1 Introduction

One of the most useful tools to deal with non-homogeneous equations is of course the volume potential. For this reason many authors have investigated the mapping properties of operators of volume potential type in different functional settings and for several partial differential operators. While the elliptic framework is better understood, parabolic volume potentials are less investigated.

For example, it is well known that if F is a \(C^{0,\alpha }\)-vector field defined on a sufficiently regular bounded open subset of \({\mathbb {R}}^n\) then the Newtonian volume potential \({{\tilde{P}}}[\textrm{div} F]\), i.e. the volume potential associated with the Laplace operator and applied on \(\textrm{div}F\), is of class \(C^{1,\alpha }\) (see, e.g., Dalla Riva, Lanza de Cristoforis and Musolino [6]). This property allows to use the Newtonian volume potential \({{\tilde{P}}}[\cdot ]\) to deal with the Poisson equation when the non-homogeneous term is the distributional divergence of a \(C^{0,\alpha }\)-vector field F, that is

$$\begin{aligned} \Delta u = \textrm{div} F. \end{aligned}$$

The parabolic analog of the above equation is

$$\begin{aligned} \partial _t u-\Delta u = \textrm{div}\, G. \end{aligned}$$
(1)

Boundary value problems for equation (1) under several Hölder regularity assumptions on the vector field G were considered in Lieberman [14] and Lunardi and Vespri [16] with different techniques. Instead, to the best of the author knowledge, the classical parabolic theory does not cover the case

$$\begin{aligned} \partial _t u -\Delta u = \partial _t f, \end{aligned}$$
(2)

where f is a \(\frac{1+\alpha }{2}\)-Hölder continuous in time and \(\beta \)-Hölder continuous in space. Motivated by the above example, in the present paper we develop a theory for the volume potential \(P[\cdot ]\) associated with the heat operator acting on the space of distributions of the form \(\partial _t f\) and as a main result we prove a mapping property of \(P[\cdot ]\) (see Theorem 3.2).

As a consequence of our main result, we show how to solve the Dirichlet and Neumann problems for equation (2) (see Theorem 5.1). We note that in principle one could also try to deal with boundary value problems for equation (2) with a semigroup approach following, e.g., Lunardi [15]. However, our aim is to consider (2) from the point of view of potential theory and develop some tools that we also plan to exploit to analyze perturbation problems for the heat equation via potential theory.

For the classical results on elliptic volume potentials we mention Gilbarg and Trudinger [8] and Miranda [19]. We also note that a potential theoretic approach has recently revealed to be very effective to deal with elliptic problems in singularly perturbed domains. For this reason, mapping properties of elliptic volume potentials have been also considered in view of applications to perturbation problems (see Dalla Riva, Lanza de Cristoforis and Musolino [4, 5]). More details on the potential theoretic approach to perturbation problems for elliptic equations and results on volume potentials can be found in the monograph by Dalla Riva, Lanza de Cristoforis and Musolino [6].

For what concerns the parabolic case, regularity properties of the heat volume potential have been considered in Friedman [7, Ch. 1, Sec. 3]. Ladyženskaja, Solonnikov, and Ural’ceva [10, Ch. IV, Sec. 2, Sec. 3] proved a series of mapping properties of the heat volume potential in parabolic Schauder and Sobolev spaces. In Cherepova [2] the author considered the heat volume potential acting on parabolic Hölder continuous functions that are allowed to blow up at the parabolic boundary. Finally, Karazym and Suragan [9] have considered the volume potential associated with a degenerate parabolic equation. However we note that, up to the author knowledge, no results for the heat volume potential on spaces of distributions are available in the literature.

The results of the present paper continue the line of the works [11, 12, 18] on the properties of integral operators of potential type appearing in the framework of parabolic theory. Incidentally, we mention that our interest in proving these kind of mapping properties for the heat volume potential is also of technical nature since an equation of type (2) arises when one tries to pull-back the heat equation from a varying domain to a fixed one requiring only optimal regularity assumptions on the domains. In a subsequent paper by Dalla Riva and the author [3] we will indeed use the results of the present paper to prove domain perturbation results for layer heat potentials and for the heat equation.

We conclude this introduction with two possibile future developments stemming from this work. First, following the lines of this paper, one can try to extend the results on parabolic Sobolev spaces of [10, Ch. IV, Sec. 3] to distributions of the form \(\partial _t f\), where f has some (low) Sobolev regularity. Second, it would be of interest to generalize our results to volume potentials supported on time-varying domains and not only on parabolic cylinders (see e.g. Lewis and Murray [13] for the use of layer heat potentials on time-varying domains).

The paper is organized as follows. In Sect. 2 we introduce the functional spaces that we need, i.e. parabolic Schauder spaces. In Sect. 3, after having introduced the heat volume potential on functions and on a suitable space of distributions, we state our main result Theorem 3.2. Moreover, we show that the heat and Newtonian volume potentials coincide, up to the sign, whenever the densities are time-independent. Section 4 contains the proof of Theorem 3.2 together with some auxiliary results needed. Finally, in Sect. 5 we apply Theorem 3.2 to solve the Dirichlet and Neumann problems for the heat equation with a distributional non-homogeneous term which is the time derivative of an Hölder continuous function. For the clarity of exposition, we have postponed a known result of functional analysis regarding quotient spaces to Appendix A.

2 Schauder spaces

Throughout the paper we fix

$$\begin{aligned} n \in {\mathbb {N}} {\setminus } \{0,1\}, \end{aligned}$$

which represents the dimension of the space variable. Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\). Let \(k \in {\mathbb {N}}\) and \(\alpha \in \mathopen ]0,1[\). For the definition of sets and functions of the Schauder class \(C^{k,\alpha }\) we refer, e.g., to Gilbarg and Trudinger [8]. Next we pass to recall the definition of the parabolic analog of Schauder spaces. Let \(T \in \mathopen ]-\infty ,+\infty ]\). For the sake of brevity, we set

$$\begin{aligned} \Omega _T \equiv \overline{\mathopen ]-\infty ,T\mathclose [} \times \Omega , \qquad \partial _T \Omega \equiv \overline{\mathopen ]-\infty ,T\mathclose [} \times \partial \Omega . \end{aligned}$$

We now introduce the definition of an anisotropic Hölder space where Hölder regularity with respect to time and space directions can differ. Let \(\alpha , \beta \in \mathopen ]0,1[\). Then \(C^{\alpha ;\beta }(\overline{\Omega _T})\) denotes the space of bounded continuous functions u from \(\overline{\Omega _T}\) to \({{\mathbb {R}}}\) such that

$$\begin{aligned} \Vert u\Vert _{C^{\alpha ;\beta }(\overline{\Omega _T})} \equiv&\sup _{ \overline{\Omega _T} }\vert u \vert +\sup _{\begin{array}{c} t_{1},t_{2}\in \overline{\mathopen ]-\infty ,T[}\\ t_{1}\ne t_{2} \end{array} }\sup _{x \in {\overline{\Omega }}} \frac{\vert u(t_{1},x) -u(t_{2},x) \vert }{ \vert t_{1}-t_{2}\vert ^{\alpha }}\\ \nonumber&\quad +\sup _{t\in \overline{\mathopen ]-\infty ,T[}} \sup _{\begin{array}{c} x_1,x_2 \in {\overline{\Omega }}\\ x_{1}\ne x_{2} \end{array}} \frac{\vert u(t,x_1) -u(t,x_2) \vert }{\vert x_{1}-x_{2}\vert ^{\beta }}<+\infty . \end{aligned}$$

We also denote by \(C^{\alpha ;0}(\overline{\Omega _T})\) the space of bounded continuous functions u from \(\overline{\Omega _T}\) to \({{\mathbb {R}}}\) such that

$$\begin{aligned} \Vert u\Vert _{C^{\alpha ;0}(\overline{\Omega _T})} \equiv&\sup _{ \overline{\Omega _T} }\vert u\vert +\sup _{\begin{array}{c} t_{1},t_{2}\in \overline{\mathopen ]-\infty ,T[}\\ t_{1}\ne t_{2} \end{array} }\sup _{x \in \overline{\Omega }} \frac{\vert u(t_{1},x) -u(t_{2},x) \vert }{\vert t_{1}-t_{2}\vert ^{\alpha }}<+\infty . \end{aligned}$$

With the aim of considering boundary value problems, we will also need higher order parabolic Hölder spaces, i.e. parabolic Schauder spaces. Let \(\alpha \in \mathopen ]0,1[\). We denote by \(C^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _{T}})\) the space of bounded continuous functions u from \(\overline{{\Omega }_{T}}\) to \({{\mathbb {R}}}\) that are continuously differentiable with respect to the space variables and such that

$$\begin{aligned} \Vert u\Vert _{C^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _{T}})} \equiv&\sup _{ \overline{{\Omega }_{T}} }\vert u\vert +\sup _{\begin{array}{c} t_{1},t_{2}\in \overline{\mathopen ]-\infty ,T[}\\ t_{1}\ne t_{2} \end{array} }\sup _{x \in \overline{\Omega }} \frac{\vert u(t_{1},x) -u(t_{2},x) \vert }{\vert t_{1}-t_{2}\vert ^{\frac{1+\alpha }{2}}}\\ \nonumber&\quad +\sum _{i=1}^n\Vert \partial _{x_i}u\Vert _{C^{\frac{\alpha }{2};\alpha }(\overline{\Omega _{T}})}<+\infty . \end{aligned}$$

If \(\Omega \) is of class \(C^{1,\alpha }\), by local parametrizations it is possible to naturally define the space \(C^{\frac{1+\alpha }{2};1+\alpha }(\partial _T\Omega )\). For more detailed definitions of parabolic Schauder spaces we refer to Baderko [1, p. 450] and Ladyženskaja, Solonnikov, and Ural’ceva [10, p. 7] (see also [11, 12]).

Since we will consider the heat volume potential on a specific space of distributions, we need the following definition. We denote by \(C^{-1+\alpha ;\beta }(\overline{\Omega _{T}})\) the space of distributions in \(\Omega _T\) that are the (distributional) time derivative of a function in \(C^{\alpha ;\beta }(\overline{\Omega _{T}})\), endowed with the quotient norm. That is

$$\begin{aligned} C^{-1+\alpha ;\beta }(\overline{\Omega _{T}}) \equiv \left\{ \partial _t u: u \in C^{\alpha ;\beta }(\overline{\Omega _{T}}) \right\} \end{aligned}$$

and

$$\begin{aligned} \Vert f\Vert _{C^{-1+\alpha ;\beta }(\overline{\Omega _{T}})} \equiv \inf \left\{ \Vert u\Vert _{C^{\alpha ;\beta }(\overline{\Omega _{T}})}: u \in C^{\alpha ;\beta }(\overline{\Omega _{T}}), f = \partial _t u \right\} . \end{aligned}$$

It can be easily seen that all the above spaces endowed with their respective norms are Banach spaces (see Theorem A.1 of the Appendix A for the case of \(C^{-1+\alpha ;\beta }(\overline{\Omega _{T}})\)). Finally, when \(T >0\), with a subscript 0 in the above spaces we mean the Banach subspace made of functions that are zero before zero. For example,

$$\begin{aligned} C_0^{\alpha ;\beta }(\overline{\Omega _T}) \equiv \left\{ u \in C^{\alpha ;\beta }(\overline{\Omega _T}): u(t,\cdot )=0 \quad \forall t\le 0 \right\} . \end{aligned}$$

The spaces \(C^{\alpha ;0}_0(\overline{\Omega _T})\), \(C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _{T}})\), and \(C_0^{\frac{1+\alpha }{2};1+\alpha }(\partial _T\Omega )\) can be defined in the same way. Similarly

$$\begin{aligned} C_0^{-1+\alpha ;\beta }(\overline{\Omega _{T}}) \equiv \left\{ \partial _t u: u \in C^{\alpha ;\beta }(\overline{\Omega _{T}}), \, \textrm{supp}\,(\partial _t u) \subseteq [0,+\infty [ \times \overline{\Omega }\right\} . \end{aligned}$$

3 The heat volume potential and statement of the main result

Let \(S_n: {\mathbb {R}}^{1+n} {\setminus } \{0,0\} \rightarrow {\mathbb {R}}\) denote the fundamental solution of the heat operator, that is

$$\begin{aligned} S_{n}(t,x)\equiv {\left\{ \begin{array}{ll} \frac{1}{(4\pi t)^{\frac{n}{2}} }e^{-\frac{\vert x\vert ^{2}}{4t}}&{}\quad {\textrm{if}}\ (t,x) \in \mathopen ]0,+\infty \mathclose [ \times {\mathbb {R}}^n, \\ 0 &{}\quad {\textrm{if}}\ (t,x) \in \left( \mathopen ]-\infty ,0] \times {\mathbb {R}}^n\right) {\setminus } \{(0,0)\}. \end{array}\right. } \end{aligned}$$

As it is well know, \(S_n \in C^\infty ({\mathbb {R}}^{1+n}{\setminus } \{(0,0)\})\) and solves the heat equation in \({\mathbb {R}}^{1+n}{\setminus } \{(0,0)\}\). We recall a known bound for \(S_n\) which can be found e.g. in Ladyzhenskaja, Solonnikov and Ural’ceva [10, p. 274]: for all \(\eta \in {\mathbb {N}}^n\) and for all \(h\in {\mathbb {N}}\) there exists a constant \(C_{\eta ,h}>0\) such that

$$\begin{aligned} \left| D_x^\eta \partial _t^hS_n(t,x)\right| \le C_{\eta ,h} t^{-\frac{n}{2}-\frac{\vert \eta \vert }{2}-h}e^{-\frac{\vert x\vert ^2}{8t}} \qquad \forall (t,x)\in \mathopen ]0,+\infty \mathclose [ \times {\mathbb {R}}^n. \end{aligned}$$
(3)

Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) and \(T \in \mathopen ]-\infty ,+\infty ]\). Let \(x_0 \in \Omega \). If \(f \in L^\infty ( \Omega _T)\), we define the heat volume potential P[f] to be the function from \(\overline{\Omega _T}\) to \({\mathbb {R}}\) defined by

$$\begin{aligned} P[f](t,x) \equiv \int _{-\infty }^{+\infty }\int _\Omega \Big (S_n(t-\tau ,x-y)- \delta _{2,n}S_n(-\tau ,x_0&-y)\Big )f(\tau ,y)\,dyd\tau \nonumber \\&\quad \forall (t,x) \in \overline{\Omega _T}, \end{aligned}$$
(4)

where \(\delta _{i,j}\) denotes the Kronecker delta. We note that the above definition, in the case \(n=2\), depends on the choice of \(x_0 \in \Omega \). Indeed, a different choice of \(x_0\) would provide a volume potential that differs by a constant. However, if \(T \in \mathopen ]0,+\infty ]\) and \(\textrm{supp}\, f \subseteq \overline{[0,T[}\times \Omega \) (this is the case needed when one considers an initial-boundary value problem with initial condition at \(t=0\)), then the volume potential P[f] no longer depends on \(x_0\) in the case \(n=2\) and

$$\begin{aligned} P[f](t,x) = \int _{0}^t\int _\Omega S_n(t-\tau ,x-y)f(\tau ,y)\,dyd\tau \qquad \forall (t,x) \in \overline{\mathopen [0,T\mathclose [} \times \overline{\Omega }, \end{aligned}$$

which is the classical definition of heat volume potential.

Remark 1

The Definition 4 of the heat volume potential contains the term \(\delta _{2,n}S_n(-\tau ,x_0-y)\). This term is needed to avoid summability issues of the kernel as \(\tau \rightarrow -\infty \) in the case \(n=2\). Indeed \(S_2(t-\tau ,x-y)\) behaves like \((t-\tau )^{-1}\) as \(\tau \rightarrow -\infty \), while \(S_2(t-\tau ,x-y)- S_2(-\tau ,x_0-y)\) does not have the same problem. To see this fact, we fix \(\tau < \textrm{min}\{0,t\}\) and \(x,y \in \Omega \). One has

$$\begin{aligned} \Big \vert S_2(t-\tau ,x-y)&- S_2(-\tau ,x_0-y)\Big \vert \le \,\,\Big \vert S_2(t-\tau ,x-y)- S_2(t-\tau ,x_0-y)\Big \vert \\&\quad +\Big \vert S_2(t-\tau ,x_0-y)- S_2(-\tau ,x_0-y)\Big \vert . \end{aligned}$$

Then, if we denote by \(\{e_j\}_{j=1,\ldots ,n}\) the standard basis of \({\mathbb {R}}^n\), the fundamental theorem of calculus and the estimates (3) for the fundamental solution \(S_n\) imply that

$$\begin{aligned}&\Big \vert S_2(t-\tau ,x-y)&- S_2(t-\tau ,x_0-y)\Big \vert \\&\quad \le \sum _{j=1}^n\vert x_j-{x_0}_j\vert \int _0^1\Big \vert \partial _{x_j}S_2(t-\tau ,\lambda x+(1-\lambda )x_0-y) \Big \vert \,d\lambda \\&\quad \le \sum _{j=1}^nC_{e_j,0}\vert x_j-{x_0}_j\vert \frac{1}{(t-\tau )^{\frac{3}{2}}}\int _0^1 e^{-\frac{\vert \lambda x+(1-\lambda )x_0-y\vert ^2}{8(t-\tau )}} \,d\lambda \\&\quad \le \sum _{j=1}^nC_{e_j,0}\vert x_j-{x_0}_j\vert \frac{1}{(t-\tau )^{\frac{3}{2}}} \end{aligned}$$

and that

$$\begin{aligned} \Big \vert S_2(t-\tau ,x_0-y)&- S_2(-\tau ,x_0-y)\Big \vert \le \vert t\vert \int _0^1\Big \vert \partial _t S_2(\lambda t-\tau ,x_0-y)\Big \vert \,d\lambda \\&\le C_{0,1}\vert t\vert \int _0^1 \frac{1}{\vert \lambda t-\tau \vert ^{2}}e^{-\frac{\vert x_0-y\vert ^2}{8(\lambda t -\tau )}}\,d\lambda \\&\le C_{0,1}\vert t\vert \frac{1}{\vert \min \{0,t\}-\tau \vert ^{2}}, \end{aligned}$$

which show that the kernel of (4) is summable for \(\tau \rightarrow -\infty \).

The goal of the present paper is to prove some properties of the heat volume potential acting on the space of distributions \(C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\). As a first step we need to specify what we mean by the heat volume potential on such a space (see Definition (7)). To this aim, we need to shows that the Definition (7) does not depend on the representative in the equivalence class and coincides with the Definition (4) when the distribution comes from a \(L^\infty \) function. We do this in the following Proposition 3.1 which we shall prove in Sect. 4.

Proposition 3.1

Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) and \(T \in \mathopen ]-\infty ,+\infty ]\). Let \(\alpha ,\beta \in \mathopen ]0,1]\). Let \(f \in C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\). Then the following statements hold.

  1. (i)

    Suppose that \(\partial _t f =0\) in the sense of distributions. Then

    $$\begin{aligned}&\int _{-\infty }^t\int _{\Omega }\partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dyd\tau =0 \qquad \forall (t,x) \in \overline{\Omega _T}. \end{aligned}$$
    (5)
  2. (ii)

    Suppose that \(\partial _t f \in L^\infty {(\Omega _T)}\). Then

    $$\begin{aligned} P[\partial _t f](t,x)=\int _{-\infty }^t\int _{\Omega }\partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dyd\tau \qquad \forall (t,x) \in \overline{\Omega _T}. \end{aligned}$$
    (6)

By Proposition 3.1 we can extend the heat volume potential \(P[\cdot ]\) to include distributions in \(C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\) via

$$\begin{aligned} P[g](t,x) \equiv \int _{-\infty }^t\int _{\Omega }\partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dyd\tau \qquad \forall (t,x) \in \overline{\Omega _T}, \end{aligned}$$
(7)

where \(g = \partial _t f\) and \(f \in C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\).

We are now ready to state our main result, whose proof is postponed to Sect. 4.

Theorem 3.2

Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) and \(T \in \mathopen ]-\infty ,+\infty ]\). Let \(\alpha ,\beta \in \mathopen ]0,1[\). Then the following statements hold.

  1. (i)

    If \(g \in C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\), then \(\partial _t P[g] - \Delta P[g] = g\) in the sense of distributions in \(\Omega _T\).

  2. (ii)

    \(P[\cdot ]\) is a bounded linear operator from \(C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\) to \(C^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\).

In order to prove Theorem 3.2 will also need the Newtonian volume potential and then we recall that the standard fundamental solution of the Laplace operator is

$$\begin{aligned} \tilde{S}_n(x) \equiv {\left\{ \begin{array}{ll} \frac{1}{s_2}\log \vert x \vert &{}\forall x \in \mathbb {R}^n\setminus \{0\}, \quad \text{ if } n=2,\\ \frac{1}{(2-n) s_n}\vert x \vert ^{2-n}&{}\forall x \in \mathbb {R}^n\setminus \{0\},\quad \text{ if } n \ge 3, \end{array}\right. } \end{aligned}$$

where

$$\begin{aligned} s_n \equiv \frac{2\pi ^\frac{n}{2}}{\Gamma \left( \frac{n}{2}\right) } \end{aligned}$$

is the \((n-1)\)-dimensional measure of the unit sphere \(\partial {\mathbb {B}}_n(0,1)\) and \(\Gamma \) denotes the Euler Gamma function. If \(h \in L^\infty (\Omega )\), the harmonic volume potential \({{\tilde{P}}}[h]\) is the function from \(\overline{\Omega }\) to \({\mathbb {R}}\) defined by

$$\begin{aligned} {{\tilde{P}}}[h](x) \equiv \int _\Omega {{\tilde{S}}}_n(x-y)h(y)\,dy \qquad \forall x \in \overline{\Omega }. \end{aligned}$$

Likewise other potential-type operators, the heat volume potential of an autonomous (i.e. time-independent) density is autonomous and coincides up to the sign with the corresponding Newtonian volume potential. That is, we have the following.

Lemma 3.3

Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) and \(T \in \mathopen ]-\infty ,+\infty ]\). Let \(x_0 \in \Omega \). Let \(h \in L^\infty (\Omega )\). Then

$$\begin{aligned} P[h](t,x) = -\big ({{\tilde{P}}}[h](x) - \delta _{2,n}{{\tilde{P}}}[h](x_0)\big ) \qquad \forall (t,x) \in \overline{\Omega _T}. \end{aligned}$$

Proof

We follow the lines of the proof of [17, Lemma A.3, Lemma A.4] where the analog relation between heat and harmonic layer potentials has been proved. Let \((t,x) \in \overline{\Omega _T}\). We first consider the case \(n=2\). Then

$$\begin{aligned} P[h](t,x) =&\int _{-\infty }^{+\infty }\int _\Omega \Big (S_2(t-\tau ,x-y)- S_2(-\tau ,x_0-y)\Big )h(y)\,dyd\tau \\ =&\lim _{\sigma \rightarrow + \infty } \int _{-\sigma }^{+\infty } \int _{\Omega } \Big (S_2(t-\tau ,x-y)- S_2(-\tau ,x_0-y)\Big ) h(y) \, dy d\tau . \end{aligned}$$

By the changes of variable \(t-\tau = \frac{\vert x-y\vert ^2}{4\xi }\) in the first term inside the integral and \(-\tau = \frac{\vert x_0-y\vert ^2}{4\xi }\) in the second term, we get

$$\begin{aligned} P[h](t,x)&= \lim _{\sigma \rightarrow +\infty } \bigg \{ \int _{\Omega }\int _{\frac{\vert x-y\vert ^2}{4(t+\sigma )}}^{+\infty } \frac{1}{4\pi \xi }e^{-\xi }h(y) \, d\xi dy -\int _{\Omega }\int _{\frac{\vert x_0-y\vert ^2}{4\sigma }}^{+\infty } \frac{1}{4\pi \xi }e^{-\xi }h(y) \, d\xi dy \bigg \} \\&= \lim _{\sigma \rightarrow +\infty } \int _{\Omega }\int _{\frac{\vert x-y\vert ^2}{4(t+\sigma )}}^{\frac{\vert x_0-y\vert ^2}{4\sigma }}\frac{1}{4\pi \xi }e^{-\xi }h(y) \, d\xi dy. \end{aligned}$$

Let g be the function from \({\mathbb {R}}\) to \({\mathbb {R}}\) defined by

$$\begin{aligned} g(\xi ) \equiv {\left\{ \begin{array}{ll} \frac{e^{-\xi }-1}{-\xi } &{} \text{ if } \xi \ne 0,\\ 1 &{} \text{ if } \xi = 0. \end{array}\right. } \end{aligned}$$

It is easy to see that g is continuous in \({\mathbb {R}}\) and that

$$\begin{aligned} \xi ^{-1} e^{-\xi } = \xi ^{-1} - g(\xi ) \qquad \forall \, \xi \in {\mathbb {R}} {\setminus } \{0\}. \end{aligned}$$

Accordingly, the dominated convergence theorem implies that

$$\begin{aligned} P[h](t,x)&= \lim _{\sigma \rightarrow +\infty } \bigg \{ \int _{\Omega }\int _{\frac{\vert x-y\vert ^2}{4(t+\sigma )}}^{\frac{\vert x_0-y\vert ^2}{4\sigma }}\frac{1}{4\pi \xi } \, d\xi \,h(y)dy -\int _{\Omega }\int _{\frac{\vert x-y\vert ^2}{4(t+\sigma )}}^{\frac{\vert x_0-y\vert ^2}{4\sigma }}\frac{g(\xi )}{4\pi } d\xi \,h(y)dy\bigg \}\\&= \lim _{\sigma \rightarrow +\infty } \bigg \{ \int _{\Omega }\frac{1}{4\pi }\log \left| \frac{\vert x_0-y\vert ^2}{4\sigma }\frac{4(t+\sigma )}{\vert x-y\vert ^2}\right| h(y)\,dy\\&\hspace{5cm}-\int _{\Omega }\int _{\frac{\vert x-y\vert ^2}{4(t+\sigma )}}^{\frac{\vert x_0-y\vert ^2}{4\sigma }}\frac{g(\xi )}{4\pi } d\xi \,h(y)dy\bigg \}\\&= \int _{\Omega }\frac{1}{4\pi }\log \left( \frac{\vert x_0-y\vert ^2}{\vert x-y\vert ^2}\right) h(y) \,dy\\&= \int _{\Omega }\frac{1}{2\pi }\log (\vert x_0-y\vert ) \, h(y)dy - \int _{\Omega }\frac{1}{2\pi }\log (\vert x-y\vert ) \, h(y)dy\\&=-\big ({{\tilde{P}}}[h](x) - {{\tilde{P}}}[h](x_0)\big ). \end{aligned}$$

Next we pass to the case \(n \ge 3\). By the change of variable \(\vert x-y\vert ^2s = 4(t-\tau )\) we have that

$$\begin{aligned} P[h](t,x)&= \int _{-\infty }^{t} \int _{\Omega }\frac{1}{(4\pi (t-\tau ))^\frac{n}{2}}e^{-\frac{\vert x-y\vert ^2}{4(t-\tau )}}h(y)\, dy d\tau \\&=\; \frac{1}{4\pi ^\frac{n}{2}}\int _{0}^{+\infty }s^{-\frac{n}{2}}e^{-\frac{1}{s}}\,ds \int _{\Omega }\frac{1}{\vert x-y\vert ^{n-2}}h(y)\, dy \\&=\;\frac{1}{4\pi ^\frac{n}{2}}\Gamma \left( \frac{n}{2}-1\right) \int _{\Omega }\frac{1}{\vert x-y\vert ^{n-2}}h(y)\, dy \\&=\;\frac{1}{(n-2) s_n} \int _{\Omega }\frac{1}{\vert x-y\vert ^{n-2}}h(y)\, dy \\&=\;-{{\tilde{P}}}[h](x), \end{aligned}$$

which proves the statement. \(\square \)

4 Proof of the main result

In the present section we consider the action of \(P[\cdot ]\) in the space \(C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\) with the final goal of proving Theorem 3.2. To this aim, we need some preliminary results.

Proposition 4.1

Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) and \(T \in \mathopen ]-\infty ,+\infty ]\). Let \(\alpha \in \mathopen ]0,1[\). Then the operator B defined by

$$\begin{aligned} B[f](t,x) \equiv \int _{-\infty }^t\int _{\Omega } \partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dyd\tau \quad \forall (t,x)\in \overline{\Omega _T} \end{aligned}$$

is linear and continuous from \(C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})\) to \(C^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\).

Proof

For convenience, for \(\gamma >0\) we set

$$\begin{aligned} K_\gamma \equiv \sup _{x \in {\overline{\Omega }}} \,\int _{\Omega }\frac{dy}{\vert x-y\vert ^{n-\gamma }}. \end{aligned}$$

It is easily seen that for \(\gamma >0\) one has

$$\begin{aligned} K_\gamma < +\infty . \end{aligned}$$
(8)

Let \((t,x) \in \overline{\Omega _T}\). By (3) there exists a constant \(C_{0,1}>0\) such that

$$\begin{aligned}&\int _{-\infty }^t \int _{\Omega } \Big \vert \partial _t S_n(t-\tau ,x-y)\big (f(\tau ,y)-f(t,y)\big )\Big \vert \,dyd\tau \nonumber \\&\quad \le C_{0,1} \;\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{-\infty }^t \int _{\Omega } (t-\tau )^{-\frac{n}{2}-1+\frac{1+\alpha }{2}} e^{-\frac{\vert x-y\vert ^2}{8(t-\tau )}}\;dyd\tau \nonumber \\ {}&\quad = 8^{\frac{n-1-\alpha }{2}}C_{0,1}\Vert f\Vert _{ C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{0}^{+\infty } s^{-\frac{n}{2}-1+\frac{1+\alpha }{2}} e^{-\frac{1}{s}}\;ds \int _{\Omega } \frac{1}{\vert x-y\vert ^{n-1-\alpha }} \;dy\nonumber \\&\quad \le 8^{\frac{n-1-\alpha }{2}}C_{0,1}K_{1+\alpha }\Gamma \left( \frac{n-1-\alpha }{2}\right) \;\Vert f\Vert _{ C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}. \end{aligned}$$

Then, by the above inequality and by the Vitali convergence theorem, \(B[\cdot ]\) is linear and continuous from \(C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})\) to \(C^{0}(\overline{\Omega _T})\).

Next we consider the Hölder continuity in the time variable. We take \(t',t'' \in \mathopen ]-\infty ,T[\), \(t' < t''\), \(x \in \Omega \). Then

$$\begin{aligned} \big \vert B[f](t',x) - B[f](t'',x) \big \vert&\le \left| \int _{t'-2\vert t''-t'\vert }^{t'} \int _{\Omega }\partial _t S_n(t'-\tau ,x-y)\big (f(\tau ,y)-f(t',y)\big )\,dyd\tau \right| \nonumber \\&\quad +\left| \int _{t'-2\vert t''-t'\vert }^{t''} \int _{\Omega }\partial _t S_n(t''-\tau ,x-y)\big (f(\tau ,y)-f(t'',y)\big )\,dyd\tau \right| \nonumber \\&\quad +\Bigg \vert \int _{-\infty }^{t'-2\vert t''-t'\vert } \int _{\Omega }\Big (\partial _t S_n(t'-\tau ,x-y)-\partial _t S_n(t''-\tau ,x-y)\Big )\nonumber \\&\quad \times \Big (f(\tau ,y)-f(t',y)\Big )\,dyd\tau \Bigg \vert \nonumber \\&\quad +\left| \int _{-\infty }^{t'-2\vert t''-t'\vert } \int _{\Omega }\partial _t S_n(t''-\tau ,x-y)\big (f(t'',y)-f(t',y)\big )\,dyd\tau \right| . \end{aligned}$$
(9)

We begin considering the first term in the right hand side of (9). The bounds (3) on \(S_n\) imply

$$\begin{aligned}&\left| \int _{t'-2\vert t''-t'\vert }^{t' } \int _{\Omega }\partial _t S_n(t'-\tau ,x-y)\big (f(\tau ,y)-f(t',y)\big )\,dyd\tau \right| \\ \nonumber&\quad \le C_{0,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{t'-2\vert t''-t'\vert }^{t' } \int _{\Omega }(t'-\tau )^{-\frac{n}{2}-\frac{1}{2}+\frac{\alpha }{2}}e^{-\frac{\vert x-y\vert ^2}{8(t'-\tau )}}\,dyd\tau \nonumber \\&\quad \le C_{0,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\int _{t'-2\vert t''-t'\vert }^{t' }(t'-\tau )^{-\frac{n}{2}-\frac{1}{2}+\frac{\alpha }{2}} \int _{{\mathbb {R}}^n}e^{-\frac{\vert x-y\vert ^2}{8(t'-\tau )}}\,dyd\tau \nonumber \\&\quad = (8\pi )^{\frac{n}{2}}C_{0,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\int _{t'-2\vert t''-t'\vert }^{t' }(t'-\tau )^{-\frac{1}{2}+\frac{\alpha }{2}} d\tau \nonumber \\&\quad = (8\pi )^{\frac{n}{2}}\frac{2^{\frac{3+\alpha }{2}}}{1+\alpha } C_{0,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert ^{\frac{1+\alpha }{2}}. \end{aligned}$$

The second term in the right hand side of (9) can be estimated in the same way by noting that \(t'-2\vert t''-t'\vert = t''-3\vert t''-t'\vert \). We then consider the third term. By (3) together with the mean value theorem one can see that there exists a constant \(C'_{0,1}>0\) such that

$$\begin{aligned} \left| \partial _t S_n(t'-\tau ,x-y)-\partial _t S_n(t''-\tau ,x-y) \right| \le C'_{0,1}\frac{\vert t'-t''\vert }{\vert t'-\tau \vert ^{\frac{n}{2}+2}} e^{-\frac{\vert x-y\vert ^2}{8(t'-\tau )}} \end{aligned}$$
(10)

for all \(x,y \in \Omega \), \(t'<t''\), \(\tau < t'-2\vert t''-t'\vert \). For an explicit derivation of (10) we refer to [11, Lem. 4.3 (iii)]. Then

$$\begin{aligned}&\Bigg \vert \int _{-\infty }^{t'-2\vert t''-t'\vert } \int _{\Omega }\Big (\partial _t S_n(t'-\tau ,x-y)-\partial _t S_n(t''-\tau ,x-y)\Big )\times \Big (f(\tau ,y)-f(t',y)\Big )\,dyd\tau \Bigg \vert \nonumber \\&\quad \le C'_{0,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert \int _{-\infty }^{t'-2\vert t''-t'\vert } (t'-\tau )^{-\frac{n}{2}-2+\frac{1+\alpha }{2}}\int _{\Omega } e^{-\frac{\vert x-y\vert ^2}{8(t'-\tau )}}\;dyd\tau \nonumber \\&\quad \le C'_{0,1}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert \int _{-\infty }^{t'-2\vert t''-t'\vert } (t'-\tau )^{\frac{-n-3+\alpha }{2}}\int _{{\mathbb {R}}^n} e^{-\frac{\vert x-y\vert ^2}{8(t'-\tau )}}\;dyd\tau \\&\quad = (8\pi )^{\frac{n}{2}} C'_{0,1}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert \int _{-\infty }^{t'-2\vert t''-t'\vert } (t'-\tau )^{\frac{-3+\alpha }{2}} d\tau \nonumber \\&\quad = \frac{2^{\frac{1+\alpha }{2}}}{1-\alpha } (8\pi )^{\frac{n}{2}} C'_{0,1}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \vert t''-t'\vert ^{\frac{1+\alpha }{2}}. \end{aligned}$$

Next, we consider the last term in the right hand side of (9).

$$\begin{aligned}&\left| \int _{-\infty }^{t'-2\vert t''-t'\vert } \int _{\Omega }\partial _t S_n(t''-\tau ,x-y)\big (f(t'',y)-f(t',y)\big )\,dyd\tau \right| \nonumber \\ {}&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert ^{\frac{1+\alpha }{2}} \int _{\Omega }\left| \int _{-\infty }^{t'-2\vert t''-t'\vert } \partial _t S_n(t''-\tau ,x-y) \,d\tau \right| dy \nonumber \\ {}&\quad = \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert ^{\frac{1+\alpha }{2}} \int _{\Omega }\Big \vert S_n(3(t''-t'),x-y)\Big \vert \,dy \nonumber \\ {}&\quad = \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert ^{\frac{1+\alpha }{2}} \int _{\Omega }\frac{1}{(12 \pi (t''-t'))^\frac{n}{2}}e^{-\frac{\vert x-y\vert ^2}{12(t''-t')}}\,dy \nonumber \\ {}&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}\vert t''-t'\vert ^{\frac{1+\alpha }{2}}. \end{aligned}$$

It remains consider the regularity in the space variables. More precisely, we need to prove that for all \(i \in \{1,\ldots ,n\}\), the map \(\partial _{x_i}B[\cdot ]\) is linear and continuous from \(C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})\) to \(C^{\frac{\alpha }{2};\alpha }(\overline{\Omega _T})\). Let \((t,x) \in \Omega _T\). Let \(i \in \{1,\ldots ,n\}\). By classical differentiation theorems for integrals depending on a parameter and by the bound (3) there exists a constant \(C_{e_i,1}>0\) such that

$$\begin{aligned} \vert \partial _{x_i}B[f](t,x)\vert&\le \;C_{e_i,1}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{-\infty }^t \int _{\Omega } (t-\tau )^{-\frac{n}{2}-\frac{3}{2}+\frac{1+\alpha }{2}} e^{-\frac{\vert x-y\vert ^2}{8(t-\tau )}}\;dyd\tau \\&= \;8^{\frac{n-\alpha }{2}}C_{e_i,1}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{0}^{+\infty } u^{-\frac{n}{2}-1+\frac{\alpha }{2}} e^{-\frac{1}{u}}\;d\tau \int _{\Omega } \frac{1}{\vert x-y\vert ^{n-\alpha }} \;dy\\&\le \;8^{\frac{n-\alpha }{2}}C_{e_i,1}K_\alpha \Gamma \left( \frac{n-\alpha }{2}\right) \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}. \end{aligned}$$

For what concerns the time \(\frac{\alpha }{2}\)-Hölder norm, it can be estimated exactly in the same way of the first part of the proof just by noting that the kernel is more singular by a term \((t-\tau )^{-\frac{1}{2}}\) but we need to obtain an \(\frac{\alpha }{2}\)-Hölder regularity instead of an \(\frac{1+\alpha }{2}\)-regularity.

Now we consider the spatial \(\alpha \)-Hölder regularity. Let \(x',x'' \in \Omega \), \(t \in \mathopen ]-\infty ,T[\). By classical differentiation theorems for integrals depending on a parameter we have

$$\begin{aligned}&\vert \partial _{x_i}B[f](t,x')-\partial _{x_i}B[f](t,x'')\vert \nonumber \\ {}&\quad = \bigg \vert \int _{0}^{+\infty } \int _{\Omega } \partial _{x_i}\partial _{t}S_n(\tau ,x'-y)\big (f(t-\tau ,y)-f(t,y)\big )\,dyd\tau \nonumber \\ {}&\qquad - \int _{0}^{+\infty } \int _{\Omega } \partial _{x_i}\partial _{t}S_n(\tau ,x''-y)\big (f(t-\tau ,y)-f(t,y)\big )\,dyd\tau \bigg \vert \nonumber \\ {}&\quad \le \left| \int _{0}^{\vert x'-x''\vert ^2} \int _{\Omega } \partial _{x_i}\partial _{t}S_n(\tau ,x'-y)\big (f(t-\tau ,y)-f(t,y)\big )\,dyd\tau \right| \nonumber \\ {}&\qquad + \left| \int _{0}^{\vert x'-x''\vert ^2} \int _{\Omega } \partial _{x_i}\partial _{t}S_n(\tau ,x''-y)\big (f(t-\tau ,y)-f(t,y)\big )\,dyd\tau \right| \nonumber \\&\qquad + \Bigg \vert \int _{\vert x'-x''\vert ^2}^{+\infty } \int _{\Omega } \Big (\partial _{x_i}\partial _{t}S_n(\tau ,x'-y)-\partial _{x_i}\partial _{t}S_n(\tau ,x''-y)\Big )\\ \nonumber&\qquad \times \Big (f(t-\tau ,y)-f(t,y)\Big )\,dyd\tau \Bigg \vert . \end{aligned}$$
(11)

We consider the first term in the right hand side of (11). By the bound (3) we have

$$\begin{aligned}&\left| \int _{0}^{\vert x'-x''\vert ^2} \int _{\Omega } \partial _{x_i}\partial _{t}S_n(\tau ,x'-y)\big (f(t-\tau ,y)-f(t,y)\big )\,dyd\tau \right| \\&\qquad \le C_{e_i,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{0}^{\vert x'-x''\vert ^2} \int _{\Omega } \tau ^{-\frac{n}{2}-\frac{3}{2}+\frac{1+\alpha }{2}} e^{-\frac{\vert x'-y\vert ^2}{8\tau }}\,dyd\tau \\&\qquad \le (8\pi )^\frac{n}{2} C_{e_i,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{0}^{\vert x'-x''\vert ^2} \tau ^{\frac{\alpha -2}{2}} \,d\tau \\&\qquad =\frac{2}{\alpha }(8\pi )^\frac{n}{2} C_{e_i,1} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \vert x'-x''\vert ^\alpha . \end{aligned}$$

The second term in the right hand side of (11) can be estimated in the same way. Finally we consider the last term. The fundamental theorem of calculus and the bound (3) imply

$$\begin{aligned}&\Bigg \vert \int _{\vert x'-x''\vert ^2}^{+\infty } \int _{\Omega } \Big (\partial _{x_i}\partial _{t}S_n(\tau ,x'-y)-\partial _{x_i}\partial _{t}S_n(\tau ,x''-y)\Big )\\&\qquad \times \Big (f(t-\tau ,y)-f(t,y)\Big )\,dyd\tau \Bigg \vert \\&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{\vert x'-x''\vert ^2}^{+\infty } \tau ^\frac{1+\alpha }{2} \\&\qquad \times \int _{\Omega } \left| \big (\partial _{x_i}\partial _{t}S_n(\tau ,x'-y)-\partial _{x_i}\partial _{t}S_n(\tau ,x''-y)\right| \,dyd\tau \\&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \sum _{j=1}^n\vert x_j'-x''_j\vert \\&\qquad \times \int _0^1\int _{\vert x'-x''\vert ^2}^{+\infty } \tau ^\frac{1+\alpha }{2} \int _{\Omega } \left| \partial _{x_j}\partial _{x_i}\partial _{t}S_n(\tau ,\lambda x'+ (1-\lambda ) x''-y) \right| \,dyd\tau d\lambda \\&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \sum _{j=1}^nC_{e_i+e_j,1}\vert x_j'-x''_j\vert \\&\qquad \times \int _0^1\int _{\vert x'-x''\vert ^2}^{+\infty } \tau ^{\frac{n}{2}-2+\frac{1+\alpha }{2}}\int _{\Omega } e^{-\frac{\vert \lambda x'+ (1-\lambda ) x''-y\vert ^2}{8\tau }} \,dyd\tau d\lambda \\&\quad \le (8\pi )^\frac{n}{2} \Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \sum _{j=1}^nC_{e_i+e_j,1}\vert x_j'-x''_j\vert \int _{\vert x'-x''\vert ^2}^{+\infty } \tau ^{\frac{-3+\alpha }{2}} \,d\tau \\&\quad \le \frac{ 2(8\pi )^\frac{n}{2}}{1-\alpha } \sum _{j=1}^nC_{e_i+e_j,1}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \vert x'-x''\vert ^{\alpha }. \end{aligned}$$

\(\square \)

Next we show that the operator B[f] of Proposition 4.1 coincides with \(\partial _t P[f]\) whenever \(f \in C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\).

Lemma 4.2

Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) and \(T \in \mathopen ]-\infty ,+\infty ]\). Let \(\alpha ,\beta \in \mathopen ]0,1[\). Let \(f \in C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\). Then P[f] is continuously differentiable with respect to t and

$$\begin{aligned} \partial _tP[f](t,x) = \int _{-\infty }^t\int _{\Omega } \partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dyd\tau \quad \forall (t,x) \in {\Omega _T}. \end{aligned}$$

Proof

Let \((t,x) \in \Omega _T\). Since f is \(\beta \)-Hölder continuous in space, by Friedman [7, Thm. 9 p. 21] the volume potential P[f] is continuously differentiable with respect to the time variable and twice continuously differentiable with respect to the space variables. Moreover, again by Friedman [7, Thm. 9 p. 21], one has

$$\begin{aligned} \partial _tP[f](t,x) = f(t,x) + \Delta P[f](t,x) . \end{aligned}$$

By the properties of the Newtonian volume potential (see e.g. Gilbarg and Trudinger [8, Sec. 4.2]), we have

$$\begin{aligned} \Delta {{\tilde{P}}}[f(t,\cdot )](x) = f(t,x). \end{aligned}$$

Since by Lemma 3.3 the heat and Newtonian volume potential coincide up to the sign, we have that

$$\begin{aligned} \partial _t P[f](t,x) = \Delta \Big (P[f](t,x) - P[f(t,\cdot )](t,x) \Big ) = \Delta P[f-f(t,\cdot )](t,x). \end{aligned}$$

The bound (3) for the fundamental solution and (8) imply that

$$\begin{aligned}&\int _{-\infty }^t \int _{\Omega } \Big \vert \Delta S_n(t-\tau ,x-y)\big (f(\tau ,y)-f(t,y)\big )\Big \vert \,dyd\tau \nonumber \\ {}&\quad \le \;\sum _{j=1}^nC_{2e_j,0}\Vert f\Vert _{C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \int _{-\infty }^t \int _{\Omega } (t-\tau )^{-\frac{n}{2}-1+\frac{1+\alpha }{2}} e^{-\frac{\vert x-y\vert ^2}{8(t-\tau )}}\;dyd\tau \\\nonumber&\quad \le \;8^\frac{n-1+\alpha }{2} \sum _{j=1}^nC_{2e_j,0}\Vert f\Vert _{ C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})} \\&\qquad \times \int _{0}^{+\infty } u^{-\frac{n}{2}-1+\frac{1+\alpha }{2}} e^{-\frac{1}{u}}\;d\tau \int _{\Omega } \frac{1}{\vert x-y\vert ^{n-1-\alpha }} \;dy\\\nonumber&\quad \le \;8^\frac{n-1+\alpha }{2} \sum _{j=1}^nC_{2e_j,0}K_{1+\alpha }\Gamma \left( \frac{n-1-\alpha }{2}\right) \Vert f\Vert _{ C^{\frac{1+\alpha }{2};0}(\overline{\Omega _T})}. \end{aligned}$$

Accordingly the statement follows by standard differentiation theorems for integral depending on a parameter and by recalling that \(S_n\) solves the heat equation in \({\mathbb {R}}^{1+n} {\setminus } \{(0,0)\}\). \(\square \)

Now we are ready to prove Proposition 3.1, which shows that the heat volume potential in \( C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\) is well-defined.

Proof of Proposition 3.1

We first consider statement i). With the notation introduced in Proposition 4.1 we need to show that

$$\begin{aligned} B[f] = 0 \qquad \text{ in } \,\overline{\Omega _T}. \end{aligned}$$

Since by Proposition 4.1 the map B[f] is continuous in \(\overline{\Omega _T}\), it suffices to show equality (5) in \(\Omega _T\). Let \((t,x) \in \Omega _T\) be fixed. Since

$$\begin{aligned} S_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y)) \end{aligned}$$

is continuous in \((\tau ,y) \in \overline{\Omega _t} {\setminus } \{(t,x)\}\), it has a distributional \(\tau \)-derivative which, since \(\partial _tf=0\) in the sense of distributions, equals

$$\begin{aligned} g(\tau ,y) \equiv -\partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y)) \qquad (\tau ,y) \in {\Omega _t} {\setminus } \{(t,x)\}. \end{aligned}$$

Let \(\varepsilon >0\). Since the function g is continuous in \(\overline{\Omega _{t-\varepsilon }}\), then

$$\begin{aligned}&-\int _{-\infty }^{t-\varepsilon }\int _{\Omega } \partial _tS_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dyd\tau \\&\quad =\int _{\Omega } S_n(\varepsilon ,x-y)(f(t-\varepsilon ,y)-f(t,y))\,dy\\&\qquad -\lim _{\tau \rightarrow -\infty }\int _{\Omega } S_n(t-\tau ,x-y)(f(\tau ,y)-f(t,y))\,dy\\&\quad =\int _{\Omega } S_n(\varepsilon ,x-y)(f(t-\varepsilon ,y)-f(t,y))\,dy. \end{aligned}$$

For all \(y \in \Omega \) we have

$$\begin{aligned}&\Big \vert S_n(\varepsilon ,x-y)(f(t-\varepsilon ,y)-f(t,y))\Big \vert \\&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})} \frac{1}{(4\pi )^{\frac{n}{2}}}\varepsilon ^{\frac{-n+1+\alpha }{2}}e^{-\frac{\vert x-y\vert ^2}{4\varepsilon }}\\&\quad \le \Vert f\Vert _{C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})}\left( \sup _{\xi >0}\xi ^{\frac{-n+\alpha }{2}}e^{-\frac{1}{4\xi }}\right) \frac{\varepsilon ^\frac{1}{2}}{\vert x-y\vert ^{n-\alpha }}. \end{aligned}$$

By the dominated convergence theorem we obtain (5) by letting \(\varepsilon \rightarrow 0\).

Statement ii) is a direct consequence of integration by parts in the time variable. \(\square \)

Now we have all the ingredients to prove our main result on the action of \(P[\cdot ]\), that is Theorem 3.2.

Proof of Theorem 3.2

We start considering i). If \(g \in C^{\frac{-1+\alpha }{2};\beta }(\overline{\Omega _T})\) and \(g = \partial _t f\) with \(f \in C^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\), then by Lemma 4.2 we have

$$\begin{aligned} P[g]= \partial _tP[f] \qquad \forall (t,x) \in \overline{\Omega _T}. \end{aligned}$$

Hence

$$\begin{aligned} \partial _t P[g] - \Delta P[g] = \partial _t \left( \partial _t P[f] - \Delta P[f] \right) = \partial _t f = g, \end{aligned}$$
(12)

in the sense of distributions. The second equality in (12) follows by classical results for the heat volume potential (see, e.g., Friedman [7, Thm. 9, p.21]).

Statement ii) follows by the definition of the heat volume potential \(P[\cdot ]\), by Proposition 4.1 and by Theorem A.1 of the Appendix. \(\square \)

5 The Dirichlet and Neumann problems

As an application, we show the solvability of the Dirichlet and Neumann problem for equation (2). Let \(\alpha , \beta \in \mathopen ]0,1[\). Let \(T>0\). Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) of class \(C^{1,\alpha }\). Let \(f \in C_0^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\), \(g \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\partial _T \Omega )\), and \(h \in C_0^{\frac{\alpha }{2};\alpha }( \partial _T \Omega )\). The Dirichlet problem for equation (2) is

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u- \Delta u = \partial _t f \qquad &{} \text{ in } ]0,T\mathclose [ \times \Omega ,\\ u = g \qquad &{} \text{ on } ]0,T\mathclose ] \times \partial \Omega ,\\ u(0,\cdot ) = 0 \qquad &{} \text{ in } {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$
(13)

while the Neumann problem reads

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t v- \Delta v = \partial _t f \qquad &{} \text{ in } ]0,T\mathclose [ \times \Omega ,\\ \partial _\nu v = h \qquad &{} \text{ on } ]0,T\mathclose ] \times \partial \Omega ,\\ v(0,\cdot ) = 0 \qquad &{} \text{ in } {\overline{\Omega }}. \end{array}\right. } \end{aligned}$$
(14)

Of course the partial differential equation in problems (13) and (14) has to be understood in the weak sense of distributions. We note that following the lines of the present section one can also show the solvability of problems for equation (2) with boundary conditions other than Dirichlet and Neumann, as for example Robin or transmission boundary conditions. For the sake of simplicity, here we treat only problems (13) and (14).

We have the following existence and uniqueness result which is an immediate consequence of classical parabolic theory together with Theorem 3.2.

Theorem 5.1

Let \(\alpha , \beta \in \mathopen ]0,1[\). Let \(T >0\). Let \(\Omega \) be a bounded open subset of \({\mathbb {R}}^n\) of class \(C^{1,\alpha }\). Let \(f \in C_0^{\frac{1+\alpha }{2};\beta }(\overline{\Omega _T})\), \(g \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\partial _T \Omega )\), and \(h \in C_0^{\frac{\alpha }{2};\alpha }( \partial _T\Omega )\). Then problem (13) admits a unique solution \(u \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\) and problem (14) admits a unique solution \(v \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\)

Proof

Since (13) and (14) are linear problems, the uniqueness of their solutions in the space \(C_0^{\frac{1+\alpha }{2};1+\alpha }( \overline{\Omega _T})\) is well known (cf, e.g., Friedman [7]).

Next we pass to consider existence of a solution. By Theorem 3.2, the map \(P[\partial _t f] \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\) and solves equation (2). Thus, the existence of a solution for problems (13) and (14) can be reduced to the existence of a solution for

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t {{\tilde{u}}}- \Delta {{\tilde{u}}} = 0\qquad &{} \text{ in } ]0,T\mathclose [ \times \Omega ,\\ {{\tilde{u}}} = g-P[\partial _t f] \qquad &{} \text{ on } ]0,T\mathclose ] \times \partial \Omega ,\\ {{\tilde{u}}}(0,\cdot ) = 0 \qquad &{} \text{ in } {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$
(15)

and for

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t {{\tilde{v}}}- \Delta {\tilde{v}} = 0\qquad &{} \text{ in } ]0,T\mathclose [ \times \Omega ,\\ \partial _\nu {{\tilde{v}}} = h-\partial _\nu P[\partial _t f] \qquad &{} \text{ on } ]0,T\mathclose ] \times \partial \Omega ,\\ {{\tilde{v}}}(0,\cdot ) = 0 \qquad &{} \text{ in } {\overline{\Omega }}, \end{array}\right. } \end{aligned}$$
(16)

respectively. Indeed, if \({{\tilde{u}}}, {{\tilde{v}}} \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\) solve respectively problems (15) and (16), then \(u = {{\tilde{u}}}+ P[\partial _t f] \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\) and \(v = {{\tilde{v}}}+ P[\partial _t f] \in C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\) solve respectively problems (13) and (14). It is classical that problems (15) and (16) admit a solution in the space \(C_0^{\frac{1+\alpha }{2};1+\alpha }(\overline{\Omega _T})\). For a proof of this result based on potential theoretic methods we refer to Baderko [1] (see also Lunardi and Vespri [16] for a proof based on semigroups theory).

\(\square \)