Some remarks on Vainikko integral operators in BV type spaces

In this paper we study Vainikko integral operators which are similar to so-called cordial integral operators and contain the classical Hardy operator, the Schur operator, and the Hilbert transform as special cases. For such operators we obtain norm estimates and equalities, mainly in BV type spaces in the sense of Jordan, Wiener, Riesz, and Waterman. Several examples are also discussed.


Introduction
Domenico Candeloro (called "Mimmo" by his friends) was one of the leading specialists in the theory, methods, and applications of integral operators. He has given many important contributions to this field, with a particular emphasis on exotic measures, non-standard integrals, and multivalued maps. In the list of references at the end we mention only the more recent papers  he wrote, in part with coauthors, in the last 20 years.
In this paper we study a class of integral operators in a much simpler setting, using only single-valued scalar functions and integrals defined by the classical Lebesgue measure on the real line. In spite of their simplicity, we are convinced that our results on mapping properties of such integral operators would have been appreciated by Mimmo. In our discussion we will give particular attention to spaces of functions of bounded variation, a topic that is also very much en vogue in the Analysis School of the University of Perugia which owes so much to Mimmo's scientific activity.

Cordial integral operators
Given a nonnegative L 1 function ϕ : (0, 1) → R, in [27] the author defines an associated Volterra integral operator V ϕ by Such operators are called cordial, the generating function ϕ the core of V ϕ . In [27] and [28] the author gives necessary and sufficient conditions for V ϕ to be bounded in the spaces C, C m and L ∞ . In the recent paper [29] he develops a parallel theory for Lebesgue spaces and proves that V ϕ is bounded in Moreover, he shows that in this case the norm V ϕ L p →L p coincides with this integral. The spectrum and essential spectrum of the operator are also calculated. Cordial integral operators have a series of remarkable properties. For example, one may show that the operator (1) has the continuum of eigenfunctions u r (t) = t r (0 ≤ r < ∞) in the space C, and so it cannot be compact. Moreover, the eigenvalues λ satisfy λI − V ϕ C→C = |λ| + ϕ L 1 ; in particular, V ϕ C→C = ϕ L 1 .

Vainikko integral operators in Lebesgue spaces
In this section we slightly modify the definition (1) to cover a wider range of examples. Given an L 1 function ϕ : (0, ∞) → R, let us call the operator V ϕ defined by Vainikko integral operator in the sequel. So the only difference between cordial and Vainikko operators is that we extend the integration in (2) over the semiaxis (0, ∞). Prominent examples for such operators are the Hardy operator which has the form (2) for the choice ϕ(s) := χ (0,1] (s), the Schur operator which has the form (2) for the choice ϕ(s) := 1/ max {1, s}, and the (strongly singular) Hilbert transform which has the form (2) for the choice ϕ(s) := 1/(1 + s).
Our first result gives a two-sided estimate for the norm of a Vainikko operator in the Lebesgue space L p [0, ∞). To this end, we will use the shortcut where |θ | ≤ 1.
Then V ϕ is bounded and satisfies the estimate Consequently, in case ϕ ≥ 0 or ϕ ≤ 0, the norm V ϕ L p →L p of V ϕ coincides with the right-hand side of (7).
Proof Although the proof may be found in [29] for L p [0, 1], we give here another proof which requires slightly different arguments, since L p (I ) is not included in L q (I ) for q ≤ p in case of an unbounded interval I . To estimate the norm V ϕ L p →L p , we use the fact that the bilinear form establishes a duality between L p and L p for p := p/( p − 1), in the sense that For x ∈ L p and y := V ϕ x we get, by Fubini's theorem . Therefore Hölder's inequality implies Using the change of variables s := tτ we get and applying again Fubini's theorem we conclude that the first integral in (8) is A similar calculation gives for the second integral in (8). Combining these equalities we end up with To illustrate Theorem 3.1 let us go back to our examples mentioned above.
which gives the precise norm H L p →L p of the Hardy operator (3) in L p ; in particular, which gives the precise norm S L p →L p of the Schur operator (4) in L p ; in particular, which gives the precise norm T L p →L p of the Hilbert transform (5) in L p ; in particular, The question arises whether or not it is possible to extend Theorem 3.1 to the extreme cases p = 1 or p = ∞. Here, the norm equalities in Theorem 3.1 illustrate the difference between the operator (3), on the one hand, and the operators (4) and (5), on the other. Since the last expression in (9) tends to 1 as p → ∞, say, one might hope that the Hardy operator also maps L ∞ into itself with H L ∞ →L ∞ = 1; this may be in fact verified by a simple calculation. On the other hand, since the last expression in (10) or (11) tends to ∞ as p → ∞, one might suspect that the Schur operator and the Hilbert transform do not map

Vainikko integral operators in Wiener spaces
In view of the importance of integral operators in spaces of functions of bounded (classical or generalized) variation, it seems reasonable to study the operator (2) in the space BV and its various generalizations. This is the purpose of this section. In contrast to L p -spaces, however, BV -type spaces have a reasonable norm only for functions on compact intervals, but not on the semiaxis [0, ∞). Since our main emphasis is on norm estimates in this paper, in what follows we consider functions x : [0, 1] → R. In this case for norm estimates we will use the fact that

Theorem 4.1
In case ϕ ∈ L p the operator V ϕ is bounded in BV p and satisfies the estimates Proof Fix x ∈ BV p and a partition P : . Then we get, by Jensen's inequality, Combining this estimate with (12) we conclude that V ϕ x BV p ≤ ϕ L p x BV p , which proves the upper estimate in (13).
For the proof of the lower estimate it suffices to take e(t) ≡ 1 and to note that, for all p ≥ 1, e BV p = 1, V ϕ e BV p = 1 0 ϕ(s) ds .
Since BV 1 = BV , we get the estimate as a special case of (13). Consequently, in case ϕ ≥ 0 or ϕ ≤ 0 a.e. on [0, 1] the norm V ϕ BV →BV coincides with the L 1 -norm of ϕ. It is not clear whether or not this is also true if ϕ changes its sign on subsets of positive measure. However, the lower and upper bounds in the estimates (13) may drift apart the more "symmetric" ϕ changes sign and has large absolute values, as the following example suggests.
So we may apply the operator V ψ c to ψ c itself and obtain, by (1),

This implies that
Thus, a comparison of (15) and (16) shows that V ψ c BV p →BV p ≥ c/3. We conclude that the lower bound in (13) is 0, while the upper bound is c which may become arbitrarily large.

Vainikko integral operators in Riesz spaces
Another generalization of the space BV is due to Riesz [26]. Given p ∈ [1, ∞) and a partition P := {t 0 , t 1 , . . . , t m−1 , t m } of [0, 1] as before, we denote by is a Banach space [26]. In spite of their similarity, the spaces BV p and R BV p have quite different properties. First of all, the scale of spaces BV p is increasing in p, while the scale of spaces R BV p is decreasing in p. Moreover, every function in the Riesz space R BV p is continuous for p > 1, but R BV 1 = BV contains of course many discontinuous functions. However, the most interesting property of Riesz spaces is that, for 1 < p < ∞, from x ∈ R BV p it follows that x is absolutely continuous with x ∈ L p and RV ar p (x) = x p L p , and vice versa [1]. This means that Riesz discovered Sobolev spaces, at least in the scalar case, 20 years before Sobolev. This fact allows us to use Theorem 3.1 for finding a condition for V ϕ to map R BV p into itself.

Theorem 5.1 In case ϕ ∈ L 1 the operator V ϕ is bounded in R BV p and satisfies the estimates
Consequently, in case ϕ ≥ 0 or ϕ ≤ 0, the norm V ϕ R BV p →R BV p of V ϕ coincides with the right-hand side of (17).
Proof The case p = 1 is covered by (14), so let 1 < p < ∞. In this case, x is absolutely continuous and x ∈ L p . Moreover, using the second integral in (2) we see that where we have used the notation (6). From ϕ ∈ L 1 it follows that also ϕ 1−1/ p ∈ L 1 ; so Vainikko's result implies that V ϕ maps R BV p into itself and, by (18), So together with (12) The lower estimate in (17) is proved exactly as before.
Observe that R BV ∞ = Li p, the linear space of all Lipschitz continuous maps with norm Consequently, in case ϕ ≥ 0 or ϕ ≤ 0, the norm V ϕ Li p→Li p of V ϕ coincides with ϕ L 1 .
Observe that this result for the space Li p may be easily proved directly. In fact, from Let us illustrate our results by means of our "test animals", the Hardy operator (3), the Schur operator (4), and the Hilbert transform (5). We close this section with another operator which depends on a real parameter α and is, in contrast to the Hilbert transform, weakly singular.

Example 5.1
For α > 0, consider the Liouville operator This operator has the form (2)

Vainikko integral operators in Waterman spaces
Finally, let us recall yet another BV -type space which was introduced by Waterman [30] and has very interesting applications.
Let Λ := (λ k ) k be a positive decreasing sequence satisfying lim k→∞ λ k = 0 and is a Banach space [30]. A typical example is of course Λ q := (k −q ) k for 0 < q ≤ 1; this has important applications to Fourier series. Thus, in [31] it was shown that, for f ∈ Λ q BV , the Fourier series of f is everywhere (C, β)-bounded for β = q − 1, and (C, α)-summable for α > q − 1. Moreover, these estimates for α and β are sharp. The starting point for the study of Waterman spaces was the choice q = 1; in this case the elements of the space Λ 1 BV =: H BV are called functions of bounded harmonic variation. Theorem 6.1 In case ϕ ∈ L 1 the operator V ϕ is bounded in ΛBV and satisfies the estimates Consequently, in case ϕ ≥ 0 or ϕ ≤ 0, the norm V ϕ Li p→Li p of V ϕ coincides with ϕ L 1 . Adding this estimate to (12) we see that V ϕ x ΛBV ≤ ϕ L 1 x ΛBV , which proves the upper estimate in (20). For the proof of the lower estimate we take the same function e(t) ≡ 1 as in the proof of Theorem 4.1.
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