement

is true.

Statement 3. Let the weight functionωtbe defined by expression2.1. The system of sines

{sinnt}_n_∈_Nis minimal inL0

p,ω, if the inequalities5.6are fulfilled. It is complete inL0p,ω, if the

inequalities5.9are fulfilled. Moreover, it forms a basis inL0_p,ω, if the inequalities

−1

p < αk< 1

q, k0, m 5.10

hold.

The basicity of system of sines inL0

p,ω, when the inequalities5.10hold, follows from

the basicity of system of exponent{eint_}

n∈Z inLp,μ, whereμt ω|t|,t ∈ −π, π. The

basicity of these systems earlier considered in papers3,4,8,9. In the similar way we prove the following statement.

Statement 4. Let the weight functionωtdefined by expression2.1. The system of cosines

1∪ {cosnt}_n_∈_N is minimalforms a basisinL0

p,ω, if the inequalities5.10are fulfilled. It is

complete inL0

p,ω, if the inequalities5.9holds.

References

1 I. I. Danilyuk, Irregular Boundary Value Problems on a Plane, Nauka, Moscow, Russia, 1975.

2 B. S. Pavlov, “Basicity of an exponential system and Muckenhoupt’s condition,” Doklady Akademii Nauk

SSSR, vol. 247, no. 1, pp. 37–40, 1979Russian.

3 K. S. Kazaryan and P. I. Lizorkin, “Multipliers, bases and unconditional bases of the weighted spaces B and SB,” Akademiya Nauk SSSR. Trudy Matematicheskogo Instituta imeni V. A. Steklova, vol. 187, pp. 111–130, 1989.

4 S. S. Pukhov and A. M. Sedletskiy, “Bases of exponents, sines and cosines in weight spaces on finite interval,” Proceedings of the Russian Academy of Sciences, vol. 425, no. 4, pp. 452–455, 2009.

5 E. I. Moiseev, “The basis property for systems of sines and cosines,” Doklady Akademii Nauk SSSR, vol. 275, no. 4, pp. 794–798, 1984.

6 E. I. Moiseev, “On the basis property of a system of sines,” Diﬀerentsial’nie Uravnenia, vol. 23, no. 1, pp. 177–179, 1987.

7 G. G. Devdariani, “The basis property of a system of functions,” Diﬀerentsial’nie Uravnenia, vol. 22, no.

1, pp. 170–171, 1986.

8 E. I. Moiseev, “On basicity of system of sines and cosines in weight space,” Diﬀerentsial’nie Uravnenia, vol. 34, no. 1, pp. 40–44, 1998.

(14)

Submit your manuscripts at

http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Journal of

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations International Journal of

Volume 2014

Applied MathematicsJournal of

Mathematical PhysicsAdvances in

Complex Analysis

Journal of Hindawi Publishing Corporation

Optimization

Journal of

Combinatorics

International Journal of Hindawi Publishing Corporation

Journal of

Function Spaces

Abstract and Applied Analysis

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation http://www.hindawi.com Volume 2014

The Scientific

World Journal

Discrete Dynamics in Nature and Society

Discrete Mathematics

Journal of