Convolution on spaces of locally summable functions

In this work we prove the existence of convolution on Marcinkiewicz spaces M(R), 1 ≤ p < ∞ , and, using pointwise approximate identities, we extend the classical definition of Hilbert transform to such spaces.


Introduction
We are interested in extending classical results in the theory of singular integrals to spaces of functions that are only locally summable.The natural environment should be the space L p loc (R), but the most typical singular integral, the Hilbert transform, does not converge therein.Therefore we need some decay condition.An interesting way to introduce mild decay conditions is based upon boundedness of integral averages over large intervals in R.This leads to Marcinkiewicz spaces M p (R) defined in Section 2. These spaces have been studied also in [3] and in [1].
In [1], the author proves that any regular bounded Borel measure gives rise to a bounded convolution operator on M p , for p ≥ 1.
In [3], K.S. Lau studies the convolution operators on M p , p ≥ 1, and on its closed subspace of regular functions, i.e. functions such that lim T →±∞ T +a T |f | p = 0 for a > 0.
Lau proves that, if μ is a bounded regular Borel measure on R, such that R |x| d|μ| < +∞, the convolution operator φ μ (defined by φ μ (f ) = μ * f ), restricted to the subspace of regular functions of M p satisfies lim where χ T is the characteristic function of [−T, T ].
In Section 2, we give a direct proof of the existence of convolution on M p , for p ≥ 1 (see Proposition 2.1), even though this also follows from the result in [1].
In Section 3, we provide examples to show that pointwise approximate identities ϕ α do not, in general, converge pointwise: convergence does not even hold at Lebesgue points for f ∈ M p (R) if the least upper monotone bounding function ψ of ϕ is not in L 1 (R).
In Section 4, we prove the existence of pointwise approximate identities on functions f ∈ M p (R) under the additional hypothesis that the least upper monotone bound of ϕ belongs to L 1 (R).We devote attention to pointwise convergence only: presumably, approximate identities that converge pointwise converge also in the norm of M p (R), but we shall return to this subject in a forthcoming paper.
In Section 5, we make use of the convergence result to show the convergence of the Hilbert transform on M p (R).

Convolution on Marcinkiewicz spaces
M p (R), 1 ≤ p < +∞, is a vector space and It is possible to prove that M p (R) , 1 ≤ p < +∞, is complete with respect to such seminorm (see [6]).
In the following proposition we prove that the convolution of a function in M p (R) with a function in L 1 (R) is finite almost everywhere.
For sufficiently large T , given ε > 0, there exists T such that

An example where convergence fails
Let f be a function in the Marcinkiewicz space M 1 (R), or more generally , and consider the approximate identity Since f is only locally integrable, convergence of the approximate identity is not granted: is it true that, for almost every Lebesgue point x 0 of f (see Definition 4.6), We show now that this is not always the case by providing a counterexample.Without loss of generality, we may assume that x 0 = 0 is a Lebesgue point: from now on we shall restrict attention to convergence at the point 0. Let χ n be the characteristic function of the interval [2 n , 2 n + 1], and let α n > 0 be such that

Consider now the following function
In what follows, it will be of interest to relate the function ϕ of this counterexample to its least upper bound In particular, we observe that, with ϕ chosen as in Proposition 3.1, if we assume {α n } non-increasing, the function ψ does not satisfy the following properties: (1) is satisfied if and only if the series +∞ n=0 2 n α n is convergent, and condition (2) if and only if its general term 2 n α n vanishes at infinity.More generally, notice that, by Proposition 3.1, if condition (1) is not satisfied, convergence fails in (1).Observe that neither condition is satisfied if, for instance, we choose α n = 2 −n , n = 1, 2, . . ., both are satisfied if α n = C2 −n /n 2 (here the normalization constant C is determined by the condition ∞ 0 α n = 1 ), and only the second one is satisfied if α n = C 2 −n /n.Therefore the second condition cannot imply convergence in (1), and we are led to investigate whether the first condition implies convergence.
To prove Proposition 4.1 we need several lemmas.
. By Jensen's inequality we have that Hence We shall need the following easy facts: By the integral mean theorem there exists ξ r such that r < ξ r < 2r and and since J r → 0 as r → 0 + and r → +∞, it follows that the same holds for rψ(r).Furthermore and so rψ(r) is bounded in R + (and hence in R).
At this point it is useful to remind the following Definition 4.6.
exists and is equal to zero is called a Lebesgue point for g .
and hence Integrating by parts, the right-handside becomes Since ψ ε is even and non-negative, we get By Lemma 4.5 we have that is uniformly bounded and hence where c is a constant depending on δ (same δ as in Lemma 4.8).
Proof.It is sufficient to take |t| > δ, since we need the inequality only for such t in order to prove Proposition 4.1.Nevertheless the statement of the Lemma holds also for |t| ≤ δ , by Lebesgue property.
By Lemma 4.2, f ∈ M 1 (R) and by Lemma 4.3 translation is continuous with respect to the M 1 − norm.Hence from the definition of M 1 − norm, it follows that there exists a constant c(δ) depending on δ such that and therefore the statement holds.
Proof of Proposition 4.1.We must show that We may take x o a Lebesgue point for f , i.e.
Let us suppose that The reader may easily verify that the same arguments hold in the case For simplicity we set Let us estimate the third integral first.
Integrating by parts and reminding that we get that since ψ ε is even.By Lemma 4.5, δψ ε (δ) is uniformly bounded and it tends to zero as ε → 0 + .By Lemma 4.8 Let us estimate I 2 now.
We need to show that for any ε if ε is sufficiently small.We write, as before Integrating by parts we get that In the first term the limit as t → +∞ is equal to zero by Lemma 4.9 and Lemma 4.5. Hence since x o is a Lebesgue point for f .Therefore for some C > 0 , by Lemma 4.5.Let us consider now the second term in (3), i.e.
By Lemma 4.9, Since ε → 0 + , we may choose ε small enough to get, for fixed ε (and hence fixed δ ), We get hence that I 2 < 2C ε for sufficiently small ε .
The estimate of I 1 is analogous to that of I 2 .Hence for all ε > 0 there exists M such that if ε is sufficiently small where M depends on δ (and hence on ε ).M and δ depend on x o , but they can be chosen bounded if x o lies in a set whose complement has arbitrary small measure.

The Hilbert Transform
Our aim is to introduce a Hilbert transform operator on Marcinkiewicz spaces.Such an operator should be defined, as in [7], by If f ∈ L 1 (R) the convergence of the integral and the norm boundedness of H are related to the theory of principal value distribution and singular integrals.Here, however, there is a further problem, because, in general, f is only locally L 1 (or L p ), and so the integral might diverge as t becomes large.
In this section we prove that H is well defined on a subset of M p (R).For this goal, we must approximate the integral in (5) by finite truncations Proposition 5.1 (cf.[4]).Let f ∈ M p (R), 1 ≤ p < +∞ be such that for all x ∈ R t 0 f x + t d t ≤ C |t| α for a fixed α ∈ (0, 1).Then the limit exists uniformly on compact subsets of C + .
Proof.Let us set Let us give an estimate of the integral We have that Since |x| ≤ T, y ≥ y o > 0 one may choose N , N such that the last quantity becomes less than ε 2 .The integral I 2 may be estimated in an analogous way.
Proposition 5.3.In the same hypotheses of Proposition 5.1 the limit exists for all x ∈ R and for all y > 0 . Proof.
Let us prove that lim
α t 2 dt and we may conclude as usual.Hence, by Proposition 5.3, we get the existence of the limitH y f (x) = lim f (x + t)Φ y (t) dt.