Boundedness of Gaussian Bessel potentials and fractional derivatives on variable Gaussian Besov$-$Lipschitz spaces

1.   Introduction

Before ^[9] the study of Besov$-$Lipschitz and Triebel$-$Lizorkin Gaussian spaces had not been extended to the context of variable exponents since there was no condition on the exponent functions $p(\cdot)$ and $q(\cdot)$ that allowed proving the independence with respect to the integer $k > \alpha$ in the definition of these spaces.

By considering the condition on $p(\cdot)$ introduced in ^[3] and the condition on $q(\cdot)$ presented in ^[4] to obtain the variable Hardy's inequalities, it was possible to prove in ^[9] that the definition of the spaces is independent of $k$ and in this way extend the results in ^[8] to the variable context.

This condition on $p(\cdot)$ is closely related to the Gaussian measure. Examples of a large family of exponent (non-constant) functions that satisfy this condition are shown in ^[3].

Once we have the Gaussian Besov$-$Lipschitz spaces with these conditions, we were able to obtain the boundedness of Gaussian Bessel potentials and fractional derivatives and thus extend the results in ^[6] to the variable context.

By changing the conditions on the exponents $p(\cdot)$ and $q(\cdot)$, we would first have to prove that the spaces corresponding to these new conditions are well defined, obtaining properties analogous to those of ^[9], and then study the boundedness of the operators.

When trying to extend these results to other function spaces, for example, for Laguerre or Jacobi measures, we would need conditions on the exponent $p(\cdot)$ that are associated with those measures, then study the properties of the new spaces and finally obtain the boundedness of the operators.

In a future work we will study the boundedness of these operators on Gaussian Triebel$-$Lizorkin spaces with variable exponents and thus extend the respective results obtained in ^[6] for Triebel$-$Lizorkin.

Our work provides new function spaces that can be the habitat for solutions of partial differential equations or differential integral equations. We still need to study the boundedness of the Gaussian Riesz potentials and their fractional derivative in these spaces as well as the Littlewood$-$Paley Gaussian function $g$.

For a deeper dive into these and other related topics, see ^[1,5,12].

In classical harmonic analysis we study the notions of semigroups, covering lemmas, maximal functions, Littlewood$-$Paley functions, spectral multipliers, fractional integrals and derivatives, singular integrals, etc., in the Lebesgue measure space $(\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}), dx)$, where $\mathcal{B}(\mathbb{R}^{d})$ is the Borel $\sigma$-algebra on $\mathbb{R}^{d}$ and consider the Laplacian operator, $\Delta_{x} = \sum\limits_{k = 1}^{d}\frac{\partial^{2}}{\partial x_{k}^{2}}$.

The Bessel potential of order $\beta > 0,$ $\mathcal{J}_\beta$, is defined as

For $f\in L^{p}(\mathbb{R}^{d})$, $\mathcal{J}_\beta(f) = G_{\beta}*f$, where the kernel $G_{\beta}$ is given by,

where for $x = (x_{1}, \cdots, x_{d})\in\mathbb{R}^{d}$, $\|x\| = \sqrt{x^{2}_{1}+ \cdots+ x^{2}_{d}}$ is the Euclidean norm on $\mathbb{R}^{d}$. The Potential spaces $\mathcal{L}^{p}_{\beta}(\mathbb{R}^{d})$ are defined by $\mathcal{L}^{p}_{\beta}(\mathbb{R}^{d}) = \mathcal{J}_\beta(L^{p}(\mathbb{R}^{d})),$ for $\beta\geq 0$ and $1\leq p\leq \infty$.

It is easy to see that $G_{\beta}\in L^{1}(\mathbb{R}^{d})$, which implies that $\|\mathcal{J}_\beta(f)\|_{p}\leq \|f\|_{p}, 1\leq p\leq \infty$.

Therefore, $\mathcal{L}^{p}_{\beta}(\mathbb{R}^{d})$ is continuously embedded in $L^{p}(\mathbb{R}^{d})$.

These spaces generalize the Sobolev space $L^{p}_{k}(\mathbb{R}^{d})$, in the sense that, for $1 < p < \infty$ and $k\in\mathbb{N}$, the potential space $\mathcal{L}^{p}_{k}(\mathbb{R}^{d})$ is equivalent to the Sobolev space $L^{p}_{k}(\mathbb{R}^{d})$.

In Gaussian harmonic analysis, we consider the Ornstein$-$Uhlenbeck second-order differential operator,

$L = \frac12\triangle_x-\langle x, \nabla _x\rangle$, where $\nabla_{x} = (\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \cdots, \frac{\partial}{\partial x_{d}})$.

In the Gaussian context, the Hermite polynomials are orthogonal with respect to the Gaussian measure and also are eigenfunctions of the Ornstein-Uhlenbeck operator $L$.

The Gaussian Bessel potential (or fractional integral) of order $\beta > 0$, which we also denote $\mathcal{J}_\beta$, is defined formally as

and Gaussian Bessel fractional derivative ${\mathcal D}^\beta$ of order $\beta > 0$, is defined by

which means that for the Hermite polynomials, we have

and

Meyer's theorem allows us to extend Gaussian Bessel potentials to a bounded operator on $L^p(\gamma_d),$ $1 < p < \infty$.

Also, from (1.4), we conclude that, ${\mathcal D}^\beta$ (as a good derivative), is not a bounded operator on $L^p(\gamma_d),$ $1\leq p\leq \infty$.

Gaussian Bessel potentials have the integral representation

where $\lbrace P_{t}\rbrace_{t\geq 0}$ is the Poisson$-$Hermite semigroup.

On the other hand, let $k$ be the smallest integer greater than $\beta$; then the fractional derivative ${\mathcal D}^\beta$ has the integral representation

with $c^k_{\beta} = \int_0^{\infty} u^{-\beta-1} (e^{-u} -1)^k du$.

There are significant differences between classical and Gaussian harmonic analysis, namely: Lebesgue measure is a doubling, translation-invariant measure. Semigroups associated with Lebesgue measure are convolution semigroups. Gaussian measure does not satisfy any of these properties. For details, see ^[11].

In ^[9], replacing $p$ and $q$ with measurable functions $p(\cdot), q(\cdot)$ taking values in $[1, \infty]$ and satisfying suitable regularity conditions, we define and study the structure of Besov$-$Lipschitz spaces $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ with variable exponents with respect to the Gaussian measure, following ^[8,11].

In this paper, we generalize some results in ^[6] for $\mathcal{J}_\beta$ and ${\mathcal D}^\beta$ on $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$. To do this, we present three sections:

● In Section 2, we give the preliminaries in the Gaussian setting and some background on variable spaces with respect to a Borel measure $\mu$.

● In Section 3, we obtain the boundedness of $\mathcal{J}_\beta$ and ${\mathcal D}^\beta$ on $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$.

● In Section 4, we give some conclusions.

2.   Preliminaries

The Gaussian measure on $\mathbb{R}^d$ is given by

For $\nu = (\nu _1, ..., \nu_d)\in \mathbb{Z}^{d}$ such that $\nu _i \geq 0, i = 1, \cdots, d$, we consider $\nu! = \prod\limits_{i = 1}^d\nu _i!,$ $\left| \nu \right| = \sum\limits_{i = 1}^d\nu _i,$ $\partial _i = \frac \partial {\partial x_i},$ with $1\leq i\leq d$ and $\partial ^\nu = \partial _1^{\nu _1}...\partial _d^{\nu _d}$. Then:

● The normalized Hermite polynomials of order $\nu$ in $d$ variables are defined by,

● The Ornstein$-$Uhlenbeck semigroup $\left\{ T_t\right\} _{t\geq 0}$ is defined by

● Poisson$-$Hermite semigroup $\left\{ P_t\right\} _{t\geq 0}$ by

where for all $t > 0$,

is the one-sided stable measure on $(0, \infty)$ of order $1/2$.

As usual, we use the notation

Now, let us obtain some background on variable Lebesgue spaces with respect to a Borel measure $\mu$.

Let $\Omega\subset\mathbb{R}^{d}$, a $\mu$-measurable function $p(\cdot):\Omega\rightarrow [1, \infty]$ is an exponent function. The set of exponent functions is denoted by $\mathcal{P}(\Omega, \mu)$. For $E\subset\Omega$ we set

Also $\Omega_{\infty} = \lbrace x\in \Omega:p(x) = \infty\rbrace$, $p_{+} = p_{+}(\Omega)$, and $p_{-} = p_{-}(\Omega)$.

Definition 2.1. Let $E\subset \mathbb{R}^{d}$ and $\alpha(\cdot):E\rightarrow \mathbb{R}$. We say that:

$i)$ $\alpha(\cdot)$ is locally log-Hölder continuous if there exists a constant $C_{1} > 0$ such that

for all $x, y\in E$. Denoted by $\alpha(\cdot)\in LH_{0}(E)$.

$ii)$ $\alpha(\cdot)$ is log-Hölder continuous at infinity with base point at $x_{0}\in \mathbb{R}^{d}$, if there exist constants $\alpha_{\infty}\in\mathbb{R}$ and $C_{2} > 0$ such that

for all $x\in E$. Denoted by $\alpha(\cdot)\in LH_{\infty}(E)$.

$iii)$ $\alpha(\cdot)$ is log-Hölder continuous if both conditions are satisfied. In this case, we say $\alpha(\cdot)\in LH(E)$.

Definition 2.2. For a $\mu$-measurable function $f:\mathbb{R}^{d}\rightarrow \overline{\mathbb{R}}$ (an extended real-valued function), we define the modular

The variable exponent Lebesgue space on $\mathbb{R}^{d}$, $L^{p(\cdot)}(\mu)$ is the set of $\mu$-measurable functions $f$ such that there exists $\lambda > 0$ with $\rho_{p(\cdot), \mu}\left(f/\lambda\right) < \infty,$ with norm given by

Theorem 2.1. (Minkowski's integral inequality for variable Lebesgue spaces) Given $\mu$ and $\nu$ complete $\sigma$-finite measures on $X$ and $Y$ respectively, $p\in \mathcal{P}(X, \mu)$. Let $f:X\times Y\rightarrow \overline{\mathbb{R}}$ be measurable with respect to the product measure on $X\times Y$, such that for almost every $y\in Y$, $f(\cdot, y)\in L^{p(\cdot)}(X, \mu)$. Then

Proof. See ^[9]. □

In the rest of the paper, $\mu$ represents the Haar measure $\mu(dt) = \frac{dt}{t}$ on $\mathbb{R}^{+}$.

Now, $\mathcal{M}_{0, \infty}$ denotes the set of measurable functions $p(\cdot): \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ that satisfy the following conditions:

i) $0\leq p_{-}\leq p_{+} < \infty$.

ii$_0$) There exists $p(0) = \lim\limits_{x\rightarrow 0}p(x)$ and $|p(x)- p(0)|\leq \frac{A}{\ln(1/x)}, 0 < x\leq 1/2$.

ii$_\infty$) There exists $p(\infty) = \lim\limits_{x\rightarrow \infty}p(x)$ and $|p(x)- p(\infty)|\leq \frac{A}{\ln(x)}, x > 2$.

$\mathcal{P}_{0, \infty}$ is the set of functions $p(\cdot)\in\mathcal{M}_{0, \infty}$ such that $p_{-}\geq 1$.

Next, we present the Hardy inequalities associated with the exponent $q(\cdot)\in\mathcal{P}_{0, \infty}$ and the measure $\mu$.

Theorem 2.2. Let $q(\cdot)\in\mathcal{P}_{0, \infty}$ and $r > 0$, then

and

Proof. See ^[9]. □

Also, we need the classical Hardy's inequalities; see ^[10]

and

with $g\geq 0, p\geq 1$ and $r > 0.$

In what follows, we only consider variable exponent Lebesgue spaces, $L^{p(\cdot)}(\gamma_d)$ with respect to the Gaussian measure $\gamma_d$.

Remark 2.1. The families $\lbrace T_{t}\rbrace_{t\geq 0}$, $\lbrace P_{t}\rbrace_{t\geq 0}$ and $\lbrace\mathcal{J}_\beta\rbrace_{\beta\geq 0}$ are bounded on $L^{p(\cdot)}(\gamma_d)$, for $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ with $1 < p_{-}\leq p_{+} < \infty$. For the proof see ^[7].

The next condition was introduced by E. Dalmasso and R. Scotto in ^[3].

Definition 2.3. Let $p(\cdot)\in\mathcal{P}(\mathbb{R}^{d}, \gamma_{d})$, we say that $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$ if there exist constants $C_{\gamma_{d}} > 0$ and $p_{\infty}\geq1$ such that

for $x\in\mathbb{R}^{d}, x\neq \mathbf{0}$.

Example 2.1. There exist non-constant functions in $\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, by considering $p(x) = p_{\infty}+ \frac{A}{(e+\|x\|)^{q}}$, $x\in\mathbb{R}^{d}$, for any $p_{\infty}\geq 1, A\geq 0$ and $q\geq 2$.

Remark 2.2. It is easy to see that $\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\subset LH_{\infty}(\mathbb{R}^{d})$.

Finally, we need some technical results

Lemma 2.1. Given $k\in \mathbb{N}$ and $t > 0$ then $\mu^{(1/2)}_t$ satisfies

For the proof see ^[11].

Lemma 2.2. Given an integer $k\geq 0$, let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ and $f\in L^{p(\cdot)}(\gamma_d)$, then

for $0 < s < t < +\infty$. Moreover,

For the proof see ^[9].

The variable Gaussian Besov$-$Lipschitz space were defined in ^[9], following ^[6,10].

Definition 2.4. Let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ and $q(\cdot)\in\mathcal{P}_{0, \infty}$. Let $\alpha \geq 0$ and $k$ be the smallest integer greater than $\alpha$. The variable Gaussian Besov$-$Lipschitz space $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$, is the set of functions $f \in L^{p(\cdot)}(\gamma_d)$ such that

with norm

The variable Gaussian Besov$-$Lipschitz space $B_{p(\cdot), \infty}^{\alpha}(\gamma_d)$, is the set of functions $f \in L^{p(\cdot)}(\gamma_d)$ for which there exists a constant $A$ such that

and the norm

where $A_{k}(f)$ is the smallest constant $A$ in the above inequality.

One of the main results in ^[9] was that the definition of $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$ is independent on the integer $k > \alpha$.

For more details about the definition of variable Gaussian Besov$-$Lipschitz spaces, we refer to ^[9].

Additionally, in ^[9] we obtained some inclusion relations between variable Gaussian Besov$-$Lipschitz spaces. These results are analogous to Proposition 10, page 153 in ^[10].

Proposition 2.1. Let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ and $q_{1}(\cdot), q_{2}(\cdot)\in\mathcal{P}_{0, \infty}$. The inclusion $B_{p(\cdot), q_1(\cdot)}^{\alpha_{1}}(\gamma_d)\subset B_{p(\cdot), q_2(\cdot)}^{\alpha_{2}}(\gamma_d)$ holds, i.e., $\|f\|_{B_{p(\cdot), q_2(\cdot)}^{\alpha_{2}}}\leq C \|f\|_{B_{p(\cdot), q_1(\cdot)}^{\alpha_{1}}}$ if:

i) $\alpha_{1} > \alpha_{2} > 0$ ($q_{1}(\cdot)$ and $q_{2}(\cdot)$ do not need to be related), or

ii) If $\alpha_{1} = \alpha_{2}$ and $q_{1}(t)\leq q_{2}(t)\quad a.e.$

As usual in this theory, $C$ represents a constant that is not necessarily the same in each occurrence.

3.   Results

The main results of the paper are the regularity properties of the Gaussian Bessel potentials and the Gaussian Bessel fractional derivatives on variable Gaussian Besov$-$Lipschitz spaces.

Let us start considering the regularity properties of the Gaussian Bessel potentials. In the following theorem we consider their action on $B_{p(\cdot), \infty}^{\alpha}(\gamma_{d})$ spaces, which is analogous to Theorem 4 in ^[6].

Theorem 3.1. Let $\alpha\geq 0, \beta > 0$ then for $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ with $1 < p_{-}\leq p_{+} < \infty$. Then, the Gaussian Bessel potential $\mathcal{J}_{\beta}$ is bounded from $B_{p(\cdot), \infty}^{\alpha}(\gamma_{d})$ into $B_{p(\cdot), \infty}^{\alpha+\beta}(\gamma_{d})$.

Proof. Let $k > \alpha+\beta$ a fixed integer and $f\in B_{p(\cdot), \infty}^{\alpha}(\gamma_{d})$, then $\mathcal{J}_{\beta}f\in L^{p(\cdot)}(\gamma_{d})$ (see ^[7]). By using the representation (1.5), the dominated convergence theorem, and the chain's rule, we obtain

Then, using Minkowski's integral inequality (2.8), and splitting the integral into two

Now, proceeding as in ^[6] by Lemma 2.2 since $t+s > t$

On the other hand, as $k > \alpha+\beta$ and again by Lemma 2.2, since $t+s > s$

Thus,

which implies that $\mathcal{J}_{\beta}f\in B_{p(\cdot), \infty}^{\alpha+\beta}(\gamma_{d})$ and $A_{k}(\mathcal{J}_{\beta}f)\leq CA_{k}(f)$.

Moreover, by Remark 2.1,

□

In the next theorem we consider the action of Gaussian Bessel potentials on $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ spaces. It is analogous to Theorem 2.4 (i) of ^[8].

Theorem 3.2. Let $\alpha\geq 0$, $\beta > 0$, $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ with $1 < p_{-}\leq p_{+} < \infty$ and $q(\cdot)\in\mathcal{P}_{0, \infty}$. Then, the Gaussian Bessel potential $\mathcal{J}_{\beta}$ is bounded from $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ into $B_{p(\cdot), q(\cdot)}^{\alpha+\beta}(\gamma_{d})$.

Proof. Let $f\in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ then $\mathcal{J}_{\beta}f\in L^{p(\cdot)}(\gamma_{d})$ since $\mathcal{J}_{\beta}$ is bounded on $L^{p(\cdot)}(\gamma_{d})$.

Let denote $U(x, t) = P_{t}{\mathcal{J}}_{\beta}f(x)$. Using the representation (2.4) of $P_t$ and the semigroup property

Now, fix $k$ and $l$ as integer greater than $\alpha$ and $\beta$, respectively. By using the dominated convergence theorem, differentiating $k$ times with respect to $t_{2}$ and $l$ times with respect to $t_{1}$, we obtain

So, taking $t = t_{1}+t_{2}$, we obtain

therefore, by using inequality (2.8), the boundedness of $T_s$ on $L^{p(\cdot)}(\gamma_{d})$ and Lemma 2.1

On the other hand, by the chain's rule

and again by inequality (2.8)

Now, since the definition of $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ is independent of the integer $k > \alpha$, take $k > \alpha+\beta$ and $l > \beta$; then $k+l > \alpha+2\beta > \alpha+\beta$.

We will show that $\mathcal{J}_{\beta}f\in B_{p(\cdot), q(\cdot)}^{\alpha+\beta}(\gamma_{d})$.

In fact, taking $t_1 = t_2 = t/2$ in (3.1) and by (3.2), we obtain

Thus, by Lemma 2.2 (since $s+\frac{t}{2} > \frac{t}{2}$) and the change of variables $u = t/2$, we have

Finally, by Lemma 2.2 (since $s+\frac{t}{2} > s$) and Hardy's inequality (2.10), with $r = k-(\alpha+\beta)$ and $g(s) = s^{{\beta}-1}\left\|\frac{\partial^{k}P_{s}f}{\partial s^{k}}\right\|_{p(\cdot), \gamma_{d}}$, we have

Therefore, $\mathcal{J}_{\beta}f\in B_{p(\cdot), q(\cdot)}^{\alpha+\beta}(\gamma_{d})$.

Moreover, $\left\|{\mathcal{J}}_{\beta}f\right\|_{B_{p(\cdot), q(\cdot)}^{\alpha+\beta}}\leq C \|f\|_{B_{p(\cdot), q(\cdot)}^{\alpha}}.$ □

Now, we will study the action of the Bessel fractional derivative $\mathcal{D}^\beta$ on variable Gaussian Besov$-$Lipschitz spaces $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$. We will use the representation (1.6) of the Bessel fractional derivative and Hardy's inequalities.

First, we need to consider the forward differences. Remember, for a given function $f$, the $k$-th order forward difference of $f$ starting at $t$ with increment $s$ is defined as

The forward differences have the following properties (see Appendix 10.9 in ^[11]): We will need the following technical result.

Lemma 3.1. For any positive integer $k$

i)$\Delta_{s}^{k}(f, t) = \Delta_{s}^{k-1}(\Delta_{s}(f, \cdot), t) = \Delta_{s}(\Delta_{s}^{k-1}(f, \cdot), t)$

ii) $\Delta_{s}^{k}(f, t) = \int_{t}^{t+s}\int_{v_{1}}^{v_{1}+s}... \int_{v_{k-2}}^{v_{k-2}+s}\int_{v_{k-1}}^{v_{k-1}+s}f^{(k)}(v_{k})dv_{k}dv_{k-1}...dv_{2}dv_{1}$.

For any positive integer $k$,

and for any integer $j > 0$,

Additionally, we obtain the next result.

Lemma 3.2. Let $p(\cdot)\in \mathcal{P}(\mathbb{R}^{d}, \gamma_{d})$, $f\in L^{p(\cdot)}(\gamma_{d})$ and $k, n\in\mathbb{N}$, then

Proof. By Lemma 3.1 ii), we have

thus, by using inequality (2.8) and Lemma 2.2 $k$-times, respectively, we obtain

□

We are ready to consider the action of Gaussian Bessel fractional derivatives on $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ spaces. This is the analogous result to Theorem 8 in ^[6].

Remark 3.1. By semigroup property of $\{P_t\}$, we have

where $v(x, t) = e^{-t} u(x, t)$.

Theorem 3.3. Let $0 < \beta < \alpha$, $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ with $1 < p_{-}\leq p_{+} < \infty$ and $q(\cdot)\in\mathcal{P}_{0, \infty}$. Then, the Gaussian Bessel fractional derivative $\mathcal{D}_{\beta}$ is bounded from $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ into $B_{p(\cdot), q(\cdot)}^{\alpha-\beta}(\gamma_{d})$.

Proof. Let $f\in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ and $k\in \mathbb{N}$ such that $k-1\leq \beta < k$. Then by Remark 3.1, inequality (2.11), the fundamental theorem of calculus, and Lemma 3.1,

and by inequality (2.8), we obtain

Again, by Lemma 3.1 and inequality (2.8),

$\left\|\Delta_{r}^{k-1}(v', r)\right\|_{p(\cdot), \gamma_{d}}\leq C^{k} \int_{r}^{2r}\int_{v_{1}}^{v_{1}+r}... \int_{v_{k-2}}^{v_{k-2}+r}\left\|v^{(k)}(\cdot, v_{k-1})\right\|_{p(\cdot), \gamma_{d}}dv_{k-1}dv_{k-2}...dv_{2}dv_{1},$

and also by Leibnitz's differentiation rule,

Thus, by Lemma 2.2, since $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$,

Therefore, by using the boundedness of $P_{t}$ on $L^{p(\cdot)}(\gamma_{d})$ for $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$, see ^[7],

Thus,

since $f\in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})\subset B_{p(\cdot), 1}^{\beta-j}(\gamma_d)$ as $\alpha > \beta > \beta-j\geq 0$, for $j\in\{0, ..., k-1\}$,

by Proposition 2.1, since $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ and $q(\cdot)\in \mathcal{P}_{0, \infty}$.

Hence, $\mathcal{D}_{\beta}f\in L^{p(\cdot)}(\gamma_d)$.

On the other hand, it is clear that

Let $n\in \mathbb{N}$ such that $n-1 \leq \alpha < n$, then

where $w(x, t) = e^{-t}u^{(n)}(x, t)$. Thus, by using the fundamental theorem of calculus,

Then, by inequality (2.11), and Lemma 3.1,

Now, proceeding as above, using again Lemma 3.1 and Leibnitz's differentiation rule

and by inequality (2.8) we obtain that

Therefore,

Now, for each $1\leq j\leq k$, $0 < \alpha-\beta+k-j\leq\alpha$. Then by Lemma 2.2 and Proposition 2.1, since $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ and $q(\cdot)\in\mathcal{P}_{0, \infty}$,

since $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})\subset B_{p(\cdot), q(\cdot)}^{\alpha-\beta+(k-j)}(\gamma_{d})$, for $j\in\lbrace 1, \cdots, k\rbrace$.

Finally, we study the case $j = 0$,

By Lemma 2.2,

and by using again Lemma 2.2 and inequality (2.10) since $q(\cdot)\in\mathcal{P}_{0, \infty}$,

Therefore,

since $f\in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$. Thus, $\mathcal{D}_{\beta}f\in B_{p(\cdot), q(\cdot)}^{\alpha-\beta}(\gamma_{d})$.

Moreover, from all the above inequalities, we conclude that

□

Remark 3.2. The boundedness of $\mathcal{D}_{\beta}$, for $0 < \beta < \alpha$, only uses the fact that $\alpha-\beta > 0$ and $\beta > 0$, It does not matter how close $\beta$ is to the ends of the interval $(0, \alpha)$ or the dimension $d$.

The boundedness of Gaussian Riesz potentials and Riesz fractional derivatives on variable Gaussian Besov$-$Lipschitz spaces and the regularity of all these operators on variable Gaussian Triebel$-$Lizorkin spaces, which were also defined in ^[9], will be considered in a forthcoming paper.

4.   Conclusions

(i) Theorems 3.1–3.3, extend some results obtained in ^[6] when we go from constant exponent to variable exponent settings if the exponent functions $p(\cdot)$, $q(\cdot)$ satisfy the regularity conditions $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$ and $q(\cdot)\in\mathcal{P}_{0, \infty}$.

(ii) The key to the proof of theorems is the generalization of Minkowski's integral inequality and Hardy's inequality to the context of variable exponents as well as Lemmas 2.1 and 2.2.

(iii) The boundedness of $\lbrace T_{t}\rbrace_{t\geq 0}$, $\lbrace P_{t}\rbrace_{t\geq 0}$, and $\lbrace \mathcal{J}_{\beta}\rbrace_{\beta\geq 0}$ on $L^{p(\cdot)}(\gamma_{d})$ was also necessary to obtain the results.

(iv) From the properties obtained in ^[9] for the Besov$-$Lipschitz spaces $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$, it was shown that these form a decreasing family of spaces (in the sense of inclusion) that are continuously immersed in $L^{p(\cdot)}(\gamma _{d})$.

(v) Boundedness of the Gaussian Bessel potentials $\mathcal{J}_{\beta}$ and fractional derivative $\mathcal{D}_{\beta}$ together with inclusion properties between the Besov$-$Lipschitz spaces $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$, indicate that when applying the Bessel potential to a function, it gains regularity while when applying the fractional derivative, it loses regularity (as a good derivative).

(vi) Currently, we do not have practical applications for the Gaussian measure, apart from having generalized the study done for a constant exponent, but we consider that there may be applications in the future in the field of partial differential equations, as occurs for the Lebesgue measure.

Author contributions

All authors of this article have contributed equally. All authors have read and approved the final version of the manuscript for publication.

Use of Generative-AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors like to thank the referees for their useful remarks and corrections that improved the presentation of the paper.

Conflict of interest

The authors declare no conflict of interest.