Variable exponent Besov-Lipschitz and Triebel-Lizorkin spaces for the Gaussian measure

1.   Introduction

Gaussian harmonic analysis is basically the study of the notions of classical harmonic analysis (such as semigroups, covering lemmas, maximal functions, Littlewood-Paley functions, Spectral multipliers, fractional integral and derivatives, singular integrals, etc.), which are formulated in the Lebesgue measure space $(\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}), dx)$, in the probability space $(\mathbb{R}^{d}, \mathcal{B}(\mathbb{R}^{d}), \gamma_d(dx))$, where $\gamma_d(dx) = \frac{e^{-\|x\|^2} }{\pi^{d/2}} dx, \, x\in\mathbb{R}^d,$ is the Gaussian probability measure in $\mathbb{R}^{d}$.

A second component of classical harmonic analysis is the Laplace operator, $\Delta_{x} = \sum_{k = 1}^{d}\frac{\partial^{2}}{\partial x_{k}^{2}}$.

In Gaussian harmonic analysis is the Ornstein-Uhlenbeck second order differential operator, $L = \frac12\triangle_x-\langle x, \nabla _x\rangle$, where $\nabla_{x} = (\frac{\partial}{\partial x_{1}}, \frac{\partial}{\partial x_{2}}, \cdots, \frac{\partial}{\partial x_{d}})$.

A third component of Gaussian harmonic analysis are the Hermite polynomials that are orthogonal with respect to the Gaussian measure and are eigenfunctions of the Ornstein-Uhlenbeck operator $L$.

Some differences between classical and Gaussian harmonic analysis, are: Lebesgue measure is a doubling, translation invariant measure. Semigroups asociated to Lebesgue measure are convolution semigroups. Gaussian measure does not satisfy any of these properties, which makes many of the proofs are completely different from the classical case. For a detailed study, see ^[16].

The structure and properties of general Lipschitz spaces in the classical case (with respect to the Lebesgue measure in $\mathbb{R}^{d}$) were studied in ^[12,14,15]. In analogous way, for $\alpha > 0$ and $1\leq p, q\leq\infty$, in ^[10,16], were introduced and studied the structure of Besov-Lipschitz $B_{p, q}^{\alpha}(\gamma_{d})$ and Triebel-Lizorkin $F_{p, q}^{\alpha}(\gamma_{d})$ spaces, with respect to the Gaussian measure $\gamma_{ d}$ in $\mathbb{R}^{d}$, that is, for expansions on Hermite polynomials. In particular, for $\alpha = 0, q = 2$, $F_{p, 2}^{0}(\gamma_{d}) = L^{p}(\gamma_{d})$ (Gaussian Lebesgue spaces) and for $p, q = \infty$, $B_{\infty, \infty}^{\alpha}(\gamma_{d}) = Lip_{\alpha}(\gamma_{d})$ (Gaussian Lipschitz spaces) ^[16], i.e., these spaces generalize known spaces. All of this, in the constant exponent setting.

Lebesgue spaces with variable exponents have been widely studied in the last three decades, see ^[2] or ^[4]. These spaces arose for a purely theoretical interest, although a short time later applications began to emerge, ^[23]. Also, recent research has been stimulated by applications in various problems, for example, elasticity theory and fluid mechanics, where electrorheological fluids are of special interest, see ^[1,11]. All of the above motivates us to define more general variable exponent spaces.

In this paper, following ^[10] or ^[16] and replacing the constants $p$ and $q$ by 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 $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ and Triebel-Lizorkin spaces $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ with variable exponents respect to the Gaussian measure, generalizing some of the results in ^[10,16] such as inclusion relations of those spaces and interpolation results for them. Therefore, for the study of variable exponent spaces, $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ and $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$, we present four 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 some technical results for the Haar measure on $\mathbb{R}^{+}$ that will be key in the proof of the main results.

● In section 4, we define and study the structure of the spaces $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ and $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$.

● In section 5, we give some conclusions.

Finally, there are some important references on variable Besov and Triebel-Lizorkin spaces in the context of Lebesgue measure, for example, ^{[6,13,17,18,19,20,21,22]}.

On the other hand, based on the results of this work, we can now study the boundedness of Riesz Potentials, Bessel Potentials and Bessel Fractional Derivatives on $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ and $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$, in order to generalize the ones presented in ^[7].

2.   Preliminaries

Let us consider the Gaussian measure

on $\mathbb{R}^d$ and the Ornstein-Uhlenbeck differential operator

Let $\nu = (\nu _1, ..., \nu_d)$ be a multi-index such that $\nu _i \geq 0, i = 1, \cdots, d$, let $\nu! = \prod_{i = 1}^d\nu _i!,$ $\left| \nu \right| = \sum_{i = 1}^d\nu _i,$ $\partial _i = \frac \partial {\partial x_i},$ for each $1\leq i\leq d$ and $\partial ^\nu = \partial _1^{\nu _1}...\partial _d^{\nu _d}$.

Consider the normalized Hermite polynomials of order $\nu$ in $d$ variables,

The Ornstein-Uhlenbeck semigroup on ${\mathbb{R} ^d}$ is defined by

Using the Bochner subordination formula

we introduce the Poisson-Hermite semigroup by

Now, taking the change of variables $s = \frac{t^2}{4u}$, $P_{t}f(x)$ can be written as

where $\mu^{(1/2)}_t(ds) = \frac{t}{2\sqrt{\pi}} e^{-t^2/4s} s^{-3/2} ds,$ is the one-sided stable measure on $(0, \infty)$ of order $1/2$, it is easy to see that $\mu^{(1/2)}_t$ is a probability measure on $(0, \infty)$.

It is well known, that Hermite polynomials are eigenfunctions of the operator $L$,

In consequence

and

i.e., Hermite polynomials are also eigenfunctions of $T_t$ and $P_t$ for any $t \geq 0$, for more details, see ^[16].

Next, we present some technical results for the measure $\mu^{(1/2)}_t$ needed in what follows.

As $\mu^{(1/2)}_t(ds) = \frac t{2\sqrt{\pi}}\frac{e^{-t^2/4s}}{s^{3/2}}ds = g(t, s) ds$, for any $k\in \mathbb{N}$, we use the notation $\frac{\partial^{k}}{\partial t^{k}}\mu_{t}^{(1/2)}(ds)$ for

Lemma 2.1. Given $k\in \mathbb{N}$,

where $\{a_{i, j}\}$ is a finite set of constants and the indexes $i\in \mathbb{Z}$, $j\in \mathbb{N}$ verifies the equation $2j-i = k$.

Lemma 2.2. Given $k\in \mathbb{N}$ and $t > 0$,

Corollary 2.1. Given $k\in \mathbb{N}$ and $t > 0$,

On the other hand, by considering the maximal function of the Ornstein-Uhlenbeck semigroup

we obtain:

Lemma 2.3. Let $f\in L^{1}(\gamma_{d}), x\in \mathbb{R}^{d}$ and $k\in \mathbb{N}$

For the proofs of the previous technical results, see ^[10] or ^[16].

Now, for completeness, we need some background on variable Lebesgue spaces with respect to a Borel measure $\mu$. A $\mu$-measurable function $p(\cdot):\Omega \subset \mathbb{R}^d \rightarrow [1, \infty]$ is said to be an exponent function. The set of all the exponent functions will be denoted by $\mathcal{P}(\Omega, \mu)$. For $E\subset\Omega$, we set $p_{-}(E) = \text{ess}\inf_{x\in E}p(x), \; p_{+}(E) = \text{ess}\sup_{x\in E}p(x)$ and $\Omega_{\infty} = \{x\in \Omega:p(x) = \infty\}$.

Also, we use the abbreviations $p_{+} = p_{+}(\Omega)$ and $p_{-} = p_{-}(\Omega)$.

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

$i)$ $p(\cdot)$ is locally log-Hölder continuous, denote by $p(\cdot)\in LH_{0}(E)$, if there exists a constant $C_{1} > 0$ such that

for all $x, y\in E$.

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

for all $x\in E$.

$iii)$ $p(\cdot)$ is log-Hölder continuous, and denote this by $p(\cdot)\in LH(E)$ if both conditions $i)$ and $ii)$ are satisfied.

The maximum, $\max\{C_{1}, C_{2}\}$ is called the log-Hölder constant of $p(\cdot)$.

Definition 2.2. Let $E\subset \mathbb{R}^{d}$, $p(\cdot)\in\mathcal{P}_{d}^{log}(E)$, if $\frac{1}{p(\cdot)}$ is log-Hölder continuous and denote by $C_{log}(p)$ or $C_{log}$ the log-Hölder constant of $\frac{1}{p(\cdot)}$.

Definition 2.3. Let $\Omega\subset \mathbb{R}^{d}$ and $p(\cdot)\in\mathcal{P}(\Omega, \mu)$. For a $\mu$-measurable function $f:\Omega\rightarrow \overline{\mathbb{R}}$, we define the modular $\rho_{p(\cdot), \mu}$ as

and the norm

Definition 2.4. The variable exponent Lebesgue space on $\Omega\subset\mathbb{R}^{d}$, $L^{p(\cdot)}(\Omega, \mu)$ consists on those $\mu\_$measurable functions $f$ for which there exists $\lambda > 0$ such that $\rho_{p(\cdot), \mu}\left(\frac{f}{\lambda}\right) < \infty,$ i.e.,

Remark 2.1. When $\mu$ is the Lebesgue measure, we write $\rho_{p(\cdot)}$ and $\|f\|_{p(\cdot)}$ instead of $\rho_{p(\cdot), \mu}$ and $\|f\|_{p(\cdot), \mu}$.

Theorem 2.1. (Norm conjugate formula) Let $\nu$ a complete, $\sigma$-finite measure on $\Omega$ and $p(\cdot)\in \mathcal{P}(\Omega, \nu)$, then

for all $f$ $\nu$-measurable on $\Omega$, where

$\|f\|^{'}_{p(\cdot), \nu} = \sup\left\{\int_{\Omega}|f||g|d\nu:g\in L^{p'(\cdot)}(\Omega, \nu), \|g\|_{p'(\cdot), \nu}\leq 1\right\}.$

Proof. See Corollary 3.2.14 in ^[4]. □

Theorem 2.2. (Hölder's inequality) Let $\nu$ a complete, $\sigma$-finite measure on $\Omega$ and $r(\cdot), q(\cdot)\in \mathcal{P}(\Omega, \nu)$. Define $p(\cdot)\in \mathcal{P}(\Omega, \nu)$ by $\frac{1}{p(x)} = \frac{1}{q(x)}+\frac{1}{r(x)}, \quad a.e.$

Then for all $f\in L^{q(\cdot)}(\Omega, \nu)$ and $g\in L^{r(\cdot)}(\Omega, \nu)$, $fg\in L^{p(\cdot)}(\Omega, \nu)$ and

Proof. See Lemma 3.2.20 in ^[4]. □

Theorem 2.3. (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 $f:X\times Y\rightarrow \overline{\mathbb{R}}$ 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. It is completely analogous to the proof of Corollary 2.38 in ^[2] by interchanging the Lebesgue measure for complete $\sigma$-finite measures $\mu$ and $\nu$ on $X$ and $Y$ respectively, and by using (2.18), Fubini's theorem and then (2.17). □

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

3.   Technical results

In this section we present some technical results regarding the Haar measure that will be key to the main results.

Remark 3.1. For a $\mu$-measurable function $f:\mathbb{R}^{+}\rightarrow \overline{\mathbb{R}}$, $q(\cdot)\in\mathcal{P}(\mathbb{R}^{+}, \mu)$, and any $\lambda > 0$:

Thus,

Next, we present a technical result for the Haar measure $\mu$.

Lemma 3.1. Let $q(\cdot)\in\mathcal{P}(\mathbb{R}^{+}, \mu)$ and $\alpha, \beta > 0$

$i)$ If $q_{+} < \infty$, then $\|t^{\alpha}e^{-t\beta}\|_{q(\cdot), \mu} < \infty.$

$ii)$ $\|t^{\alpha}\chi_{(0, 1]}\|_{q(\cdot), \mu} < \infty.$

$iii)$ $\|t^{-\alpha}\chi_{(1, \infty)}\|_{q(\cdot), \mu} < \infty.$

$iv)$ For any $t_{0} > 0, \; (\ln 2)^{\frac{1}{q_{-}}}\leq\|\chi_{[t_{0}/2, t_{0}]}\|_{q(\cdot), \mu}\leq 1.$

Proof. Let us prove $i)$. Set $f(t) = t^{\alpha}e^{-t\beta}$,

Now,

since $\alpha, \beta > 0$ and $0\leq t\leq 1.$ On the other hand, by making the change of variables $u = t\beta q_{-}$

Thus, $\rho_{q(\cdot), \mu}(f) < \infty$, and therefore $\|t^{\alpha}e^{-t\beta}\|_{q(\cdot), \mu} < \infty.$ The proofs of $ii)$ and $iii)$ are immediate.

Now, in order to prove $iv)$, set $g = \chi_{[t_{0}/2, t_{0}]}$,

Then, $\lambda\geq 1$ implies $\rho_{q(\cdot), \mu}(\frac{g}{\lambda})\leq \rho_{q(\cdot), \mu}(g)\leq 1$. Thus, $\|g\|_{q(\cdot), \mu}\leq 1$.

On the other hand, taking $0 < \lambda < 1$

So, $\lambda < (\ln 2)^{1/q_{-}}$ implies $\rho_{q(\cdot), \mu}(\frac{g}{\lambda}) > 1$. Thus, $\rho_{q(\cdot), \mu}(\frac{g}{\lambda})\leq 1$ implies $\lambda\geq(\ln 2)^{1/q_{-}}$.

Therefore, $\|g\|_{q(\cdot), \mu}\geq (\ln 2)^{1/q_{-}}$. □

In the case $\Omega = \mathbb{R}^{+}$, we denote $\mathcal{M}_{0, \infty}$ the set of all measurable functions $p(\cdot): \mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ which satisfy the following conditions:

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

$ii_{0})$ there exists $p(0) = \lim_{x\rightarrow 0}p(x)$ and $|p(x)- p(0)|\leq \frac{A}{\ln(1/x)}, 0 < x\leq 1/2$.

$ii_{\infty})$ there exists $p(\infty) = \lim_{x\rightarrow \infty}p(x)$ and $|p(x)- p(\infty)|\leq \frac{A}{\ln(x)}, x > 2$.

We denote $\mathcal{P}_{0, \infty}$ the subset of functions $p(\cdot)$ such that $p_{-}\geq 1$.

Let $\alpha(\cdot), \beta(\cdot)\in LH(\mathbb{R}^{+})$, bounded with

and

Theorem 3.1. Let $p(\cdot)\in\mathcal{P}_{0, \infty}$, $\alpha(\cdot), \beta(\cdot)\in LH(\mathbb{R}^{+})$, bounded. Then Hardy-type inequalities

and

are valid, if and only if, $\alpha(\cdot), \beta(\cdot)$ satisfy conditions $(3.2)$ and $(3.3)$.

Proof. For the proof see Theorem 3.1 and Remark 3.2 in ^[5]. □

As a consequence, we obtain the Hardy's inequalities associated to the exponent function $q(\cdot)\in\mathcal{P}_{0, \infty}$ and the measure $\mu$.

Corollary 3.1. Let $q(\cdot)\in\mathcal{P}_{0, \infty}$ and $r > 0$, then

and

Proof. Let $\alpha(t) = -r+\frac{1}{q'(t)} = -r+1-\frac{1}{q(t)},$ for any $t\in \mathbb{R}^{+}$, $f(y) = y^{\alpha(y)}g(y),$ for any $y\in \mathbb{R}^{+}$ then $\alpha(\cdot)\in LH(\mathbb{R}^{+})$ and bounded, $\alpha(0) = -r+\frac{1}{q'(0)} < \frac{1}{q'(0)}$ and $\alpha(\infty) = -r+\frac{1}{q'(\infty)} < \frac{1}{q'(\infty)}$. Then, using (3.1) and (3.4)

On the other hand, by taking $\beta(t) = r-\frac{1}{q(t)}, f(y) = y^{\beta(y)+1}g(y), t, y\in \mathbb{R}^{+}$ then $\beta(\cdot)\in LH(\mathbb{R}^{+})$ and the proof of (3.7) is completely analogous. □

4.   Main results

In this section we are going to introduce variable Gaussian Besov-Lipschitz spaces and variable Gaussian Triebel-Lizorkin spaces. In what follows we will consider only variable Lebesgue spaces with respect to the Gaussian measure $\gamma_d.$ The next condition was introduced by E. Dalmasso and R. Scotto in ^[3].

Definition 4.1. 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 $x\in\mathbb{R}^{d}\setminus\{(0, 0, \ldots, 0)\}.$

Example 4.1. Consider $p(x) = p_{\infty}+ \frac{A}{(e+\|x\|)^{q}}$, $x\in\mathbb{R}^{d}$, for any $p_{\infty}\geq 1$, $A\geq 0$ and $q\geq 2$. Then $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$.

Remark 4.1. It can be proved that if $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})$, then $p(\cdot)\in LH_{\infty}(\mathbb{R}^{d})$.

In fact, by fixing $x_{0}\in \mathbb{R}^{d}$, such that $\|x_{0}\| = 1$, we get $\log(e+\|x-x_{0}\|)\leq C\|x\|^{2}$, for all $x\in \mathbb{R}^{d}$.

Lipschitz spaces can be generalized of the following way (see, for example ^{[10,12,14,15]}), using the Poisson-Hermite semigroup. We are ready to define variable Gaussian Besov-Lipschitz spaces $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$, also called Gaussian Besov-Lipschitz spaces with variable exponents or variable Besov-Lipschitz spaces for expansions in Hermite polinomials.

Definition 4.2. 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 $\alpha \geq 0$, $k$ the smallest integer greater than $\alpha$. The variable Gaussian Besov-Lipschitz space $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$ is defined as the set of functions $f \in L^{p(\cdot)}(\gamma_d)$ such that

The norm of $f \in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$ is defined as

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

and then the norm of $f \in B_{p(\cdot), \infty}^{\alpha}(\gamma_d)$ is defined as

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

The following lemmas show that we could have replaced $k$ with any other integer $l$ greater than $\alpha$ and the resulting norms are equivalents. Next, we denote $u(\cdot, t) = P_{t}f$.

Lemma 4.1. Let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$, $f\in L^{p(\cdot)}(\gamma_d), \alpha\geq 0$ and $k, l$ integers greater than $\alpha$, then

Moreover, if $A_{k}(f), A_{l}(f)$ are the smallest constants in the inequalities above then there exist constants $A_{k, l, \alpha, p(\cdot)}$ and $D_{k, l, \alpha}$ such that

for all $f\in L^{p(\cdot)}(\gamma_d)$.

Proof. Let us suppose without loss of generality that $k\geq l$. We start by proving the direct implication. For this we use the representation of the Poisson-Hermite semigroup (2.6), this is,

Then, by differentiating $k$-times with respect to $t$ and by using the dominated convergence theorem, we get

By using Lemma 2.3, it's easy to prove that for all $m\in \mathbb{N}$

Now, given $n\in \mathbb{N}$, $n > \alpha$

Thus, for Minkowski's integral inequality (2.19)

Therefore

and since $n > \alpha$ is arbitrary, then, by using the above result $k-l$ times, we obtain

To prove the converse implication, we use again the representation (2.6) and we obtain that

Thus, taking $t = t_{1}+t_{2}$ and differentiating $l$ times with respect to $t_{2}$ and $k-l$ times with respect to $t_{1}$, we get

Then, by Corollary 2.1, Minkowski's integral inequality (2.19) and the $L^{p(\cdot)}$-boundedness of the Ornstein-Uhlenbeck semigroup (see ^[8]), we get

Therefore, taking $t_{1} = t_{2} = \frac{t}{2}$,

Thus, $A_{k}(f)\leq 4C_{p(\cdot)}\frac{C_{k-l}}{2^{-k+\alpha}} A_{l}(f).$ □

Lemma 4.2. 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 $k, l$ integers greater than $\alpha$. Then

if and only if

Moreover, there exist constants $A_{k, l, \alpha, p(\cdot)}$ and $D_{k, l, \alpha, q(\cdot)}$ such that

for all $f\in L^{p(\cdot)}(\gamma_d)$.

Proof. Suppose without loss of generality that $k\geq l$. We prove first the converse implication; by proceeding as in Lemma 4.1, taking $t_{1} = t_{2} = \frac{t}{2}$, we have

Thus

with $A_{k, l, \alpha, p(\cdot)} = 4C_{p(\cdot)}C_{k-l} 2^{k-\alpha}$.

For the direct implication, given $n\in \mathbb{N}$, $n > \alpha$, again, as in the above lemma

Therefore, from this and by Hardy's inequality (3.7)

Now, since $n > \alpha$ is arbitrary, by using the previous result $k-l$ times, we obtain

where $D_{k, l, \alpha, q(\cdot)} = 4^{k-l}C_{l, \alpha, q(\cdot)}\cdots C_{k-1, \alpha, q(\cdot)}$. □

Now, we define variable Gaussian Triebel-Lizorkin spaces $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$, which represent another way to measure regularity of functions, proceeding as in ^[10,14,15].

Definition 4.3. 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 $k$ the smallest integer greater than $\alpha$. The variable Gaussian Triebel-Lizorkin space $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ is the set of functions $f\in L^{p(\cdot)}(\gamma_d)$ such that

the norm of $f \in F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$ is defined as

The following lemma shows that the definition of $F_{p(\cdot), q(\cdot)}^{\alpha}$ is independent of the integer $k > \alpha$ chosen and the resulting norms are equivalents.

Lemma 4.3. 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 $k, l$ integers greater than $\alpha$. Then

if and only if

Moreover, there exist constants $A_{k, l, \alpha, p(\cdot)}, D_{k, l, \alpha, q(\cdot)}$ such that

for all $f\in L^{p(\cdot)}(\gamma_d)$.

Proof. Suppose without loss of generality that $k\geq l$. Let $n\in \mathbb{N}$ such that $n > \alpha$, we can prove that

Then, by the Hardy's inequality (3.7),

Now, since $n > \alpha$ is arbitrary, by iterating the previous argument $k-l$ times, we obtain

where $C_{k, l, \alpha, q(\cdot)} = C_{l, \alpha, q(\cdot)} C_{l+1, \alpha, q(\cdot)}\cdots C_{k-1, \alpha, q(\cdot)}.$ Thus, $D_{k, l, \alpha, q(\cdot)}\left\|\left\|t^{l-\alpha}\left|\frac{\partial^{l}}{\partial t^{l}}P_{t}f\right|\right\|_{q(\cdot), \mu}\right\|_{p(\cdot), \gamma_d}\leq\left\|\left\|t^{k-\alpha}\left| \frac{\partial^{k}}{\partial t^{k}}P_{t}f\right|\right\|_{q(\cdot), \mu}\right\|_{p(\cdot), \gamma_d}$, with, $D_{k, l, \alpha, q(\cdot)} = 1/C_{k, l, \alpha, q(\cdot)}.$ The other inequality is obtained from the case $k = l+1$ by an inductive argument. Let $t_{1}, t_{2} > 0$ and take $t = t_{1}+t_{2}$, from (4.5) we get

and since, $\frac{\partial }{\partial t_{1}}\mu_{t_{1}}^{(1/2)} (ds) = \left(t_{1}^{-1}-\frac{t_{1}}{2s}\right)\mu_{t_{1}}^{(1/2)}(ds)$, we obtain

Therefore

Now, by using Minkowski's integral inequality twice (2.19) (since $T_{s}$ is an integral transformation with positive kernel) and the fact that $\mu_{t_{1}}^{(1/2)}(ds)$ is a probability measure, we get

For $(II)$, we proceed in analogous way, and by using Lemma 2.2 we get

Now, since $T^{\ast}$ is defined as a supremum, we get

Then, taking $t_{1} = t_{2} = \frac{t}{2}$ and the change of variable $s = \frac{t}{2}$, we have

and

Thus, by the $L^{p(\cdot)}(\gamma_d)$-boundedness of $T^*$ (see ^[8]),

Therefore,

□

Next, we need a technical result for the $L^{p(\cdot)}(\gamma_d)$-norms of the derivatives of the Poisson-Hermite semigroup:

Lemma 4.4. Let $p(\cdot)\in\mathcal{P}_{\gamma_{d}}^{\infty}(\mathbb{R}^{d})\cap LH_{0}(\mathbb{R}^{d})$. Suppose that $f\in L^{p(\cdot)}(\gamma_d)$, then for any integer $k$, $\left\|\frac{\partial^{k}}{\partial t^{k}}P_t f\right\|_{p(\cdot), \gamma_{d}}\leq C_{p(\cdot)}\left\|\frac{\partial^{k}}{\partial s^{k}}P_s f\right\|_{p(\cdot), \gamma_{d}}$, for whatever $0 < s\leq t < +\infty$. Moreover,

Proof. First, let us consider the case $k = 0$. Fixed $t_{1}, t_{2} > 0$, by using the semigroup property of $\{P_{t}\}$, we get

Thus, by the $L^{p(\cdot)}$-boundedness of $\{P_{t}\}$ (see ^[8]),

In order to prove the general case, $k > 0$, using the dominated convergence theorem and differentiating the identity $u(x, t_{1}+t_{2}) = P_{t_{1}}(u(x, t_{2}))$ $k$-times with respect to $t_{2}$, we obtain

and then we proceed as in the previous argument.

Finally, to prove (4.8), we use again the representation (2.6) of the Poisson-Hermite semigroup and differentiating $k$-times with respect to $t$, we get

Thus, by the Minkowski's integral inequality, the $L^{p(\cdot)}$-boundedness of the Ornstein-Uhlenbeck semigroup (see ^[8]) and the Corollary 2.1,

Hence, $\left\|\frac{\partial^{k}u(\cdot, t)}{\partial t^{k}}\right\|_{p(\cdot), \gamma_d}\leq\frac{C_{k, p(\cdot)}}{t^{k}} \|f\|_{p(\cdot), \gamma_d}$, $t > 0$. □

Now, let us study some inclusion relations between variable Gaussian Besov-Lipschitz spaces. The next result is analogous to Proposition 10, page 153 in ^[12] (see also ^[10] or Proposition 7.36 in ^[16]).

Proposition 4.1. 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. $\left\| f \right\|_{B_{p(\cdot), q_{2}(\cdot)}^{\alpha_{2}}(\gamma_d)}\leq C \left\| f \right\|_{B_{p(\cdot), q_{1}(\cdot)}^{\alpha_{1}}(\gamma_d)}$ if:

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

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

Proof. To prove part $ii)$, let us take $\alpha$ the common value of $\alpha_{1}$ and $\alpha_{2}$.

Let $f\in B_{p(\cdot), q_1(\cdot)}^{\alpha}$ and set $A = \left\|t^{k-\alpha}\left\|\frac{\partial^{k}P_t f}{\partial t^{k} }\right\|_{p(\cdot), \gamma_d}\right\|_{q_{1}(\cdot), \mu}$.

Fixed $t_{0} > 0$,

However, by Lemma 4.4,

Thus, we obtain

Therefore,

and by Lemma 3.1

Then,

and since $t_{0}$ is arbitrary

In other words, $f\in B_{p(\cdot), q_1(\cdot)}^{\alpha}$ implies that $f\in B_{p(\cdot), \infty}^{\alpha}$.

Now, let us take $g(t) = t^{k-\alpha}\left\|\frac{\partial^{k}P_t f}{\partial t^{k} }\right\|_{p(\cdot), \gamma_d}$, then $\rho_{q_{1}(\cdot), \mu}(g) < \infty$, since $f\in B_{p(\cdot), q_1(\cdot)}^{\alpha}$.

Thus, as $q_{2}(t)\geq q_{1}(t)\quad a.e.$,

Hence, $f\in B_{p(\cdot), q_2(\cdot)}^{\alpha}$.

In order to prove part $i)$, by Lemma 4.4, we obtain

Now, given $f\in B_{p(\cdot), q_1(\cdot)}^{\alpha_{1}}$, again by setting

we obtain, as in part $ii)$,

Therefore,

Now, again by Lemma 3.1 we get,

and also by Lemma 3.1,

Hence, $\left\|t^{k-\alpha_{2}}\left\|\frac{\partial^{k}P_t f}{\partial t^{k} }\right\|_{p(\cdot), \gamma_d}\right\|_{q_{2}(\cdot), \mu} < +\infty$, and then $f\in B_{p(\cdot), q_2(\cdot)}^{\alpha_{2}}$. □

Remark 4.2. Variable Gaussian Besov-Lipschitz and variable Gaussian Triebel-Lizorkin spaces are, by construction, subspaces of $L^{p(\cdot)}(\gamma_d)$. Moreover, since trivially $\left\| f \right\|_{p(\cdot), \gamma_d} \leq \left\| f \right\|_{B_{p(\cdot), q(\cdot)}^{\alpha}}$ and $\left\| f \right\|_{p(\cdot), \gamma_d} \leq \left\| f \right\|_{F_{p(\cdot), q(\cdot)}^{\alpha}}$, the inclusions are continuous.

On the other hand, from (2.9) it is clear that for all $t > 0$ and $k\in \mathbb{N}$,

and again by Lemma 3.1,

Thus, $h_{\beta}\in B^{\alpha}_{p(\cdot), q(\cdot)}(\gamma_{d})$ and

In a similar way, $h_{\beta}\in F^{\alpha}_{p(\cdot), q(\cdot)}(\gamma_{d})$ and

Hence, the set of all polynomials ${\mathcal P}$ is contained in $B^{\alpha}_{p(\cdot), q(\cdot)}(\gamma_{d})$ and in $F^{\alpha}_{p(\cdot), q(\cdot)}(\gamma_{d})$.

Also, we have an inclusion result for variable Gaussian Triebel-Lizorkin spaces, which is analogous to Proposition 4.1, see also ^[10] or Proposition 7.40 in ^[16].

Proposition 4.2. 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

$F_{p(\cdot), q_1(\cdot)}^{\alpha_{1}}(\gamma_d)\subset F_{p(\cdot), q_2(\cdot)}^{\alpha_{2}}(\gamma_d)$ holds i.e., $\left\| f \right\|_{F_{p(\cdot), q_{2}(\cdot)}^{\alpha_{2}}(\gamma_d)}\leq C \left\| f \right\|_{F_{p(\cdot), q_{1}(\cdot)}^{\alpha_{1}}(\gamma_d)}$,

for $\alpha_1 > \alpha_2 > 0$ and $q_1(t) > q_2(t)\quad a.e.$

Proof. Let us consider $f\in F_{p(\cdot), q_1(\cdot)}^{\alpha_{1}}$, then

Now, since $q_{1}(t) > q_{2}(t)\quad a.e.$, by taking $r(t) = \frac{q_{1}(t)q_{2}(t)}{q_{1}(t)-q_{2}(t)}$, we obtain that $r(\cdot)\geq 1\quad a.e.$ and $\frac{1}{r(\cdot)}+\frac{1}{q_{1}(\cdot)} = \frac{1}{q_{2}(\cdot)}$. Thus, by Hölder's inequality (2.18) and Lemma 3.1

Now, for the second term $(II)$, by using Lemmas 3.1 and 2.3, we get

Then, by using the $L^{p(\cdot)}(\gamma_d)$ boundedness of $T^{\ast}$ (see ^[8]),

Therefore, $f\in F_{p(\cdot), q_{2}(\cdot)}^{\alpha_{2}}.$ □

Finally, we are going to consider some interpolation results for the Gaussian variable Besov-Lipschitz and the variable Triebel-Lizorkin spaces. We will use the following results for general variable Lebesgue spaces $L^{p(\cdot)}(X, \nu)$.

Lemma 4.5. Let $p(\cdot)\in \mathcal{P}(\Omega, \nu)$ and $s > 0$ such that $sp^{-}\geq 1$. Then

$\||f|^{s}\|_{p(\cdot), \nu} = \|f\|^{s}_{sp(\cdot), \nu}$.

Proof. It is the same proof of Lemma 3.2.6 in ^[4]. □

Lemma 4.6. Let $\nu$ a complete $\sigma$-finite measure on $X$. $r_{j}(\cdot)\in \mathcal{P}(X, \nu)$, $1 < r^{-}_{j}, r^{+}_{j} < \infty$, $j = 0, 1$. For all $0 < \lambda < 1$, if $f\in L^{r_{j}(\cdot)}(X, \nu)$, $j = 0, 1$ then $f\in L^{r(\cdot)}(X, \nu)$ where $\frac{1}{r(y)} = \frac{1-\lambda}{r_{0}(y)}+\frac{\lambda}{r_{1}(y)},$ a.e. $y\in X$ and

Proof. It is a consequence of Hölder's inequality (2.18) and Lemma 4.5. □

Now, we present the interpolation result.

Theorem 4.1. Let $p_{j}(\cdot)\in \mathcal{P}(\mathbb{R}^{d}, \gamma_{d})$, $q_{j}\in \mathcal{P}(\mathbb{R}^{+}, \mu)$, with $j = 0, 1$. Suppose that $1 < p^{-}_{j}, q^{-}_{j}$, $p^{+}_{j}, q^{+}_{j} < +\infty$ and $\alpha_{j}\geq 0$. For all $0 < \theta < 1$, let us take

$i)$ if $f\in B_{p_{j}(\cdot), q_{j}(\cdot)}^{\alpha_{j}}(\gamma_d)$, $j = 0, 1,$ then $f\in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$.

$ii)$ if $f\in F_{p_{j}(\cdot), q_{j}(\cdot)}^{\alpha_{j}}(\gamma_d)$, $j = 0, 1,$ then $f\in F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$.

Proof. $i)$ Let $k$ be any integer greater than $\alpha_{0}$ and $\alpha_{1}$, by using Lemma 4.6, we obtain for $\alpha = \alpha_0 (1- \theta) +\alpha_1 \theta$,

Thus, by Hölder's inequality (2.18) and Lemma 4.5,

that is, $f\in B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$.

$ii)$ Analogously, by Hölder's inequality (2.18) and Lemma 4.5, we obtain for $\alpha = \alpha_0 (1- \theta) +\alpha_1 \theta$,

Therefore

and again by Hölder's inequality and Lemma 4.5,

that is, $f\in F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_d)$. □

In a forthcoming paper ^[9], we establish boundedness properties on $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ for some operators associated with the Gaussian measure, such as Riesz Potentials, Bessel Potentials and Bessel Fractional Derivatives.

5.   Conclusions

$i)$ Lemmas 4.1–4.3 showed that the definitions of $B_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ and $F_{p(\cdot), q(\cdot)}^{\alpha}(\gamma_{d})$ are independent of the integer $k$ greater than $\alpha$ considered and the corresponding norms are equivalent.

$ii)$ Lemma 3.1 was the key in the proof of Proposition 4.1.

$iii)$ The boundedness of the maximal function of the Ornstein-Uhlenbeck semigroup $T^{*}$ on $L^{p(\cdot)}(\gamma_{d})$ (see ^[8]) was crucial in the proof of Lemma 4.4 and Proposition 4.2.

$iv)$ The structure and properties of the Besov-Lipschitz and Triebel-Lizokin Gaussian spaces are preserved when we go from constant exponent to variable exponent setting 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 $q(\cdot)\in\mathcal{P}_{0, \infty}$.

Use of 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.

Also we like to thank Professor Elvis Aponte for his carefully reading and corrections of the manuscript.

Conflict of interest

The authors declare no conflict of interest.