Mixed Sobolev-like inequalities in Lebesgue spaces of variable exponents and in Orlicz spaces

Abstract

In this short article we show a particular version of the Hedberg inequality which can be used to derive, in a very simple manner, functional inequalities involving Sobolev and Besov spaces in the general setting of Lebesgue spaces of variable exponents and in the framework of Orlicz spaces.

Appendix

An alternative proof of Theorem 2 relies in the characterization of the Lebesgue and Sobolev spaces of variable exponents using the Littlewood–Paley decomposition. In the classical setting, the representation of these spaces using dyadic blocs is well known (see the books [2, 13]). For spaces of variable exponents, this theory is given in [1], see also Chapter 12 of the book [10]. Let us point out that the identification of the spaces given by a Littlewood–Paley decomposition with the spaces used here (in the non-homogeneous case) is done in the article [11].

Let us briefly recall the Littlewood–Paley decomposition: consider $\varphi \in {\mathcal {S}}({\mathbb {R}}^n, {\mathbb {R}})$ such that ${\widehat{\varphi }}(\xi )=1$ if $|\xi |\le 1/2$ and ${\widehat{\varphi }}(\xi )=0$ if $|\xi |>1$ and for $j\in {\mathbb {Z}}$ define the function $\varphi _j$ by the expression $\varphi _j(x)=2^{-jn}\varphi (2^{-j}x)$. Consider the function $\psi$ which is given by the formula ${\widehat{\psi }}(\xi )={\widehat{\varphi }}(\xi /2)-{\widehat{\varphi }}(\xi )$ and for all $j\in {\mathbb {Z}}$ we set ${\widehat{\psi }}_j(\xi )={\widehat{\psi }}(2^{-j}\xi )={\widehat{\varphi }}_{j+1}(\xi )-{\widehat{\varphi }}_{j}(\xi )$, we have then

$$\begin{aligned} \sum _{j\in {\mathbb {Z}}}{\widehat{\psi }}_j(\xi )\equiv 1\quad \text{ for } \text{ all } \xi \ne 0. \end{aligned}$$

Now, for all $j\in {\mathbb {Z}}$, the dyadic-bloc operator $\Delta _j$ is defined by the formula $\Delta _j(f)=f*\psi _j$ and we have the formula

$$\begin{aligned} f=\sum _{j\in {\mathbb {Z}}}\Delta _j(f), \end{aligned}$$

where the convergence of the sum must be considered in ${\mathcal {S}}'({\mathbb {R}}^n)$ modulo the polynomials ${\mathbb {C}}[X]$.

Now for $p \in {\mathcal {P}}({\mathbb {R}}^n)$ such that $1<p^-\le p^+<+\infty$ and for $0<s<+\infty$, we have the following characterizations of variable exponents spaces: for all $f\in {\mathcal {S}}'/{\mathbb {C}}[X]$ we have

$$\begin{aligned}&\Vert f\Vert _{L^{p(\cdot )}({\mathbb {R}}^n)}\simeq \left\| \left( \displaystyle {\sum _{j\in {\mathbb {Z}}}}|\Delta _{j}(f)(x)|^{2}\right) ^{\frac{1}{2}}\right\| _{L^{p(\cdot )}({\mathbb {R}}^n)}\text{ and } \quad \\&\quad \Vert f\Vert _{\dot{{\mathcal {W}}}^{s,p(\cdot )}({\mathbb {R}}^n)}\simeq \left\| \left( \displaystyle {\sum _{j\in {\mathbb {Z}}}}2^{2sj}|\Delta _{j}(f)(x)|^{2}\right) ^{\frac{1}{2}}\right\| _{L^{p(\cdot )}({\mathbb {R}}^n)}. \end{aligned}$$

Note that the spaces $\dot{{\mathcal {W}}}^{s,p(\cdot )}({\mathbb {R}}^n)$ given above are defined modulo the polynomials and thus they are not equivalent to the spaces ${\dot{W}}^{s,p(\cdot )}({\mathbb {R}}^n)$ given in (2.4). However, in the framework of the Theorem 2, we are considering functions that also belong to the Besov space ${\dot{B}}^{-\beta ,\infty }_{\infty }({\mathbb {R}}^n)$, which can be characterized by the equivalent quantity

$$\begin{aligned} \Vert f\Vert _{{\dot{B}}^{-\beta ,\infty }_{\infty }({\mathbb {R}}^n)}\simeq \underset{j\in {\mathbb {Z}}}{\sup \;} 2^{-\beta j}\Vert \Delta _{j}(f)\Vert _{L^\infty ({\mathbb {R}}^n)}, \end{aligned}$$

(A.1)

and this fact allows us to avoid this unpleasant issue related to the different definitions of homogeneous spaces as we have the equivalence of spaces²${\dot{W}}^{s,p(\cdot )}\cap {\dot{B}}^{-\beta ,\infty }_{\infty }\simeq \dot{{\mathcal {W}}}^{s,p(\cdot )}\cap {\dot{B}}^{-\beta ,\infty }_{\infty }$.

With this short introduction, we can proof Theorem 2 using the tools related to the Littlewood–Paley decomposition. We start with the following interpolation result

Lemma A.1

Let $(a_{j})_{j\in {\mathbb {Z}}}$ be a sequence and set $s=(1-\theta ) s_0+ \theta s_{1}$ with $0<\theta < 1$ and $s_0\ne s_{1}$. Then for all $r,r_{1},r_{2}\in [1,+\infty ]$ we have the interpolation estimate:

$$\begin{aligned} \Vert 2^{js}a_{j}\Vert _{\ell ^{r}}\le C\Vert 2^{js_{0}}a_{j}\Vert _{\ell ^{r_{1}}}^{1-\theta }\Vert 2^{js_{1}}a_{j}\Vert _{\ell ^{r_{2}}}^{\theta }. \end{aligned}$$

See [3] for a proof of this interpolation inequality.

If we apply this lemma to the dyadic blocs $\Delta _{j}(f)$ with $s_1=(1-\theta )s+\theta (-\beta )$ and $r=r_{1}=2$ and $r_{2}=+\infty$, we obtain

$$\begin{aligned}&\left( \sum _{j\in {\mathbb {Z}}}2^{2s_1j}|\Delta _{j}(f)(x)|^{2}\right) ^{\frac{1}{2(1-\theta )}}\\&\quad \le C \left( \sum _{j\in {\mathbb {Z}}}2^{2sj}|\Delta _{j}(f)(x)|^{2}\right) ^{\frac{1}{2}} \left( \underset{j\in {\mathbb {Z}}}{\sup \;} 2^{-\beta j}|\Delta _{j}(f)(x)|\right) ^{\frac{\theta }{1-\theta }}\\&\quad \le C \left( \sum _{j\in {\mathbb {Z}}}2^{2sj}|\Delta _{j}(f)(x)|^{2}\right) ^{\frac{1}{2}}\Vert f\Vert _{{\dot{B}}^{-\beta ,\infty }_\infty ({\mathbb {R}}^n)}^{\frac{\theta }{1-\theta }}, \end{aligned}$$

where in the last line we used the characterization of the Besov space ${\dot{B}}^{-\beta ,\infty }_\infty ({\mathbb {R}}^n)$ in terms of the dyadic blocs given in (A.1). Now, we take the Luxemburg $L^{p(\cdot )}$-norm to get

$$\begin{aligned}&\left\| \left( \sum _{j\in {\mathbb {Z}}}2^{2s_1j}|\Delta _{j}(f)|^{2}\right) ^{\frac{1}{2} \frac{1}{1-\theta }}\right\| _{L^{p(\cdot )}({\mathbb {R}}^n)}\\&\quad \le C \left\| \left( \sum _{j\in {\mathbb {Z}}}2^{2sj}|\Delta _{j}(f)|^{2}\right) ^{\frac{1}{2}} \right\| _{L^{p(\cdot )}({\mathbb {R}}^n)}\Vert f\Vert _{{\dot{B}}^{-\beta ,\infty }_\infty ({\mathbb {R}}^n)}^{\frac{\theta }{1-\theta }}. \end{aligned}$$

Then, using the property (2.3) we obtain

$$\begin{aligned}&\left\| \left( \sum _{j\in {\mathbb {Z}}}2^{2s_1j}|\Delta _{j}(f)|^{2}\right) ^{\frac{1}{2} }\right\| _{L^{\frac{p(\cdot )}{1-\theta }}({\mathbb {R}}^n)}^{\frac{1}{1-\theta }}\\&\quad \le C \left\| \left( \sum _{j\in {\mathbb {Z}}}2^{2sj}|\Delta _{j}(f)|^{2}\right) ^{\frac{1}{2}} \right\| _{L^{p(\cdot )}({\mathbb {R}}^n)}\Vert f\Vert _{{\dot{B}}^{-\beta ,\infty }_\infty ({\mathbb {R}}^n)}^{\frac{\theta }{1-\theta }}. \end{aligned}$$

To finish, we recall that $q(\cdot )=\frac{p(\cdot )}{1-\theta }$ and using the characterization of Sobolev spaces via the Littlewood–Paley theory we can write

$$\begin{aligned} \Vert f\Vert _{\dot{{\mathcal {W}}}^{s_1,q(\cdot )}({\mathbb {R}}^n)}\le C \Vert f\Vert _{\dot{{\mathcal {W}}}^{s,p(\cdot )}({\mathbb {R}}^n)}^{1-\theta }\Vert f\Vert _{{\dot{B}}^{-\beta ,\infty }_{\infty }({\mathbb {R}}^n)}^\theta . \end{aligned}$$

$\square$

Remark A.1

Let us note here that this second proof of the Sobolev-like estimates relies in the Littlewood–Paley theory which is available in the setting of Lebesgue spaces of variable exponent over the euclidean space ${\mathbb {R}}^n$. But this is not always the case if we consider general spaces over other spaces than ${\mathbb {R}}^n$. In this sense the first proof based in the modified Hedberg inequality (1.19) seems more robust and simple to display.