1 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.

$$\begin{aligned}{} & {} \text {Let}\ {{\mathcal {N}}} \ \text { be a }\sigma \text {-algebra of parts of}\ Y \,, {{\mathcal {B}}}_Y\subseteq {{\mathcal {N}}}\,. \nonumber \\{} & {} \text {Let}\ \nu \ \text {be measure on}\ {{\mathcal {N}}} \,. \nonumber \\{} & {} \text {Let}\ \nu (B(x,r)\cap Y)<+\infty \qquad \forall (x,r)\in X\times ]0,+\infty [\,, \end{aligned}$$
(1.1)

where \({{\mathcal {B}}}_Y\) denotes the \(\sigma \)-algebra of the Borel subsets of Y and

$$\begin{aligned} B(\xi ,r)\equiv \left\{ \eta \in M:\, d(\xi ,\eta )<r\right\} \qquad \forall (\xi ,r)\in M\times ]0,+\infty [\,. \end{aligned}$$
(1.2)

We assume that \(\upsilon _Y\in ]0,+\infty [\) and we consider two types of assumptions on \(\nu \). The first assumption is that Y is upper \(\upsilon _Y\)-Ahlfors regular with respect to X, i.e., that

$$\begin{aligned}{} & {} \text {there exist}\ r_{X,Y,\upsilon _Y}\in ]0,+\infty ]\,,\ c_{X,Y,\upsilon _Y}\in ]0,+\infty [\ \text {such that} \nonumber \\{} & {} \nu ( B(x,r)\cap Y )\le c_{X,Y,\upsilon _Y} r^{\upsilon _Y} \nonumber \\{} & {} \text {for all}\ x\in X\ \text {and}\ r \in ]0,r_{X,Y,\upsilon _Y}[ \,. \end{aligned}$$
(1.3)

In case \(X=Y\), we just say that Y is upper \(\upsilon _Y\)-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 [15]. Namely, we assume that Y is strongly upper \(\upsilon _Y\)-Ahlfors regular with respect to X, i.e., that

$$\begin{aligned}{} & {} \text {there exist}\ r_{X,Y,\upsilon _Y}\in ]0,+\infty ]\,,\ c_{X,Y,\upsilon _Y}\in ]0,+\infty [\ \text {such that} \nonumber \\{} & {} \nu ( (B(x,r_2)\setminus B(x,r_1))\cap Y )\le c_{X,Y,\upsilon _Y}(r_2^{\upsilon _Y}-r_1^{\upsilon _Y}) \nonumber \\{} & {} \text {for all}\ x\in X\ \text {and}\ r_1,r_2\in [0,r_{X,Y,\upsilon _Y}[ \ \text {with}\ r_1<r_2\,, \end{aligned}$$
(1.4)

where we understand that \(B(x,0)\equiv \emptyset \) (in case \(X=Y\), we just say that Y is strongly upper \(\upsilon _Y\)-Ahlfors regular). So, for example, if Y is the boundary of a bounded open Lipschitz subset of \(M={{\mathbb {R}}}^n\), then Y is upper \((n-1)\)-Ahlfors regular with respect to \({{\mathbb {R}}}^n\) and if Y is the boundary of an open bounded subset of \(M={{\mathbb {R}}}^n\) of class \(C^1\), then Y is strongly upper \((n-1)\)-Ahlfors regular with respect to Y (cf. [17, Rmk. 2]). Here and throughout the paper,

$$\begin{aligned} n\in {{\mathbb {N}}}, n\ge 2\,. \end{aligned}$$

We plan to consider off-diagonal kernels K from \((X\times Y)\setminus D_{X\times Y}\) to \({{\mathbb {C}}}\), where

$$\begin{aligned} D_{X\times Y}\equiv \left\{ (x,y)\in X\times Y:\,x=y \right\} \end{aligned}$$

denotes the diagonal set of \(X\times Y\) and we introduce the following class of ‘potential type’ kernels as in the following definition, which is a generalisation of related classes as in Gegelia, Basheleishvili and Burchuladze [14] (see also Dondi and the author [3], where such classes have been introduced in a form that generalizes those of Giraud [12], Gegelia [11] and Gegelia, Basheleishvili and Burchuladze [14, Chap. IV]).

Definition 1.5

Let (M, d) be a metric space. Let X, \(Y\subseteq M\). Let \(s_1\), \(s_2\), \(s_3\in {{\mathbb {R}}}\). We denote by the symbol \({{\mathcal {K}}}_{s_1, s_2, s_3} (X\times Y)\) the set of continuous functions K from \((X\times Y)\setminus D_{X\times Y}\) to \({{\mathbb {C}}}\) such that

$$\begin{aligned} \begin{aligned} \left\| K \right\| _{\mathrm{{\mathcal {K}}}_{s_1 ,s_2 ,s_3 } (X \times Y)} \equiv&\sup \{ d(x,y)^{s_1 } |K(x,y)|:(x,y) \in X \times Y,x \ne y\} \\&\quad + \sup \left\{ {\frac{{d(x^{\prime } ,y)^{s_2 } }}{{d(x^{\prime } ,x^{{\prime } {\prime } } )^{s_3 } }}|K(x^{\prime } ,y) - K(x^{{\prime } {\prime } } ,y)|} \right. \\&\quad \left. {:x^{\prime } ,x^{{\prime } {\prime } } \in X,x^{\prime } \ne x^{{\prime } {\prime } } ,y \in Y\mathrm{{ \setminus }}B(x^{\prime } ,2d(x^{\prime } ,x^{{\prime } {\prime } } ))} \right\} < + \infty {\hspace{0.55542pt}} . \\ \end{aligned} \end{aligned}$$

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 T1 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

$$\begin{aligned} Q[Z,g,1](x)\equiv \int _YZ(x,y)(g(x)-g(y))\,d\nu (y)\qquad \forall x\in X\,. \end{aligned}$$
(1.6)

where Z belongs to a class \({{\mathcal {K}}}_{ s_1, s_2, s_3 } (X\times Y)\) as in Definition 1.5 and g is a \({{\mathbb {C}}}\)-valued function in \(X\cup Y\). 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 [15, 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.

Definition 1.7

Let (M, d) be a metric space. Let X, \(Y \subseteq M\). Let \(\nu \) be as in (1.1). Let \(s_{1}\), \(s_{2}\), \(s_{3}\in {{\mathbb {R}}}\). We set

$$\begin{aligned}{} & {} { {{\mathcal {K}}}_{{s_{1}, s_{2}, s_{3}} }^\sharp (X\times Y) \equiv \biggl \{ K\in {{\mathcal {K}}}_{ s_{1}, s_{2}, s_{3} }(X\times Y):\, } \\{} & {} \ \ K(x,\cdot )\ \text {is}\ \nu -\text {integrable in}\ Y\setminus B(x,r) \ \text {for all}\ (x,r)\in X\times ]0,+\infty [\,, \\{} & {} \ \ \sup _{x\in X}\sup _{r\in ]0,+\infty [} \left| \int _{Y\setminus B(x,r)}K(x,y)\,d\nu (y)\right| <+\infty \biggr \} \end{aligned}$$

and

$$\begin{aligned}{} & {} { \Vert K\Vert _{{{\mathcal {K}}}_{ s_1, s_2, s_3 }^\sharp (X\times Y)} \equiv \Vert K\Vert _{{{\mathcal {K}}}_{ s_1, s_2, s_3 }(X\times Y)} } \\{} & {} + \sup _{x\in X}\sup _{r\in ]0,+\infty [} \left| \int _{Y\setminus B(x,r)}K(x,y)\,d\nu (y) \right| \quad \forall K\in {{\mathcal {K}}}_{ s_1, s_2, s_3 }^\sharp (X\times Y)\,. \end{aligned}$$

Clearly, \(({{\mathcal {K}}}^\sharp _{ s_{1},s_{2},s_{3} }(X\times Y),\Vert \cdot \Vert _{ {{\mathcal {K}}}^\sharp _{s_{1},s_{2},s_{3} }(X\times Y) })\) is a normed space and the space \({{\mathcal {K}}}^\sharp _{ s_{1},s_{2},s_{3} }(X\times Y)\) is continuously embedded into \({{\mathcal {K}}}_{ s_{1},s_{2},s_{3} }(X\times Y)\). Then one can establish the Hölder continuity of Q[Z, g, 1] in the ‘singular’ case \(s_1=\upsilon _Y\) by means of the following statement, that extends some work of Gatto [10, Proof of Thm. 3] (cf. [15, Prop. 6.3 (ii)]).

Proposition 1.8

Let (M, d) be a metric space. Let X, \(Y\subseteq M\). Let

$$\begin{aligned} \upsilon _Y\in ]0,+\infty [\,, \ \beta \in ]0,1]\,, \ s_2\in [\beta , +\infty [\,, \ s_3\in ]0,1]\,. \end{aligned}$$

Let \(\nu \) be as in (1.1), \(\nu (Y)<+\infty \). Then the following statements hold.

  1. (b)

    If \(s_2-\beta >\upsilon _Y\), \(s_2< \upsilon _Y+\beta +s_3\) and Y is upper \(\upsilon _Y\)-Ahlfors regular with respect to X, then the bilinear map from

    $$\begin{aligned} {{\mathcal {K}}}^\sharp _{ \upsilon _Y,s_{2},s_{3} }(X\times Y)\times C^{0,\beta } (X\cup Y) \quad \text {to}\quad C_b^{0,\min \{ \beta , \upsilon _Y+s_3+\beta -s_2 \}}(X)\,, \end{aligned}$$

    which takes (Z, g) to Q[Z, g, 1] is continuous.

  2. (bb)

    If \(s_2-\beta =\upsilon _Y\) and Y is strongly upper \(\upsilon _Y\)-Ahlfors regular with respect to X, then the bilinear map from

    $$\begin{aligned} {{\mathcal {K}}}^\sharp _{\upsilon _Y,s_{2},s_{3} }(X\times Y)\times C^{0,\beta }(X\cup Y) \quad \text {to}\quad C_b^{0,\max \{ r^\beta ,\omega _{s_3}(r) \}}(X)\,, \end{aligned}$$

    which takes (Z, g) to Q[Z, g, 1] is continuous.

  3. (bbb)

    If \(s_2-\beta <\upsilon _Y\) and Y is upper \(\upsilon _Y\)-Ahlfors regular with respect to X, then the bilinear map from

    $$\begin{aligned} {{\mathcal {K}}}^\sharp _{\upsilon _Y,s_{2},s_{3} }(X\times Y)\times C^{0,\beta }(X\cup Y) \quad \text {to}\quad C_b^{0,\min \{ \beta , s_3 \}}(X)\,, \end{aligned}$$

    which takes (Z, g) to Q[Z, g, 1] is continuous.

Here \(C^{0,\beta } (X\cup Y)\) denotes the semi-normed space of \(\beta \)-Hölder continuous functions on \(X\cup Y\), \(C^{0,\gamma }_b(X)\) denotes the normed space of bounded \(\gamma \)-Hölder continuous functions on X for all \(\gamma \in ]0,1]\) and \(C_b^{0,\max \{ r^\beta ,\omega _{s_3}(r) \}}(X)\) denotes the normed space of bounded generalized Hölder continuous functions on X with modulus of continuity \(\max \{ r^\beta ,\omega _{s_3}(r) \}\) (see notation around (2.4)–(2.5) below). Now the assumptions of Proposition 1.8 require that the kernel Z belongs to \({{\mathcal {K}}}^\sharp _{\upsilon _Y,s_{2},s_{3} }(X\times Y)\) and the membership in \({{\mathcal {K}}}^\sharp _{\upsilon _Y,s_{2},s_{3} }(X\times Y)\) requires an estimation of the maximal function of the kernel Z, i.e., to show that

$$\begin{aligned} \sup _{x\in X}\sup _{r\in ]0,+\infty [} \left| \int _{Y\setminus B(x,r)}Z(x,y)\,d\nu (y) \right| <+\infty \end{aligned}$$
(1.9)

and here we wonder whether such assumption is actually necessary. Indeed, under the only assumption that the kernel Z belongs to \({{\mathcal {K}}}_{\upsilon _Y,s_{2},s_{3} }(X\times Y)\) and that g is \(\beta \)-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 \(Q[Z,\cdot ,1]\) when \(Z\in {{\mathcal {K}}}_{\upsilon _Y,s_{2},s_{3} }(X\times Y)\). Namely, we prove the validity of (2.9)–(2.11), that are conditions on the growth of the integral in 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 \(\nu \) (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 \(Q[Z,\cdot ,1]\) equals \(C_b^{0,\beta }(X)\).

To do so, in case of statement (b), one can further require that \(s_2\le \upsilon _Y+s_3\), an assumption that is compatible with the condition \(s_2=\upsilon _Y+s_3\) of standard kernels.

Instead, in case of statement (bb), one can further require that \(s_3>\beta \) and in case of statement (bbb), one can further require that \(s_3\ge \beta \), but such requirements are not compatible with the condition \(s_2=\upsilon _Y+s_3\) of standard kernels.

In all cases (b)–(bbb), we are saying that when \(Q[Z,\cdot ,1]\) is bounded with no decrease of regularity, then the necessary condition (1.9) holds true. Instead when \(Q[Z,\cdot ,1]\) 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 Sect. 3, we present an application of Proposition 2.6 (i) in case Y is a compact differentiable manifold.

2 Analysis of the Operator Q

We first introduce the following two technical lemmas, that generalize to case \(X\ne Y\) the corresponding statements of Gatto [10, p. 104]. For a proof, we refer to [15, Lem. 3.2, 3.4, 3.6].

Lemma 2.1

Let (M, d) be a metric space. Let X, \(Y\subseteq M\). Let \(\upsilon _Y\in ]0,+\infty [\). Let \(\nu \) be as in (1.1). Let Y be upper \(\upsilon _Y\)-Ahlfors regular with respect to X. Then the following statements hold.

  1. (i)

    \(\nu (\{x\})=0\) for all \(x\in X\cap Y\).

  2. (ii)

    Let \(\nu (Y)<+\infty \). If \(s\in ]0,\upsilon _Y[\), then

    $$\begin{aligned} c'_{s,X,Y}\equiv \sup _{x\in X}\int _Y\frac{d\nu (y)}{d(x,y)^s} \le \nu (Y)a^{-s}+c_{X,Y,\upsilon _Y}\frac{\upsilon _Y}{\upsilon _Y-s}a^{\upsilon _Y-s} \end{aligned}$$
    (2.2)

    for all \(a\in ]0,r_{X,Y,\upsilon _Y}[\). If \(s=0\), then

    $$\begin{aligned} c'_{0,X,Y}\equiv \sup _{x\in X}\int _Y\frac{d\nu (y)}{d(x,y)^0} = \nu (Y) \,. \end{aligned}$$
  3. (iii)

    Let \(\nu (Y)<+\infty \) whenever \(r_{X,Y,\upsilon _Y}<+\infty \). If \(s\in ]-\infty ,\upsilon _Y[\), then

    $$\begin{aligned} c''_{s,X,Y}\equiv \sup _{(x,t)\in X\times ]0,+\infty [} t^{s-\upsilon _Y}\int _{B(x,t)\cap Y}\frac{d\nu (y)}{d(x,y)^s}<+\infty \,. \end{aligned}$$

Then we have the following Lemma.

Lemma 2.3

Let (M, d) be a metric space. Let X, \(Y\subseteq M\). Let \(\upsilon _Y\in ]0,+\infty [\). Let \(\nu \) be as in (1.1), \(\nu (Y)<+\infty \). Then the following statements hold.

  1. (i)

    Let Y be upper \(\upsilon _Y\)-Ahlfors regular with respect to X. If \(s\in ]\upsilon _Y,+\infty [\), then

    $$\begin{aligned} c'''_{s,X,Y}\equiv \sup _{(x,t)\in X\times ]0,+\infty [} t^{s-\upsilon _Y}\int _{Y\setminus B(x,t) }\frac{d\nu (y)}{d(x,y)^s}<+\infty \,. \end{aligned}$$
  2. (ii)

    Let Y be strongly upper \(\upsilon _Y\)-Ahlfors regular with respect to X. Then

    $$\begin{aligned} c^{iv}_{X,Y}\equiv \sup _{(x,t)\in X\times ]0,1/e[} \vert \ln t\vert ^{-1}\int _{Y\setminus B(x,t) }\frac{d\nu (y)}{d(x,y)^{\upsilon _Y}}<+\infty \,. \end{aligned}$$

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

$$\begin{aligned} B(X)\equiv \left\{ f\in {{\mathbb {C}}}^X:\,f\ \text {is bounded} \right\} \,,\quad \Vert f\Vert _{B(X)}\equiv \sup _X\vert f\vert \qquad \forall f\in B(X)\,. \end{aligned}$$

If (M, d) is a metric space, then \(C^0(M)\) denotes the set of continuous functions from M to \({{\mathbb {C}}}\) and we introduce the subspace \( C^0_b(M)\equiv C^0(M)\cap B(M) \) of B(M). Let \(\omega \) be a function from \([0,+\infty [\) to itself such that

$$\begin{aligned}{} & {} \qquad \qquad \omega (0)=0,\qquad \omega (r)>0\qquad \forall r\in ]0,+\infty [\,, \nonumber \\{} & {} \qquad \qquad \omega \ {\text {is increasing,}}\ \lim _{r\rightarrow 0^{+}}\omega (r)=0\,, \nonumber \\{} & {} \qquad \qquad {\text {and}}\ \sup _{(a,t)\in [1,+\infty [\times ]0,+\infty [} \frac{\omega (at)}{a\omega (t)}<+\infty \,. \end{aligned}$$
(2.4)

If f is a function from a subset \({{\mathbb {D}}}\) of a metric space (M, d) to \({{\mathbb {C}}}\), then we denote by \(\vert f:{{\mathbb {D}}}\vert _{\omega (\cdot )}\) the \(\omega (\cdot )\)-Hölder constant of f, which is delivered by the formula

$$\begin{aligned} \vert f:{{\mathbb {D}}}\vert _{\omega (\cdot ) } \equiv \sup \left\{ \frac{\vert f( x )-f( y)\vert }{\omega (d( x, y)) }: x, y\in {{\mathbb {D}}} , x\ne y\right\} \,. \end{aligned}$$

If \(\vert f:{{\mathbb {D}}}\vert _{\omega (\cdot )}<+\infty \), we say that f is \(\omega (\cdot )\)-Hölder continuous. Sometimes, we simply write \(\vert f\vert _{\omega (\cdot )}\) instead of \(\vert f:{{\mathbb {D}}}\vert _{\omega (\cdot )}\). The subset of \(C^{0}({{\mathbb {D}}} ) \) whose functions are \(\omega (\cdot )\)-Hölder continuous is denoted by \(C^{0,\omega (\cdot )} ({{\mathbb {D}}})\) and \(\vert f:{{\mathbb {D}}}\vert _{\omega (\cdot )}\) is a semi-norm on \(C^{0,\omega (\cdot )} ({{\mathbb {D}}})\). Then we consider the space \(C^{0,\omega (\cdot )}_{b}({{\mathbb {D}}} ) \equiv C^{0,\omega (\cdot )} ({{\mathbb {D}}} )\cap B({{\mathbb {D}}} ) \) with the norm

$$\begin{aligned} \Vert f\Vert _{ C^{0,\omega (\cdot )}_{b}({{\mathbb {D}}} ) }\equiv \sup _{x\in {{\mathbb {D}}} }\vert f(x)\vert +\vert f\vert _{\omega (\cdot )}\qquad \forall f\in C^{0,\omega (\cdot )}_{b}({{\mathbb {D}}} )\,. \end{aligned}$$

In the case in which \(\omega (\cdot )\) is the function \(r^{\alpha }\) for some fixed \(\alpha \in ]0,1]\), a so-called Hölder exponent, we simply write \(\vert \cdot :{{\mathbb {D}}}\vert _{\alpha }\) instead of \(\vert \cdot :{{\mathbb {D}}}\vert _{r^{\alpha }}\), \(C^{0,\alpha } ({{\mathbb {D}}})\) instead of \(C^{0,r^{\alpha }} ({{\mathbb {D}}})\), \(C^{0,\alpha }_{b}({{\mathbb {D}}})\) instead of \(C^{0,r^{\alpha }}_{b} ({{\mathbb {D}}})\), and we say that f is \(\alpha \)-Hölder continuous provided that \(\vert f:{{\mathbb {D}}}\vert _{\alpha }<+\infty \). Next we introduce a function that we need for a generalized Hölder norm. For each \(\theta \in ]0,1]\), we define the function \(\omega _{\theta }(\cdot )\) from \([0,+\infty [\) to itself by setting

$$\begin{aligned} \omega _{\theta }(r)\equiv \left\{ \begin{array}{ll} 0 &{}r=0\,, \\ r^{\theta }\vert \ln r \vert &{}r\in ]0,r_{\theta }]\,, \\ r_{\theta }^{\theta }\vert \ln r_{\theta } \vert &{} r\in ]r_{\theta },+\infty [\,, \end{array} \right. \end{aligned}$$
(2.5)

where \( r_{\theta }\equiv e^{-1/\theta } \) for all \(\theta \in ]0,1]\). Obviously, \(\omega _{\theta }(\cdot ) \) is concave and satisfies condition (2.4). We also note that if \({{\mathbb {D}}}\subseteq M\), then the continuous embeddings

$$\begin{aligned} C^{0, \theta }_b({{\mathbb {D}}})\subseteq C^{0,\omega _\theta (\cdot )}_b({{\mathbb {D}}})\subseteq C^{0,\theta '}_b({{\mathbb {D}}}) \end{aligned}$$

hold for all \(\theta '\in ]0,\theta [\). 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 \({{\mathcal {K}}}^\sharp _{\upsilon _Y,s_{2},s_{3} }(X\times Y)\) and the membership in \({{\mathcal {K}}}^\sharp _{\upsilon _Y,s_{2},s_{3} }(X\times Y)\) 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, \(Y\subseteq M\). Assume that

$$\begin{aligned}{} & {} \text {there exists}\ a\in ]0,+\infty [\ \text {such that for each}\ \rho \in ]0,a[\ \text {and}\ x'\in X, \nonumber \\{} & {} \text {there exists}\ x''\in X\ \text {such that}\ d(x',x'')=\rho \,. \end{aligned}$$
(2.7)

Let

$$\begin{aligned} \upsilon _Y\in ]0,+\infty [\,, \ \beta \in ]0,1]\,, \ s_2\in [\beta , +\infty [\,, \ s_3\in ]0,1]\,. \end{aligned}$$

Let \(\nu \) be as in (1.1), \(\nu (Y)<+\infty \). Let \(Z\in {{\mathcal {K}}}_{\upsilon _Y,s_{2},s_{3} }(X\times Y)\). Then

$$\begin{aligned} Z(x,\cdot )\ \text {is}\ \nu -\text {integrable in}\ Y\setminus B(x,r) \ \text {for all}\ (x,r)\in X\times ]0,+\infty [ \end{aligned}$$
(2.8)

and the following statements hold.

  1. (i)

    Let \(s_2-\beta >\upsilon _Y\), \(s_2< \upsilon _Y+\beta +s_3\). Let Y be upper \(\upsilon _Y\)-Ahlfors regular with respect to X. If the linear map from \(C^{0,\beta } (X\cup Y)\) to \(C_b^{0,\min \{ \beta , \upsilon _Y+s_3+\beta -s_2 \}}(X)\) that takes g to Q[Z, g, 1] is continuous, then

    $$\begin{aligned} \qquad \sup _{x\in X}\sup _{r\in ]0,e^{-1/s_3}[} \left| \int _{Y\setminus B(x,r)}Z(x,y)\,d\nu (y) \right| \frac{r^\beta }{\max \{r^\beta ,r^{\upsilon _Y+s_3+\beta -s_2} \}} <+\infty \,.\nonumber \\ \end{aligned}$$
    (2.9)

    In particular, if \(s_2\le \upsilon _Y+s_3\), then \(\min \{ \beta , \upsilon _Y+s_3+\beta -s_2 \}=\beta \) and inequality (1.9) holds true.

  2. (ii)

    Let \(s_2-\beta =\upsilon _Y\). Let Y be strongly upper \(\upsilon _Y\)-Ahlfors regular with respect to X. If the linear map from \(C^{0,\beta } (X\cup Y)\) to \(C_b^{0,\max \{ r^\beta ,\omega _{s_3}(r) \}}(X)\) that takes g to Q[Z, g, 1] is continuous, then

    $$\begin{aligned} \sup _{x\in X}\sup _{r\in ]0,e^{-1/s_3}[} \left| \int _{Y\setminus B(x,r)}Z(x,y)\,d\nu (y) \right| \frac{r^\beta }{\max \{ r^\beta ,\omega _{s_3}(r) \}} <+\infty \,. \end{aligned}$$
    (2.10)

    In particular, if \(s_3>\beta \), then \(C_b^{0,\max \{ r^\beta ,\omega _{s_3}(r) \}}(X)=C_b^{0,\beta } (X)\) and inequality (1.9) holds true.

  3. (iii)

    Let \(s_2-\beta <\upsilon _Y\). Let Y be upper \(\upsilon _Y\)-Ahlfors regular with respect to X. If the linear map from \(C^{0,\beta } (X\cup Y)\) to \(C_b^{0,\min \{ \beta , s_3 \}}(X)\) that takes g to Q[Z, g, 1] is continuous, then

    $$\begin{aligned} \sup _{x\in X}\sup _{r\in ]0,e^{-1/s_3}[} \left| \int _{Y\setminus B(x,r)}Z(x,y)\,d\nu (y) \right| \frac{r^\beta }{\max \{ r^\beta ,r^{s_3} \}} <+\infty \,. \end{aligned}$$
    (2.11)

    In particular, if \(s_3\ge \beta \), then \(C_b^{0,\min \{ \beta , s_3 \}}(X)=C^{0,\beta }_b (X)\) and inequality (1.9) holds true.

Proof

Since \(Z\in {{\mathcal {K}}}_{\upsilon _Y,s_{2},s_{3} }(X\times Y)\), we have

$$\begin{aligned} \vert Z(x,y)\vert \le \frac{\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_{2},s_{3} }(X\times Y)}}{d(x,y)^{\upsilon _Y}} \le \frac{\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_{2},s_{3} }(X\times Y)}}{r^{\upsilon _Y}}\qquad \forall y\in Y\setminus B(x,r)\nonumber \\ \end{aligned}$$
(2.12)

for all \((x,r)\in X\times ]0,+\infty [\). On the other hand, Z is continuous and \(\nu (Y)\) is finite and thus condition (2.8) is satisfied. Inequality (2.12) also shows that it suffices to show inequalities (1.9), (2.9)–(2.11), for \(r\in ]0, \min \{a,e^{-1/s_3}\}/2[\) (\(\subseteq ]0,\min \{a,e^{-1}\}/2[\)). In order to consider cases (i)–(iii) at the same time, we find convenient to set

$$\begin{aligned} \phi (r)\equiv \left\{ \begin{array}{ll} \max \{ r^\beta , r^{\upsilon _Y+s_3+\beta -s_2} \} &{}\text {if}\ s_2-\beta >\upsilon _Y, s_2< \upsilon _Y+\beta +s_3\,, \\ \max \{ r^\beta ,\omega _{s_3}(r) \} &{}\text {if}\ s_2-\beta =\upsilon _Y\,, \\ \max \{ r^\beta ,r^{s_3} \}&{}\text {if}\ s_2-\beta <\upsilon _Y \end{array} \right. \end{aligned}$$

for all \(r\in [0,+\infty [\). The idea consists in proving an upper estimate for

$$\begin{aligned} \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y)\,d\nu (y) \right| \vert g(x')-g(x'')\vert \end{aligned}$$

in terms of \(\phi (d(x',x''))\), of the norm of the kernel Z and of the seminorm of g when \(g\in C^{0,\beta } (X\cup Y)\), \(x'\), \(x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and then in making a suitable choice of g that enables to establish an upper bound for \(\left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y)\,d\nu (y) \right| \) that is independent of \(x'\) and \(x''\) and then to choose \(x''\) so that \(2d(x',x'')=r\) and to obtain the upper bound we need for \(\left| \int _{Y\setminus B(x',r)}Z(x',y)\,d\nu (y) \right| \). To do so, we first look at the following elementary equality

$$\begin{aligned}{} & {} { Q[Z,g,1](x')-Q[Z,g,1](x'') } \\{} & {} \qquad = \int _{Y\cap B(x',2d(x',x''))} Z(x',y)(g(x')-g(y))\,d\nu (y) \\{} & {} \qquad \quad -\int _{Y\cap B(x',2d(x',x''))}Z(x'',y)(g(x'')-g(y))\,d\nu (y) \\{} & {} \qquad \quad +\int _{Y\setminus B(x',2d(x',x''))}Z(x',y) [(g(x')-g(y)) -(g(x'')-g(y)) ]\,d\nu (y) \\{} & {} \qquad \quad +\int _{Y\setminus B(x',2d(x',x''))}[Z(x',y) -Z(x'',y)](g(x'')-g(y))\,d\nu (y) \end{aligned}$$

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and \(g\in C^{0,\beta } (X\cup Y)\). Next we note that under each of the assumptions of statements (i)–(iii), the linear operator \(Q[Z,\cdot ,1]\) is continuous from \(C^{0,\beta } (X\cup Y)\) to \(C^{0,\phi (\cdot )}_b (X)\). Then we find convenient to introduce the symbol \({{\mathcal {L}}}(C^{0,\beta } (X\cup Y),C^{0,\phi (\cdot )}_b (X))\) for the normed space of linear and bounded operators from \(C^{0,\beta } (X\cup Y)\) to \(C^{0,\phi (\cdot )}_b (X)\). Then Lemma 2.1 and the inclusion

$$\begin{aligned} B(x',2d(x',x''))\subseteq B(x'',3d(x',x'')) \end{aligned}$$

imply that

$$\begin{aligned}{} & {} { \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) [g(x'')- g(x') ]\,d\nu (y)\right| } \nonumber \\{} & {} \le \Vert Q[Z,\cdot ,1]\Vert _{{{\mathcal {L}}}(C^{0,\beta } (X\cup Y),C^{0,\phi (\cdot )}_b (X))} \vert g:\,X\cup Y\vert _\beta \phi (d(x',x'')) \nonumber \\{} & {} \quad +\int _{Y\cap B(x',2d(x',x''))} \vert Z(x',y)\vert \,\vert g(x')-g(y)\vert \,d\nu (y) \nonumber \\{} & {} \quad +\int _{Y\cap B(x',2d(x',x''))}\vert Z(x'',y)\vert \,\vert g(x'')-g(y)\vert \,d\nu (y) \nonumber \\{} & {} \quad +\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_2,s_3}(X\times Y) } \vert g:X\cup Y\vert _\beta \nonumber \\{} & {} \qquad \quad \int _{Y\setminus B(x',2d(x',x''))} \frac{d(x',x'')^{s_3}}{d(x',y)^{s_2}}d(y,x'')^\beta \,d\nu (y) \nonumber \\{} & {} \le \Vert Q[Z,\cdot ,1]\Vert _{{{\mathcal {L}}}(C^{0,\beta } (X\cup Y),C^{0,\phi (\cdot )}_b (X))} \vert g:\,X\cup Y\vert _\beta \phi (d(x',x'')) \nonumber \\{} & {} \quad +\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_2,s_3}(X\times Y) } \vert g:X\cup Y\vert _\beta \biggl \{\biggr . \int _{ Y\cap B(x',2d(x',x'')) } \frac{d\nu (y)}{d(x',y)^{\upsilon _Y-\beta }} \nonumber \\{} & {} \quad + \int _{ Y\cap B(x'',3d(x',x'')) } \frac{d\nu (y)}{d(x'',y)^{\upsilon _Y-\beta }} \biggl .\biggr \} \nonumber \\{} & {} \quad +\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_2,s_3}(X\times Y) } \vert g:X\cup Y\vert _\beta \nonumber \\{} & {} \quad \times \int _{Y\setminus B(x',2d(x',x''))} \frac{d(x',x'')^{s_3}}{d(x',y)^{s_2}}d(y,x'')^\beta \,d\nu (y) \nonumber \\{} & {} \le \Vert Q[Z,\cdot ,1]\Vert _{{{\mathcal {L}}}(C^{0,\beta } (X\cup Y),C^{0,\phi (\cdot )}_b (X))} \vert g:\,X\cup Y\vert _\beta \phi (d(x',x'')) \nonumber \\{} & {} \quad +\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_2,s_3}(X\times Y) } \vert g:X\cup Y\vert _\beta c_{\upsilon _Y-\beta ,X,Y}'' \nonumber \\{} & {} \quad \times \biggl \{\biggr . (2d(x',x''))^{\upsilon _Y-(\upsilon _Y-\beta ) } + (3d(x',x''))^{\upsilon _Y-(\upsilon _Y-\beta ) } \biggl .\biggr \} \nonumber \\{} & {} \quad +\Vert Z\Vert _{{{\mathcal {K}}}_{\upsilon _Y,s_2,s_3}(X\times Y) } \vert g:X\cup Y\vert _\beta \nonumber \\{} & {} \quad \times \int _{Y\setminus B(x',2d(x',x''))} \frac{d(x',x'')^{s_3}}{d(x',y)^{s_2}}d(y,x'')^\beta \,d\nu (y) \end{aligned}$$
(2.13)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and \(g\in C^{0,\beta } (X\cup Y)\). Next we note that the triangular inequality implies that

$$\begin{aligned} d(y,x'')\le d(y,x')+d(x',x'') \le 2d(x',y)\qquad \forall y\in Y\setminus B(x',2d(x',x'')) \end{aligned}$$

and that accordingly

$$\begin{aligned}{} & {} { \int _{Y\setminus B(x',2d(x',x'')} \frac{d(x',x'')^{s_3}}{d(x',y)^{s_2}}d(y,x'')^\beta \,d\nu (y) } \nonumber \\{} & {} \qquad \le 2^\beta \int _{Y\setminus B(x',2d(x',x'')} \frac{d\nu (y)}{d(x',y)^{s_2-\beta } }d(x',x'')^{s_3} \end{aligned}$$
(2.14)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\). We now distinguish three cases. If \(s_2-\beta >\upsilon _Y\) as in statement (i), then Lemma 2.3 (i) implies that

$$\begin{aligned}{} & {} { \int _{Y\setminus B(x',2d(x',x''))} \frac{d\nu (y)}{ d(x',y)^{s_2-\beta } }d(x',x'')^{s_3} } \nonumber \\{} & {} \qquad \le c'''_{s_2-\beta ,X,Y}(2d(x',x''))^{\upsilon _Y-(s_2-\beta )}d(x',x'')^{s_3} \nonumber \\{} & {} \qquad \le c'''_{ s_2-\beta ,X,Y}d(x',x'')^{\upsilon _Y+s_3+\beta -s_2 } \end{aligned}$$
(2.15)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\). If \(s_2-\beta =\upsilon _Y\) as in statement (ii), then Lemma 2.3 (ii) implies that

$$\begin{aligned}{} & {} { \int _{Y\setminus B(x',2d(x',x''))} \frac{d\nu (y)}{ d(x',y)^{s_2-\beta } }d(x',x'')^{s_3} } \nonumber \\{} & {} \qquad \le c^{iv}_{X,Y}\vert \ln (2d(x',x''))\vert \,d(x',x'')^{s_3} \nonumber \\{} & {} \qquad \le c^{iv}_{X,Y}\vert \ln d(x',x'') \vert \,d(x',x'')^{s_3} \left( 1+\frac{\ln 2}{\vert \ln d(x',x'') \vert } \right) \nonumber \\{} & {} \qquad \le 2c^{iv}_{X,Y}\vert \ln d(x',x'') \vert \,d(x',x'')^{s_3} \end{aligned}$$
(2.16)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\). If \(s_2-\beta <\upsilon _Y\) as in statement (iii), then Lemma 2.1 (i) implies that

$$\begin{aligned} \int _{Y\setminus B(x',2d(x',x''))} \frac{d\nu (y)}{ d(x',y)^{s_2-\beta } }d(x',x'')^{s_3} \le c'_{s_2-\beta ,X,Y}d(x',x'')^{s_3} \end{aligned}$$
(2.17)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\).

Under the assumptions of statement (i), we have \( s_2-\beta >\upsilon _Y \) and inequalities (2.13), (2.14), (2.15) imply that there exist \(c_{(i)}\in ]0,+\infty [\) such that

$$\begin{aligned}{} & {} { \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) [g(x'')- g(x') ]\,d\nu (y)\right| } \\{} & {} \qquad \le c_{(i)} \vert g:X\cup Y\vert _\beta \\{} & {} \qquad \quad \times \max \left\{ \phi (d(x',x'')), d(x',x'')^\beta ,d(x',x'')^{\upsilon _Y+s_3+\beta -s_2} \right\} \\{} & {} \qquad =c_{(i)} \vert g:X\cup Y\vert _\beta \phi (d(x',x'')), \end{aligned}$$

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and \(g\in C^{0,\beta } (X\cup Y)\).

Under the assumptions of statement (ii), we have \(s_2-\beta =\upsilon _Y\) and inequalities (2.13), (2.14), (2.16) imply that there exist \(c_{(ii)}\in ]0,+\infty [\) such that

$$\begin{aligned}{} & {} { \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) [g(x'')- g(x') ]\,d\nu (y)\right| } \\{} & {} \qquad \le c_{(ii)} \vert g:X\cup Y\vert _\beta \\{} & {} \qquad \quad \times \max \left\{ \phi (d(x',x'')),d(x',x'')^\beta ,\vert \ln d(x',x'') \vert \,d(x',x'')^{s_3} \right\} \\{} & {} \qquad =c_{(ii)} \vert g:X\cup Y\vert _\beta \phi (d(x',x'')) \end{aligned}$$

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and \(g\in C^{0,\beta } (X\cup Y)\).

Under the assumptions of statement (iii), we have \(s_2-\beta <\upsilon _Y\) and inequalities (2.13), (2.14), (2.17) imply that there exist \(c_{(iii)}\in ]0,+\infty [\) such that

$$\begin{aligned}{} & {} { \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) [g(x'')- g(x') ]\,d\nu (y)\right| } \\{} & {} \qquad \le c_{(iii)} \vert g:X\cup Y\vert _\beta \\{} & {} \qquad \quad \times \max \left\{ \phi (d(x',x'')), d(x',x'')^\beta , d(x',x'')^{s_3} \right\} \\{} & {} \qquad =c_{(iii)} \vert g:X\cup Y\vert _\beta \phi (d(x',x'')) \end{aligned}$$

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and \(g\in C^{0,\beta } (X\cup Y)\).

Hence, in all cases (i)–(iii), there exists \(c\in ]0,+\infty [\) such that

$$\begin{aligned}{} & {} { \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) \,d\nu (y)\right| \,\vert g(x'')- g(x')\vert } \nonumber \\{} & {} \qquad \qquad \qquad \le c \vert g:X\cup Y\vert _\beta \phi (d(x',x'')) \end{aligned}$$
(2.18)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\) and \(g\in C^{0,\beta } (X\cup Y)\). Next we make a choice of g that depends upon \(x'\). Namely, we set

$$\begin{aligned} g_{x'}(y)\equiv d(x',y)^\beta \qquad \forall y\in X\cup Y \end{aligned}$$

for all \(x'\in X\). By the elementary inequality

$$\begin{aligned} \vert g_{x'}(u_1)-g_{x'}(u_2)\vert =\vert d(x',u_1)^\beta -d(x',u_2)^\beta \vert \le d(u_1,u_2)^\beta \ \forall u_1, u_2\in X\cup Y\,, \end{aligned}$$

that follows by the Yensen inequality, we have

$$\begin{aligned} \vert g_{x'}:X\cup Y\vert _\beta \le 1\qquad \forall x'\in X\,. \end{aligned}$$

(cf., e.g., Folland [6, Prop. 6.11]). Then \(g_{x'}\in C^{0,\beta } (X\cup Y)\) and inequality (2.18) implies that

$$\begin{aligned}{} & {} { \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) \,d\nu (y)\right| d(x',x'')^\beta } \nonumber \\{} & {} \qquad = \left| \int _{Y\setminus B(x',2d(x',x''))}Z(x',y) \,d\nu (y)\right| \,\vert g_{x'}(x'')- g_{x'}(x')\vert \nonumber \\{} & {} \qquad \le c \vert g_{x'}:X\cup Y\vert _\beta \phi (d(x',x'')) \le c \phi (d(x',x'')) \end{aligned}$$
(2.19)

for all \(x',x''\in X\), \(d(x',x'')\le \min \{a,e^{-1/s_3}\}/2\). Thus if \((x',r)\) belongs to \( X\times ]0,\min \{a,e^{-1/s_3}\}/2[\) our assumption implies that there exists \(x''\in X\) such that \(d(x',x'')=r/2\) and thus inequality (2.18) implies that

$$\begin{aligned} \left| \int _{Y\setminus B(x',r)}Z(x',y) \,d\nu (y)\right| r^\beta 2^{-\beta } \le c \phi (r/2)\le c \phi (r) \end{aligned}$$

and thus the proof is complete. The last part of statements (i)–(iii) is an immediate consequence of the corresponding first parts. \(\Box \)

We note that the condition of existence of a implies that each point of X is an accumulation point for X.

3 An Application in Case Y is a Compact Differentiable Manifold

Since a compact manifold Y of class \(C^1\) that is embedded in \({{\mathbb {R}}}^n\) is \((n-1)\)-upper Ahlfors regular (cf. [17, Rmk. 2]), we now present an application of Proposition 2.6 (i) in case Y is a compact manifold of class \(C^1\) that is embedded in \({{\mathbb {R}}}^n\). To do so, we need the following elementary statement, that shows that Y satisfies the technical condition (2.7).

We also set

$$\begin{aligned} {{\mathbb {B}}}_{n}(x,\rho )\equiv \left\{ y\in {{\mathbb {R}}}^{n}:\,\vert x-y\vert <\rho \right\} \end{aligned}$$

for all \(\rho >0\), \( x\in {{\mathbb {R}}}^{n}\).

Proposition 3.1

Let \(n\in {\mathbb {N}}; n \ge 2\). Let Y be a compact manifold of class \(C^0\) that is embedded in \({{\mathbb {R}}}^n\) and of dimension m. Let \(m\ge 1\). 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 \(x'\in Y\) and \(\rho \in ]0,a[\), the set \(Y\cap \overline{{{\mathbb {B}}}_n(x',\rho )} \) 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 \({{\mathbb {R}}}^m\) that is not empty, A cannot be compact. Since \(Y\cap \overline{{{\mathbb {B}}}_n(x',\rho )} \) is compact, \(Y\cap \overline{{{\mathbb {B}}}_n(x',\rho )} \) cannot be equal to A and thus the set \(A\setminus (Y\cap \overline{{{\mathbb {B}}}_n(x',\rho )})\) cannot be empty. Since \(Y\cap {{\mathbb {B}}}_n(x',\rho )\) contains \(x'\), it is not empty. Since the set \(Y\cap {{\mathbb {B}}}_n(x',\rho )\) is open in A and is not empty and A is connected, we conclude that \(Y\cap \partial {{\mathbb {B}}}_n(x',\rho )\) cannot be empty and condition (2.7) with \(X=Y\) holds true.

Next we introduce the following. For the definition of tangential gradient \( {\textrm{grad}}_{Y}\), we refer e.g., to Kirsch and Hettlich [13, A.5], Chavel [1, Chap. 1]. For a proof, we refer to [16, Thm. 6.2]

Theorem 3.2

Let \(n\in {{\mathbb {N}}}\), \(n\ge 2\). Let Y be a compact manifold of class \(C^1\) that is embedded in \({{\mathbb {R}}}^n\). Let \(s_1\in [0,(n-1)[\). Let \(\beta \in ]0,1]\), \(t_1\in ]0,(n-1)+\beta [\). Let the kernel \(K\in {{\mathcal {K}}}_{s_1,s_1+1,1}(Y\times Y)\) satisfy the following assumptions

$$\begin{aligned}{} & {} K(\cdot ,y)\in C^1(Y\setminus \{y\}) \qquad \forall y\in Y\,, \qquad \int _YK(\cdot ,y)\,d\sigma _y\in C^1(Y) \,, \\{} & {} {\textrm{grad}}_{Y,x}K(\cdot ,\cdot )\in \left( {{\mathcal {K}}}_{t_1,Y\times Y} \right) ^n\,, \end{aligned}$$

where \({\textrm{grad}}_{ Y ,x}K(\cdot ,\cdot )\) denotes the tangential gradient of \(K(\cdot ,\cdot )\) with respect to the first variable. Let \(\mu \in C^{0,\beta }_b(Y)\). Then the function \(\int _YK(\cdot ,y)\mu (y)\,d\sigma _y\) is of class \(C^1(Y) \), the function \( [{\textrm{grad}}_{Y,x} K(x,y)](\mu (y)-\mu (x))\) is integrable in \(y\in Y\) for all \(x\in Y\) and formula

$$\begin{aligned}{} & {} { {\textrm{grad}}_{Y} \int _YK(x,y)\mu (y)\,d\sigma _y } \nonumber \\{} & {} \ \ =\int _Y[{\textrm{grad}}_{Y,x}K(x,y)](\mu (y)-\mu (x))\,d\sigma _y\nonumber \\{} & {} \quad +\mu (x){\textrm{grad}}_{Y} \int _{Y}K(x,y) \,d\sigma _y \,, \end{aligned}$$
(3.3)

for all \(x\in Y\), for the tangential gradient of \(\int _YK(\cdot ,y)\mu (y)\,d\sigma _y\) holds true.

By combining Proposition 1.8 (b) and Theorem 3.2, one can prove the following continuity Theorem 3.4 for the integral operator with kernel K and with values into a Schauder space on a compact manifold Y of class \(C^1\) (cf. [16, Thm. 6.3 (ii) (b)]). For the definition of the Schauder space \(C^{1,\beta }(Y)\) of functions \(\mu \) of class \(C^1\) on Y such that the tangential gradient of \(\mu \) is \(\beta \)-Hölder continuous or for an equivalent definition based on a finite family of parametrizations of Y, we refer for example to [2, §2.20].

Theorem 3.4

Let \(n\in {{\mathbb {N}}}\), \(n\ge 2\). Let Y be a compact manifold of class \(C^1\) that is embedded in \({{\mathbb {R}}}^n\). Let \(s_1\in [0,(n-1)[\). Let \(\beta \in ]0,1]\), \(t_1\in [\beta ,(n-1)+\beta [\), \(t_2\in [ \beta ,+\infty [\), \(t_3\in ]0,1]\). Let the kernel \(K\in {{\mathcal {K}}}_{s_1,s_1+1,1}(Y\times Y)\) satisfy the following assumption

$$\begin{aligned} K(\cdot ,y)\in C^1(Y\setminus \{y\}) \quad \forall y\in Y\,. \end{aligned}$$

Let \({\textrm{grad}}_{ Y ,x}K(\cdot ,\cdot )\) denote the tangential gradient of \(K(\cdot ,\cdot )\) with respect to the first variable. Let \(t_1=(n-1)\), \({\textrm{grad}}_{Y,x}K(\cdot ,\cdot )\in \left( {{\mathcal {K}}}^\sharp _{t_1,t_2,t_3}(Y\times Y) \right) ^n\), \(t_2-\beta >(n-1)\), \(t_2<(n-1)+\beta +t_3\) and

$$\begin{aligned} \int _YK(\cdot ,y)\,d\nu (y)\in C^{1,\min \{ \beta , (n-1)+t_3+\beta -t_2\} }(Y)\,. \end{aligned}$$

Then the map from \(C^{0,\beta }_b(Y)\) to \(C^{1,\min \{ \beta , (n-1)+t_3+\beta -t_2\}}_b(Y)\) that takes \(\mu \) to the function \(\int _YK(\cdot ,y)\mu (y)\,d\sigma _y\) is linear and continuous.

By combining Proposition 2.6 (i) and Theorem 3.2, we can now prove the following converse result of the continuity Theorem 3.4 with the additional restriction that \(t_2\le (n-1)+t_3\), a case in which \(C^{1,\min \{ \beta , (n-1)+t_3+\beta -t_2\} }(Y)=C^{1,\beta }(Y)\).

Theorem 3.5

Let \(n\in {{\mathbb {N}}}\), \(n\ge 2\). Let Y be a compact manifold of class \(C^1\) that is embedded in \({{\mathbb {R}}}^n\). Let \(s_1\in [0,(n-1)[\). Let \(\beta \in ]0,1]\), \(t_1\in [\beta ,(n-1)+\beta [\), \(t_2\in [ \beta ,+\infty [\), \(t_3\in ]0,1]\). Let the kernel \(K\in {{\mathcal {K}}}_{s_1,s_1+1,1}(Y\times Y)\) satisfy the following assumption

$$\begin{aligned} K(\cdot ,y)\in C^1(Y\setminus \{y\}) \quad \forall y\in Y\,. \end{aligned}$$

Let \({\textrm{grad}}_{ Y ,x}K(\cdot ,\cdot )\) denote the tangential gradient of \(K(\cdot ,\cdot )\) with respect to the first variable. Let \(t_1=(n-1)\), \({\textrm{grad}}_{Y,x}K(\cdot ,\cdot )\in \left( {{\mathcal {K}}}_{t_1,t_2,t_3}(Y\times Y) \right) ^n\), \(t_2-\beta >(n-1)\), \(t_2\le (n-1)+t_3\). If the map from \(C^{0,\beta }(Y)\) to \(C^{1,\beta }(Y)\) that takes \(\mu \) to the function \(\int _YK(\cdot ,y)\mu (y)\,d\sigma _y\) is linear and continuous, then inequality

$$\begin{aligned} \sup _{x\in Y}\sup _{r\in ]0,+\infty [} \left| \int _{Y\setminus {{\mathbb {B}}}_n(x,r)}{\textrm{grad}}_{Y,x}K(x,y)\,d\sigma _y \right| <+\infty \end{aligned}$$
(3.6)

holds true.

Proof

One can easily verify that Y is upper \((n-1)\)-Ahlfors regular with respect to Y (cf. [17, Rmk. 2]). By Proposition 3.1, Y satisfies condition (2.7). Since \(1\in C^{0,\beta }(Y)\), our assumption implies that

$$\begin{aligned} \int _YK(\cdot ,y)\,d\sigma _y\in C^{1,\beta }(Y)\,. \end{aligned}$$

By Theorem 3.2, the function \(\int _YK(\cdot ,y)\mu (y)\,d\sigma _y\) is of class \(C^1(Y)\) for all \(\mu \in C^{0,\beta }(Y)\) and formula (3.3) for the tangential gradient of \(\int _YK(\cdot ,y)\mu (y)\,d\sigma _y\) holds true. The continuity of the pointwise product in \(C^{0,\beta }(Y)\) ensures that the map from

$$\begin{aligned} C^{0,\beta } (Y)\qquad \text {to}\qquad C^{0,\beta }(Y)\,, \end{aligned}$$

which takes \(\mu \) to \(\mu (\cdot ) {\textrm{grad}}_{Y}\int _YK(\cdot ,y)\,d\sigma _y\) is linear and continuous. Then our assumption and formula (3.3) implies that that map from

$$\begin{aligned} C^{0,\beta } (Y)\qquad \text {to}\qquad C^{0,\beta }(Y)\,, \end{aligned}$$

which takes \(\mu \) to the function \(\int _Y[{\textrm{grad}}_{Y,x}K(x,y)](\mu (y)-\mu (x))\,d\sigma _y\) is linear and continuous. Since the kernel \(Z(x,y)\equiv {\textrm{grad}}_{Y,x}K(x,y)\) satisfy the assumptions of Proposition 2.6 (i), we conclude that inequality (1.9) holds true for \(X=Y\), \(Z(x,y)\equiv {\textrm{grad}}_{Y,x}K(x,y)\) and thus the proof is complete. \(\square \)