A necessary condition on a singular kernel for the continuity of an integral operator in H¨older spaces

: We prove that a condition of boundedness of the maximal function of a singular integral operator, that is known to be suﬃcient for the continuity of a corresponding integral operator in H¨older spaces, is actually also necessary in case the action of the integral operator does not decrease the regularity of a function. We do so in the frame of metric measured spaces with a measure satisfying certain growth conditions that include nondoubling measures. Then we present an application to the case of an integral operator deﬁned on a compact diﬀerentiable manifold.


Introduction
This paper concerns the analysis of a singular integral operator in Hölder spaces on subsets of measured metric spaces with a measure that may be non-doubling, in the spirit of a research project that has been initiated by García-Cuerva and Gatto, [7], [8], Gatto [9], [10] and that is here being developed with the aim of specific applications to potential theory.Let (M, d) be a metric space and let X, Y be subsets of M .
In case X = Y , we just say that Y is upper υ Y -Ahlfors regular and this is the assumption that has been considered by García-Cuerva and Gatto [7], [8], Gatto [9], [10] in case X = Y = M .See also Edmunds, Kokilashvili and Meskhi [5,Chap. 6] in the frame of Lebsgue spaces.
Then we consider a stronger version of the upper Ahlfors regularity as in [12].Namely, we assume that Y is strongly upper υ Y -Ahlfors regular with respect to X, i.e., that there exist r X,Y,υY ∈]0, +∞] , c X,Y,υY ∈]0, +∞[ such that ν((B(x, r 2 ) \ B(x, r 1 )) ∩ Y ) ≤ c X,Y,υY (r υY 2 − r υY 1 ) for all x ∈ X and r 1 , r 2 ∈ [0, r X,Y,υY [ with r 1 < r 2 , ( where we understand that B(x, 0) ≡ ∅ (in case X = Y , we just say that Y is strongly upper υ Y -Ahlfors regular).We plan to consider off-diagonal kernels K from (X × Y ) \ D X×Y to C, where denotes the diagonal set of X × Y and we introduce the following class of 'potential type' kernels (see also [3]).
For s 2 = s 1 + s 3 one has the so-called class of standard kernels that is the case in which García-Cuerva and Gatto [7], [8], Gatto [10] have proved T 1 Theorems for the integral operators with kernel K in case of weakly singular, singular and hyper-singular integral operators with X = Y .We plan to analyze the integral operator where Z belongs to a class K s1,s2,s3 (X × Y ) as in Definition 1.5 and g is a C-valued function in X ∪ Y .We exploit the operator in (1.6) in the analysis of the double layer potential and we note that operators as in (1.6) appear in the applications (cf.e.g., [3, §8]).As in (cf.[12,Prop. 6.3 (ii)]), we can estimate the Hölder quotient of Q[Z, g, 1] by introducing a more restrictive class of kernels.Namely the following.
X, then the bilinear map from and Y is strongly upper υ Y -Ahlfors regular with respect to X, then the bilinear map from Y is upper υ Y -Ahlfors regular with respect to X, then the bilinear map from Here C 0,β (X ∪ Y ) denotes the semi-normed space of β-Hölder continuous functions on X ∪ Y , C 0,γ b (X) denotes the normed space of bounded γ-Hölder continuous functions on X for all γ ∈]0, 1] and C 0,max{r β ,ωs 3 (r)} b (X) denotes the normed space of bounded generalized Hölder continuous functions on X with modulus of continuity max{r β , ω s3 (r)} (see notation around (2.4)-(2.5)below).Now the assumptions of Proposition 1.8 require that the kernel Z belongs to K ♯ υY ,s2,s3 (X × Y ) and the membership in K ♯ υY ,s2,s3 (X × Y ) requires an estimation of the maximal function of the kernel Z, i.e., to show that Z(x, y) dν(y) < +∞ (1.9) and here we wonder whether such assumption is actually necessary.Indeed, under the only assumption that the kernel Z belongs to K υY ,s2,s3 (X × Y ) and that g is β-Hölder continuous, the integral that defines Q[Z, g, 1] is weakly singular.Thus the main purpose of the present paper is to establish what conditions on the maximal function of Z as in (1.9) are actually necessary for the continuity of Q[Z, •, 1] when Z ∈ K υY ,s2,s3 (X × Y ).Namely, we prove the validity of (2.9)- (2.11), that are conditions on the growth of the left hand side of (1.9) if r is small (see Proposition 2.6 (i)-(iii)).In order to prove such necessary conditions, we formulate an extra assumption on the set X, but no extra assumptions on the measure ν (see condition (2.7)).
We also note that one can formulate some extra restrictions on the parameters in order that conditions (2.9)-(2.11)imply the validity of condition (1.9) and that the target space of Q[Z, •, 1] equals C 0,β b (X).To do so, in case of statement (b), one can further require that s 2 ≤ υ Y + s 3 , an assumption that is compatible with the condition s 2 = υ Y + s 3 of standard kernels.
Instead, in case of statement (bb), one can further require that s 3 > β and in case of statement (bbb), one can further require that s 3 ≥ β, but such requirements are not compatible with the condition s 2 = υ Y + s 3 of standard kernels.
In all cases (b)-(bbb), we are saying that when Q[Z, •, 1] is bounded with no decrease of regularity, then the necessary condition (1.9) holds true.Instead when Q[Z, •, 1] is bounded with some decrease of regularity, then condition (1.9) is replaced by a growth condition of the maximal function in the variable r.
In section 3, we present an application of Proposition 2.6 (i) in case Y is a compact differentiable manifold.

Analysis of the operator Q
We first introduce the following two technical lemmas, that generalize to case X = Y the corresponding statements of Gatto [10, p. 104].For a proof, we refer to [12,Lem. 3.2,3.4,3.6].(1.1).Let Y be upper υ Y -Ahlfors regular with respect to X. Then the following statements hold.
Then we have the following Lemma.
(ii) Let Y be strongly upper υ Y -Ahlfors regular with respect to X. Then Since we plan to analyze the operator Q of (1.6) in Hölder spaces, we now introduce some notation.Let X be a set.Then we set If (M, d) is a metric space, then C 0 (M ) denotes the set of continuous functions from M to C and we introduce the subspace and sup If f is a function from a subset D of a metric space (M, d) to C, then we denote by |f : D| ω(•) the ω(•)-Hölder constant of f , which is delivered by the formula In the case in which ω(•) is the function r α for some fixed α ∈]0, 1], a so-called Hölder exponent, we simply write , and we say that f is α-Hölder continuous provided that |f : D| α < ∞.Next we introduce a function that we need for a generalized Hölder norm.For each θ ∈]0, 1], we define the function ω θ (•) from [0, +∞[ to itself by setting where r θ ≡ e −1/θ for all θ ∈]0, 1].Obviously, ω θ (•) is concave and satisfies condition (2.4).We also note that if D ⊆ M , then the continuous embeddings hold for all θ ′ ∈]0, θ[.We now turn to analyze the integral operator Q of (1.6).As we have already said, the sufficient conditions of Proposition 1.8 require that the kernel Z belongs to K ♯ υY ,s2,s3 (X × Y ) and the membership in K ♯ υY ,s2,s3 (X × Y ) requires an estimation of the maximal function of the kernel Z, i.e., to show the validity of condition (1.9) and we now prove the following 'converse' statement that shows that under certain assumptions condition (1.9) is actually necessary.Proposition 2.6 Let (M, d) be a metric space.Let X, Y ⊆ M .Assume that there exists a ∈]0, +∞[ such that for each ρ ∈]0, a[ and x ′ ∈ X, there exists x ′′ ∈ X such that d(x ′ , x ′′ ) = ρ . (2.7) and the following statements hold.

3
[13,pplication in case Y is a compact differentiable manifold ′ , ρ)) cannot be empty.Since the Y ∩ B n (x ′ , ρ) contains x ′ it is not empty.Since the Y ∩ B n (x ′ , ρ) is open in A and is not empty and A is connected, we conclude that Y ∩ ∂B n (x ′ , ρ) cannot be empty and condition (2.7) with X = Y holds true.✷Next we introduce the following.For the definition of tangential gradient grad Y , we refer e.g., to Kirsch and Hettlich [11, A.5], Chavel[1, Chap.1].For a proof, we refer to[13, Thm.6.2] Since a compact manifold Y of class C 1 that is imbedded in R n is (n − 1)-upper Ahlfors regular, we now present an application of Proposition 2.6 (i) in case Y is a compact manifold of class C 1 that is imbedded in R n .To do so, we need the following elementary lemma, that shows that Y satisfies the technical condition (2.7).Lemma 3.1 Let n ∈ N \ {0}.Let Y be a compact manifold of class C 0 that is imbedded in R n and of dimension m.Let m ≥ 1.Then Y satisfies condition (2.7) with X = Y .Proof.Since Y is a compact manifold of class C 0 , Y can be covered by a finite number of open connected domains of charts, each of which cannot be equal to Y .Then by taking a to be one half of a Lebesgue number for such a finite open cover, for each x ′ ∈ Y and ρ ∈]0, a[, the set Y ∩ B n (x ′ , ρ) is contained in at least one open connected domain of chart of the finite cover of Y , say A (cf. e.g., Dugundji [4, Theorem 4.5, Chap.XI]).Since A is homeomorphic to a open subset of R m that is not empty, A cannot be compact.Since Y ∩ B n (x ′ , ρ) is compact, Y ∩ B n (x ′ , ρ) cannot be equal to A and thus the set A \ (Y ∩ B n (x