We investigated the s-number of the modified convolution operator and obtained the following results
where , , G is a set of all segments Q from , F is a set of all compacts from , is the measure of a set Q.
MSC: 42A45, 44A35.
Keywords:Fourier series multipliers; convolution operator; s-number; Lorentz space; Besov space
Let , . We denote by the space of all compact operators A, acting in the space of all 1-periodic functions square integrable on for s-numbers such that the following quasinorm is finite
if , and
Recall that the sequence (s-numbers of operator A) are numerated eigenvalues of the operator .
We consider the convolution operator
acting in . Given a function , we consider also the modified convolution operator
We say that φ belongs to the space , if for , and
where depends only on , , , .
This means that the linear operator defined by the equality is bounded from to . Let
Given that the eigenvalues of the operator coincide with the Fourier coefficients of the kernel K with respect to the trigonometric system, in the case this problem reduces to the well-known problem of Fourier series multipliers. Let and be the sequence of its Fourier coefficients with respect to the trigonometric system . It is assumed that K is such that , . Let . The problem is to determine conditions on the function φ ensuring the boundedness of the operator .
We obtain sufficient conditions on a multiplier φ ensuring that it belongs to the space . These conditions are expressed in terms of Lorentz and Besov spaces. We also construct examples showing the sharpness of the obtained constants for corresponding embedding theorems.
2 Main results
Let f be a μ measurable function which takes finite values almost everywhere and let
be its distribution function. The function
is a nonincreasing rearrangement of f.
We say that a function f belongs to the Lorentz space if f is measurable and
Theorem 1Let , , , and . If , then and
In the following theorem the cases , are considered. The upper and the lower estimates of the norm ( ) are obtained.
Theorem 2Let , . LetGbe a set of all segmentsQfrom , Fbe a set of all compacts from , then
where is the measure of a setQ.
We shall define the class of generalized monotone functions for which the upper and the lower estimates coincide.
We say that function f is a generalized monotone function, if there exists a constant such that for every the inequality
holds. The class of such functions is denoted by .
Corollary 1Let . If , then if and only if
In case parameters , are both either less or greater than 2, we use the space of smooth functions.
Let , . We denote by the space of all measurable functions f on such that
for , and
for . Here are the Fourier coefficients of the function f by trigonometric system , , and is the integer part of .
This class is called the Nikol’skii-Besov space.
Theorem 3Let , , ,
If , then and
In the case , Karadzhov’s result (see ) follows from Theorem 3:
Now consider the case .
Theorem 4Let , , , , , .
3 Properties of class
To prove the properties of class we need the following lemma. We first define a discrete Lorentz space. is called a discrete Lorentz space whose elements are sequences of numbers with the only limit point 0 such that
where nonincreasing rearrangement of the sequence .
Lemma 1 (See )
Let , . Then
where , .
Let , where , are Banach spaces, be a compatible pair. We define the functional for and by the following formula:
We denote by the space , where is a functional defined on nonnegative functions φ by formula
Let and be the spaces obtained by the method of real interpolation of Banach pairs of spaces , respectively.
Lemma 2 (See )
Let , , , , . IfTis a bilinear operator:
Here , , , , .
Remark Since the s-numbers of convolution operator A coincide with the modules of the Fourier coefficients of the kernel K, the problem of estimating the s-numbers of “transformed” operator can be reduced to the study of the following inequality
and we have to describe the class of those functions φ with Fourier coefficients , for which Inequality (1) holds.
(1) Let , , , . Then
(2) Let , , , then
Proof The proof of the first statement follows from Remark and from the fact that , where is a complex conjugate of the function φ. Now we prove (2).
Let , then by (1) it follows that , and
where . According to Lemma 1, the operator
is bounded. Using (1) we have
Further, applying the theorem on bilinear interpolation (Lemma 2) we find that the operator
is also bounded, i.e., , where
for every . Eliminating θ from this equation, we obtain that
and the condition implies the condition , where .
The proof is complete. □
By (2), in particular, the following proposition follows.
Let , , then
where and are any.
4 Proof of main results
For a given pair we consider the space consisting of all functions f bounded and continuous in the strip
with values in . Moreover, f are analytic in the open strip
and such that the mapping ( ) is a continuous function on the real axis with values in ( ) which tends to 0 for . It is clear that is a vector space. We endow Γ with the norm
The space consists of all elements such that for some function . The norm on is equal to
In order to prove our main result, we need two lemmas in .
Lemma 3 (Bilinear interpolation, the complex method, see )
LetTbe a bilinear operator such that
where , , are the spaces obtained by the method of complex interpolation of Banach pairs of spaces , , respectively.
Lemma 4 (Bilinear interpolation, the real method, see )
LetTbe a bilinear operator such that
with the norms , respectively. Then
where . Moreover,
Proof of Theorem 1 First we prove the inequality:
where are Fourier coefficients of the function φ.
If , Inequality (2) follows by Lemma 1 and Hardy-Littlewood-Paley inequality . Indeed, since , by the Hardy-Littlewood-Paley theorem, we have and the following inequality holds
Taking , we get
Now let . Let , then by Parseval’s equality we get
i.e., . From Lemma 1, using Parseval’s equality we have
Thus, for the bilinear operator we obtain
Applying the method of complex interpolation (Lemma 3), we obtain Inequality (2). Now we shall prove the inequality
Let and be fixed in Inequality (2). Taking , choose parameters such that
Then from Inequality (2) we have
Using Marcinkiewicz-Calderón interpolation theorem (see ), we get
where , i.e., .
Now we apply Lemma 2 with fixed parameters r, s and parameters , , satisfying (4) and the inequality of type (5). We have:
where , , , i.e., , .
Since the parameters , , are arbitrary in Inequality (5), it guarantees the arbitrary of the corresponding parameters in Inequality (4).
Thus, the following inequality holds:
where are Fourier coefficients of the function φ and , . According to Remark, this inequality is equivalent to the statement of Theorem 1. □
Proof of Theorem 2 Let and Q be an arbitrary segment in ,
Applying Theorem 5 from  and using (6), we obtain:
Since the interval Q is arbitrary, we get
where constants c and depend only on parameters p and q.
The proof of obtaining an upper estimate follows from Theorem 1 and the embedding , for .
Indeed, from Theorem 1 it follows
Proof of Corollary 1 Let Q be an arbitrary compact from F.
From the condition of generalized monotonicity of the function φ we have
Taking into account that is arbitrary, we have
Thus, from Theorem 2 we get
Proof of Theorem 3 Let . For a sequence of numbers and a function we consider the mapping T of the form , where is the sequence of Fourier coefficients on the trigonometric system of functions φ. This map is bilinear and from Karadzhov’s theorem  and Remark it follows that it is bounded from to .
Since , the mapping
is also bounded. Thus, for the operator T, the following is true
Then, by Lemma 4 on bilinear interpolation, we get
i.e., the operator T is bounded from to . In the paper  it is shown that , where . Thus, taking into account Theorem 5, we will get
From Minkowski’s inequality and Parseval’s equality we get
Thus, for the operator T, the following is true
Then, by Lemma 3 we get
i.e., T is a bounded mapping from to , where , , . The arbitrary choice of parameters guarantees the arbitrary of the parameters available in the theorem. □
The case follows from the statements proved above and the property .
Proof of Theorem 4 Let . Let us consider the bilinear mapping , where is the sequence of Fourier coefficients of the function φ. The mapping
is bounded according to Theorem 1. Here , , , . The result of Theorem 3, in the case , can be written as
Applying Lemma 3 on the bilinear interpolation to (8) and (9), and taking into account the properties of the embedding of the spaces and , we have:
where parameters r, α, , satisfy the following conditions:
Let in (11) parameter r be fixed. Using Lemma 4 on bilinear interpolation and taking into account that
where , , .
Therefore, with fixed and r we obtain that
where , , .
Using Marcinkiewicz-Calderón interpolation theorem we have
To complete the proof we fix the function φ and the parameters r, s, α and we choose the parameters , , satisfying (11). We use Lemma 2 to get . □
The case , as in the proof of Theorem 3, will follow from .
5 Examples demonstrating the sharpness of the results
Proposition 1Let , , . If , then for any there exists such that , if there exists such that .
Proof Let ε be an arbitrary positive number, and numbers , be such that
Thus, . Since
, and therefore . Since Fourier coefficients of are the sequence it follows that .
To prove the second part of the proposition, we take . Let numbers and be such that
(note that the last inequality does not contradict the previous two). Choosing
we can show that
Hence , and therefore . At the same time taking into account the monotonicity of the sequence and Hardy-Littlewood theorem, we have that . The statement is proved. □
Theorem 6Let , , . Then for any there exist and such that , .
Proof Let and numbers , be such that
Let and , where
It is obvious that , and belongs to .
It is easy to show that
and consequently, . Therefore .
To construct the function , it is sufficient to consider the sequences
. The proof that , is similar to the proof of the first part. □
The authors declare that they have no competing interests.
All authors contributed equally to writing this article. All authors read and approved the final manuscript.
The authors thank the referees for their careful reading and valuable comments on how to improve the article. This paper was partly supported by the grant MTM 2011-27637.
Hirshman, II: On multiplier transformations. Duke Math. J.. 26, 221–242 (1959). Publisher Full Text
Birman, MS, Solomyak, MZ: Quantitative analysis in Sobolev embedding theorems, application to spectral theory. Tenth Mathematical School (Summer School, Kaciveli/Nalchik, 1972), Izdanie Inst. Mat. Akad. Nauk Ukrain. SSR, Kiev (1974)
O’Neil, RO: Convolution operators and spaces. Duke Math. J.. 30, 129–142 (1963). Publisher Full Text
Stein, EM: Interpolation of linear operators. Trans. Am. Math. Soc.. 83, 482–492 (1956). Publisher Full Text
Liflyand, E, Tikhonov, S: Extended solution of Boas’ conjecture on Fourier transforms. C. R. Math. Acad. Sci. Paris. 346(21-22), 1137–1142 (2008). Publisher Full Text