Trace principle for Riesz potentials on Herz-type spaces and applications

Abstract

We establish trace inequalities for Riesz potentials on Herz-type spaces and examine the optimality of conditions imposed on specific parameters. We also present some applications in the form of Sobolev-type inequalities, including the Gagliardo–Nirenberg–Sobolev inequality and the fractional integration theorem in the Herz space setting. In addition, we obtain a Sobolev embedding theorem for Herz-type Sobolev spaces.

1 Introduction and preliminaries

The Riesz potential operator $I_{\gamma}$ is an integral operator defined by the convolution of a function f with the Riesz kernel $K_{\gamma}(x):=|x|^{\gamma -n}$. More precisely, for $n \in \mathbb{N}$ and $0 < \gamma < n$,

$$ I_{\gamma}f(x):= \int _{\mathbb{R}^{n}}\frac{f(y)}{ \vert x-y \vert ^{n-\gamma}}\,dm(y),\quad x \in \mathbb{R}^{n}, $$

where f is a suitable function, for example, a locally integrable function on $\mathbb{R}^{n}( L^{1}_{\mathrm{loc}}(\mathbb{R}^{n}) )$ or a function with sufficiently rapid decay at infinity, particularly, if $f \in L^{p}(\mathbb{R}^{n})$ with $1 \leq p < \frac{n}{\gamma}$, and m is the Lebesgue measure on $\mathbb{R}^{n}$. If $\gamma = 2 \neq n$, then this integral operator is called the Newtonian potential and is used to describe the potential energy distribution of a system of point masses in classical mechanics or the electrostatic potential created by a charge distribution in physics.

The trace problem for Riesz potentials deals with finding nonnegative (positive) Borel measures μ on $\mathbb{R}^{n}$ such that $I_{\gamma}$ maps $\mathcal{F}(\mathbb{R}^{n}, m)$ boundedly into $\mathcal{F}^{\prime}(\mathbb{R}^{n}, \mu )$, where $\mathcal{F}(\mathbb{R}^{n},m)$ and $\mathcal{F}^{\prime}(\mathbb{R}^{n},\mu )$ are function spaces defined over $\mathbb{R}^{n}$ with respect to measures m and μ, respectively. Adams [1, 2] proved that for $1< p_{1}< p_{2}<\infty$ and $0<\gamma <\frac{n}{p_{1}}$,

$$ \Vert I_{\gamma}f \Vert _{L^{p_{2}}(\mathbb{R}^{n}, \mu )} \lesssim \Vert f \Vert _{L^{p_{1}}(\mathbb{R}^{n}, m)} $$

(1)

if and only if $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}}- \frac{\gamma}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$. Here we have used the standard notation $\zeta \lesssim \rho$ (or, equivalently, $\rho \gtrsim \zeta$) to express that there exists a positive constant c, independent of relevant variables, such that $\zeta \leq c \rho$. Inequality (1) is not true when $p_{1} = p_{2}$ (see, for example, [3]). Inequalities involving Riesz potentials often provide an important tool for estimating functions in terms of the norms of their derivatives. The wide-ranging applicability of trace inequalities for Riesz potentials has sparked significant interest in recent studies; see, for instance, [7, 9, 15, 24] and references therein. For Morrey–Lorentz spaces, the following theorem has been established in [7].

Theorem 1.1

Let $1< p_{1} \leq q_{1} < \infty$ and $1< p_{2} \leq q_{2}< \infty$ satisfy $\frac{p_{2}}{q_{2}} \leq \frac{p_{1}}{q_{1}}$ for all $1< p_{1}< p_{2}<\infty$. Then the inequality

$$ \Vert I_{\gamma}f \Vert _{\mathcal{M}_{p_{2},r_{2}}^{q_{2}}( \mathbb{R}^{n}, \mu )} \lesssim \Vert f \Vert _{ \mathcal{M}_{p_{1},r_{1}}^{q_{1}}(\mathbb{R}^{n}, m)} $$

holds if and only if the measure μ satisfies $\mu (B) \lesssim [m(B)]^{q_{2} (\frac{1}{q_{1}}- \frac{\gamma}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$, given that $n (\frac{1}{q_{1}}-\frac{1}{q_{2}} ) \leq \gamma < \frac{n}{q_{1}}$ and $1 \leq r_{1} < r_{2} \leq \infty$ (or $r_{1}=r_{2}=\infty$).

In particular, this yields the following outcome for Lorentz spaces (see Definition 1.3).

Corollary 1.2

If $1< p_{1}< p_{2}<\infty$, $n (\frac{1}{p_{1}}-\frac{1}{p_{2}} ) \leq \gamma < \frac{n}{p_{1}}$, and $1 \leq r_{1} < r_{2} \leq \infty$ (or $r_{1}=r_{2}=\infty$). Then

$$ \Vert I_{\gamma}f \Vert _{L^{p_{2},r_{2}}(\mathbb{R}^{n}, \mu )} \lesssim \Vert f \Vert _{L^{p_{1},r_{1}}( \mathbb{R}^{n}, m)} $$

if and only if the measure μ satisfies $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}}- \frac{\gamma}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$.

1.1 Function spaces

In this subsection, we fix some notations and recall definitions of certain function spaces required for the subsequent discussion. We begin with Lorentz spaces.

Definition 1.3

A Lorentz space $L^{p,r}(\Omega , \nu )$ defined over a σ-finite measure space $(\Omega , \Sigma , \nu )$ consists of all ν-measurable functions on Ω for which the functional $\|f\|_{L^{p,r}(\Omega , \nu )}$ is finite, where

$$ \Vert f \Vert _{L^{p,r}(\Omega , \nu )}:=\textstyle\begin{cases} ( \int _{0}^{\infty } ( t^{\frac{1}{p}}f^{\ast } ( t ) ) ^{r}\frac{dt}{t} )^{1/r} & \text{if } 0< p< \infty ,0< r< \infty , \\ \sup_{t>0} t^{\frac{1}{p}}f^{\ast } ( t ) & \text{if } 0< p\leq \infty ,r=\infty ,\end{cases} $$

and $f^{*}(t) :=\operatorname{inf}\{s \geq 0 : \nu (\{x \in \Omega : |f|>s \}) \leq t \}$, $t \geq 0$, is the decreasing (or nonincreasing) rearrangement of f.

Note that $L^{p,p} = L^{p}$. It is important to emphasize that $\|\cdot \|_{L^{p,r}(\Omega , \nu )}$ is not always a norm, but rather a quasi-norm (see [5, p. 216]). However, we can define a functional $\|\cdot \|_{L^{(p,r)}(\Omega , \nu )}$ on $L^{p,r}(\Omega , \nu )$ as follows:

$$ \Vert f \Vert _{L^{ ( p,r )}(\Omega , \nu )}:= \textstyle\begin{cases} ( \int _{0}^{\infty} ( t^{\frac{1}{p}}f^{\ast \ast } ( t ) )^{r}\frac{dt}{t} )^{1/r} & \text{if } 0< p< \infty ,0< r< \infty , \\ \sup_{t>0} t^{\frac{1}{p}}f^{\ast \ast } ( t ) & \text{if } 0< p\leq \infty ,r=\infty , \end{cases} $$

where the function $f^{**}(t):=\frac{1}{t}\int _{0}^{t} f^{*}(t)\,dt$ is referred to as the maximal average function. Fortunately, this functional is subadditive. Consequently, $L^{(p,r)}(\Omega , \nu ) := ( L^{p,r}(\Omega , \nu ),\|\cdot \|_{L^{(p,r)}(\Omega , \nu )} )$ is a normed space for $1< p< \infty$, $1 \leq r \leq \infty$, or $p=r=\infty$. Since $f^{*} \leq f^{**}$, we have $L^{(p,r)} \hookrightarrow L^{p,r}$. Moreover, if $1< p \leq \infty$ and $1 \leq r \leq \infty$, then $L^{p,r} \hookrightarrow L^{(p,r)}$ (see [5, Lemma 4.5, p. 219]). The substitution of $L^{p,1}$ for $L^{p_{1}}$ on the right-hand side of inequality (1) retains its validity in the limiting case $p_{1}=p_{2}=p$ (see [13, 14]).

Another important generalization of Lebesgue spaces is the classical Herz space, introduced by Herz [12] as a suitable environment for the action of Fourier transform on a Lipschitz class. Although the Herz spaces are defined in various equivalent ways, we adopt the formulation presented in [11, 19] with a slightly changed notation for our convenience.

Let $( \Omega _{t} )_{t \in \mathbb{Z}}$ be the dyadic decomposition of $\mathbb{R}^{n}$, i.e., $\Omega _{t}=\{x\in \mathbb{R}^{n}:2^{t-1} \leq |x|<2^{t}\}$ for $t\in \mathbb{Z}$. We denote $\tilde{\chi}_{\Omega _{t}}=\chi _{\Omega _{t}}$ for $t\in \mathbb{Z}_{+}$, and $\tilde{\chi}_{\Omega _{-1}}=\chi _{B(0,\frac{1}{2})}$, where $B(0,\frac{1}{2})$ represents the ball centered at the origin with radius $\frac{1}{2}$ in $\mathbb{R}^{n}$.

Definition 1.4

Let $\lambda \in \mathbb{R}$ and $0< p,q\leq \infty$, and let ν be a positive measure on $\mathbb{R}^{n}$.

• (i) The homogeneous Herz space $\dot{K}_{\lambda ,q}^{p}(\mathbb{R}^{n}, \nu )$ is defined by

  $$ \dot{K}_{\lambda ,q}^{p}\bigl(\mathbb{R}^{n}, \nu \bigr):= \bigl\{ f \in L_{loc}^{p}\bigl( \mathbb{R}^{n} \setminus \{0\}, \nu \bigr): \Vert f \Vert _{\dot{K}_{ \lambda ,q}^{p}(\mathbb{R}^{n}, \nu )}< \infty \bigr\} , $$

  where

  $$ \Vert f \Vert _{\dot{K}_{\lambda ,q}^{p}(\mathbb{R}^{n}, \nu )}:= \biggl( \sum_{t\in \mathbb{Z}} 2^{t \lambda q} \Vert f\chi _{\Omega _{t}} \Vert _{{L^{p}(\mathbb{R}^{n}, \nu )}}^{q} \biggr) ^{\frac{1}{q}}. $$

• (ii) The inhomogeneous Herz space $K_{\lambda ,q}^{p}(\mathbb{R}^{n}, \nu )$ is defined by

  $$ K_{\lambda ,q}^{p}\bigl(\mathbb{R}^{n}, \nu \bigr):=\bigl\{ f\in L_{loc}^{p}\bigl( \mathbb{R}^{n}, \nu \bigr): \Vert f \Vert _{K_{\lambda ,q}^{p}(\mathbb{R}^{n}, \nu )}< \infty \bigr\} , $$

  where

  $$ \Vert f \Vert _{K_{\lambda ,q}^{p}(\mathbb{R}^{n})}:= \Biggl( \sum_{t=-1}^{ \infty }2^{t \lambda q} \Vert f\tilde{\chi}_{\Omega _{t}} \Vert _{{L^{p}( \mathbb{R}^{n}, \nu )}}^{q} \Biggr) ^{\frac{1}{q}}. $$

If p and/or q are infinite, then the usual modifications are made.

It is obvious that $\dot{K}_{0,p}^{p}(\mathbb{R}^{n}, \nu )= K_{0,p}^{p}(\mathbb{R}^{n}, \nu )= L^{p}(\mathbb{R}^{n}, \nu )$. In recent years, there has been substantial advancement in the development of Herz spaces, primarily driven by their wide range of applications (see, for instance, [4, 8, 10, 16, 22, 23] and references therein). However, Herz spaces alone are insufficient to describe some fine properties of functions and operators. Consequently, defining the Lorentz–Herz spaces $\dot{HL}_{\lambda ,q}^{p,r}(\mathbb{R}^{n}, \nu )$ and $HL_{\lambda ,q}^{p,r}(\mathbb{R}^{n}, \nu )$ emerges as a natural progression. These spaces are derived simply by amalgamating Lorentz spaces with Lebesgue sequence spaces, essentially replacing the functionals $\| \cdot \|_{L^{p}(\mathbb{R}^{n}, \nu )}$ with $\| \cdot \|_{L^{p,r}(\mathbb{R}^{n}, \nu )}$ in Definition 1.4. The properties of these spaces, even in more general settings, have been investigated in [6].

The trace principle for Riesz potentials on Herz spaces and their extensions remains absent from the academic literature. This absence is particularly worth noting, given the pivotal role that inequalities associated with Riesz potentials are indispensable tools for estimating functions in terms of their gradients, commonly referred to as Sobolev inequalities. These inequalities are considered as cornerstones of the Sobolev theory in partial differential equations.

To establish such estimates within the Herz-type setting, the derivation of trace inequalities is of paramount importance. In the ensuing sections, we present rigorous proofs of trace inequalities for both Herz and Lorentz–Herz spaces. It is important to mention that our focus here is on homogeneous spaces; however, analogous proofs for nonhomogeneous spaces can be conducted similarly. Additionally, we engage in a comprehensive discussion on the optimality of specific parametric conditions inherent in these trace inequalities. The resulting trace theorems subsequently facilitate the proof of Sobolev inequalities within Herz space settings, providing succinct estimates for functions in relation to their gradients. As a consequential outcome, we establish a Sobolev embedding theorem for Herz-type Sobolev spaces.

2 Trace theorems

We begin this section by presenting the trace theorem for Herz spaces. To handle convolution operators with kernels having singularities at the origin, we adopt a conventional and widely used approach. It involves decomposing the summation into distinct components, systematically accounting for the presence of singularity. This well-established technique has found pervasive application in several research papers. Hereafter, if the measure associated with a particular norm is not explicitly mentioned, then it is to be understood as the Lebesgue measure on $\mathbb{R}^{n}$.

Theorem 2.1

Assume that $1 < p_{1} < p_{2} < \infty$, $1 \leq q_{1} \leq q_{2} < \infty$, $0 < \gamma < \frac{n}{p_{1}}$, and $\gamma - \frac{n}{p_{1}} < \lambda < n - \frac{n}{p_{1}}$. If $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}} - \frac{\gamma}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$, then

$$ \Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} \lesssim \Vert f \Vert _{\dot{K}_{ \lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)}. $$

(2)

Proof

Let $f \in \dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)$. Since $0< \frac{q_{1}}{q_{2}} \leq 1$, we have

$$\begin{aligned} \Vert I_{\gamma}f \Vert ^{q_{1}}_{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} = & \biggl[ \sum_{t \in \mathbb{Z}} 2^{t \lambda q_{2}} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{q_{1}}{q_{2}}} \\ \leq & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{ \frac{q_{1}}{p_{2}}} \biggr]. \end{aligned}$$

By setting $f_{s}=f\chi _{\Omega _{s}}$ for $s \in \mathbb{Z}$ we have $f=\sum_{s \in \mathbb{Z}} f_{s}$. Using Minkowski’s inequality, we get

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} \leq & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum _{s \in \mathbb{Z}} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{1}{p_{2}}} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ \leq & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl( \sum_{s \leq t-2} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{1}{p_{2}}} \biggr)^{q_{1}} \biggr]^{ \frac{1}{q_{1}}} \\ &{}+ \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl( \sum_{t-1 \leq s \leq t+1} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{1}{p_{2}}} \biggr)^{q_{1}} \biggr]^{ \frac{1}{q_{1}}} \\ &{}+ \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl( \sum_{s \geq t+2} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{1}{p_{2}}} \biggr)^{q_{1}} \biggr]^{ \frac{1}{q_{1}}} \\ := & E_{1}+E_{2}+E_{3}. \end{aligned}$$

(3)

Now we estimate the terms $E_{1}$, $E_{2}$, and $E_{3}$ one by one.

Estimation of $E_{1}$: For $s \leq t-2$ and a.e. $x \in \Omega _{t}$, we have

$$\begin{aligned} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert \lesssim & 2^{-t(n-\gamma )} \biggl\vert \int _{ \mathbb{R}^{n}}f_{s}(y)\,dm(y) \biggr\vert \\ \leq & 2^{-t(n-\gamma )} \Vert f_{s} \Vert _{L^{p_{1}}} \Vert \chi _{\Omega _{s}} \Vert _{L^{p'_{1}}}. \end{aligned}$$

Thus

$$\begin{aligned} E_{1} = & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum_{s \leq t-2} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{1}{p_{2}}} \biggr)^{q_{1}} \biggr]^{ \frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum_{s \leq t-2} 2^{-t(n-\gamma )} \Vert f_{s} \Vert _{L^{p_{1}}} \Vert \chi _{\Omega _{s}} \Vert _{L^{p'_{1}}} \Vert \chi _{\Omega _{t}} \Vert _{L^{p_{2}}( \mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} \biggl(\sum _{s \leq t-2} 2^{s \lambda} \Vert f_{s} \Vert _{L^{p_{1}}} \cdot 2^{\alpha (t-s)} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}}, \end{aligned}$$

where $\alpha =\frac{n}{p_{1}}-n+ \lambda <0$. Using Hölder’s inequality for inner sum and changing order of summations, we get

$$\begin{aligned} E_{1} \lesssim & \biggl[\sum_{t \in \mathbb{Z}} \biggl(\sum_{s \leq t-2} 2^{s \lambda q_{1}} \Vert f_{s} \Vert ^{q_{1}}_{L^{p_{1}}} \cdot 2^{ \frac{\alpha q_{1}}{2}(t-s)} \biggl\{ \sum_{s \leq t-2} 2^{ \frac{\alpha q'_{1}}{2}(t-s)} \biggr\} ^{\frac{q_{1}}{q'_{1}}} \biggr) \biggr]^{\frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} \sum _{s \leq t-2} 2^{s \lambda q_{1}} \Vert f_{s} \Vert ^{q_{1}}_{L^{p_{1}}} \cdot 2^{ \frac{\alpha q_{1}}{2}(t-s)} \biggr]^{\frac{1}{q_{1}}} \\ =& \biggl[\sum_{s \in \mathbb{Z}} 2^{s \lambda q_{1}} \Vert f_{s} \Vert ^{q_{1}}_{L^{p_{1}}} \sum _{t \geq s+2} 2^{\frac{\alpha q_{1}}{2}(t-s)} \biggr]^{ \frac{1}{q_{1}}} \\ \lesssim & \Vert f \Vert _{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n},m)}. \end{aligned}$$

Estimation of $E_{2}$: Applying Minkowski’s inequality and (1), we have

$$\begin{aligned} E_{2} \leq & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum_{t-1 \leq s \leq t+1} \Vert I_{\gamma} f_{s} \Vert _{L^{p_{2}}( \mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ \leq & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \Vert I_{ \gamma} f_{t-1} \Vert ^{q_{1}}_{L^{p_{2}}(\mathbb{R}^{n}, \mu )} \biggr]^{ \frac{1}{q_{1}}} + \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \Vert I_{\gamma} f_{t} \Vert ^{q_{1}}_{L^{p_{2}}(\mathbb{R}^{n}, \mu )} \biggr]^{\frac{1}{q_{1}}} \\ &{}+ \biggl[\sum _{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \Vert I_{\gamma} f_{t+1} \Vert ^{q_{1}}_{L^{p_{2}}(\mathbb{R}^{n}, \mu )} \biggr]^{\frac{1}{q_{1}}} \\ \lesssim & \Vert f \Vert _{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n},m)}. \end{aligned}$$

Estimation of $E_{3}$: For $s \geq t+2$ and a.e $x \in \Omega _{t}$, by using a similar technique as in estimation of $E_{1}$, we get

$$\begin{aligned} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert \lesssim & 2^{-s(n-\gamma )} \Vert f_{s} \Vert _{L^{p_{1}}} \Vert \chi _{\Omega _{s}} \Vert _{L^{p'_{1}}}. \end{aligned}$$

Therefore

$$\begin{aligned} E_{3} = & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum_{s \geq t+2} \biggl( \int _{\Omega _{t}} \bigl\vert I_{\gamma}f_{s}(x) \bigr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{1}{p_{2}}} \biggr)^{q_{1}} \biggr]^{ \frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum_{s \geq t+2} 2^{-s(n-\gamma )} \Vert f_{s} \Vert _{L^{p_{1}}} \Vert \chi _{\Omega _{s}} \Vert _{L^{p'_{1}}} \Vert \chi _{\Omega _{t}} \Vert _{L^{p_{2}}( \mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} \biggl(\sum _{s \geq t+2} 2^{s \lambda} \Vert f_{s} \Vert _{L^{p_{1}}} \cdot 2^{\beta (t-s)} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}}, \end{aligned}$$

where $\delta =\frac{n}{p_{1}}- \gamma +\lambda > 0$. Now using Hölder’s inequality for the inner sum and then interchanging the order of summations, we obtain

$$\begin{aligned} E_{3} \lesssim & \biggl[\sum_{t \in \mathbb{Z}} \biggl(\sum_{s \geq t+2} 2^{s \lambda q_{1}} \Vert f_{s} \Vert ^{q_{1}}_{L^{p_{1}}} \cdot 2^{ \frac{\delta q_{1}}{2}(t-s)} \biggl\{ \sum_{s \geq t+2} 2^{ \frac{\delta q'_{1}}{2}(t-s)} \biggr\} ^{\frac{q_{1}}{q'_{1}}} \biggr) \biggr]^{\frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} \sum _{s \geq t+2} 2^{s \lambda q_{1}} \Vert f_{s} \Vert ^{q_{1}}_{L^{p_{1}}} \cdot 2^{ \frac{\delta q_{1}}{2}(t-s)} \biggr]^{\frac{1}{q_{1}}} \\ =& \biggl[\sum_{s \in \mathbb{Z}} 2^{s \lambda q_{1}} \Vert f_{s} \Vert ^{q_{1}}_{L^{p_{1}}} \sum _{t \leq s-2} 2^{\frac{\delta q_{1}}{2}(t-s)} \biggr]^{ \frac{1}{q_{1}}} \\ \lesssim & \Vert f \Vert _{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n},m)}. \end{aligned}$$

This completes the proof. □

In the case of $p_{i}=q_{i}$ for $i=1,2$ and $\lambda =0$, the converse of the above theorem holds (see (1)). However, in general, the question of its converse remains an open problem. Nevertheless, we establish a partial answer for a particular set of parameters.

Proposition 2.2

Let $p_{1}$, $p_{2}$, $q_{1}$, $q_{2}$, and γ be as in Theorem 2.1. Suppose $p_{1} \leq q_{1} \leq q_{2} \leq p_{2}$ and $\lambda =0$. If $\Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} \lesssim \Vert f \Vert _{\dot{K}_{ \lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)}$, then $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}}- \frac{\gamma}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$.

Proof

For a given ball $B \subset \mathbb{R}^{n}$, set $f(x)= \chi _{B}(x)$. Then

$$\begin{aligned} \Vert f \Vert _{\dot{K}_{0,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)} = & \biggl[ \sum _{t \in \mathbb{Z}} \biggl( \int _{\Omega _{t}} \bigl\vert \chi _{B}(x) \bigr\vert ^{p_{1}}\,dm(x) \biggr)^{\frac{q_{1}}{p_{1}}} \biggr]^{\frac{1}{q_{1}}} \\ \leq & \biggl[ \sum_{t \in \mathbb{Z}}{\bigl[m(B \cap \Omega _{t})\bigr]} \biggr]^{\frac{1}{p_{1}}} \\ = & \bigl[m(B)\bigr]^{\frac{1}{p_{1}}}. \end{aligned}$$

(4)

Moreover,

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{K}_{0,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} = & \biggl[ \sum _{t \in \mathbb{Z}} \biggl( \int _{\Omega _{t}} \biggl\vert \int _{\mathbb{R}^{n}} \frac{\chi _{B}(y)}{ \vert x-y \vert ^{n-\gamma}}\,dm(y) \biggr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}} \\ \gtrsim & \biggl[ \sum_{t \in \mathbb{Z}} \biggl( \int _{\Omega _{t} \cap B} \biggl\vert \int _{B}\frac{1}{ \vert x-y \vert ^{n-\gamma}}\,dm(y) \biggr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}} . \end{aligned}$$

Since $x,y \in B$, we have $|x-y| \leq 2r$, where r is the radius of the ball. Thus

$$ \frac{1}{ \vert x-y \vert ^{n-\gamma}} \gtrsim \frac{1}{(r^{n})^{1-\frac{\gamma}{n}}} \gtrsim \bigl[m(B) \bigr]^{ \frac{\gamma}{n}-1}. $$

Hence

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{K}_{0,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} \gtrsim & \biggl[ \sum_{t \in \mathbb{Z}} \bigl[m(B)\bigr]^{ \frac{q_{2}\gamma}{n}} \bigl[\mu (\Omega _{t} \cap B)\bigr]^{ \frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}} \\ \gtrsim & \bigl[m(B)\bigr]^{\frac{\gamma}{n}} \biggl[ \sum _{t \in \mathbb{Z}} \bigl[ \mu (\Omega _{t} \cap B)\bigr] \biggr]^{\frac{1}{p_{2}}} \\ = & \bigl[m(B)\bigr]^{\frac{\gamma}{n}} \bigl[\mu (B)\bigr]^{\frac{1}{p_{2}}}. \end{aligned}$$

(5)

From (4) and (5) we get $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}}- \frac{\gamma}{n} )}$. □

Next, we present the trace inequality for homogeneous Lorentz–Herz spaces. Since the proof is similar to that of Theorem 2.1, we only provide the necessary steps and point out the differences in the arguments.

Theorem 2.3

Let $p_{1}$, $p_{2}$, $q_{1}$, $q_{2}$, μ be as in Theorem 2.1, and let $1 \leq r_{1} < r_{2} \leq \infty$ (or $r_{1} = r_{2} = \infty$). Suppose $n (\frac{1}{p_{1}}-\frac{1}{p_{2}} ) \leq \gamma < \frac{n}{p_{1}}$ and $\gamma -\frac{n}{p_{1}}<\lambda <n-\frac{n}{p_{1}}$. Then

$$ \Vert I_{\gamma}f \Vert _{\dot{HL}_{\lambda ,q_{2}}^{p_{2},r_{2}}( \mathbb{R}^{n}, \mu )} \lesssim \Vert f \Vert _{\dot{HL}_{ \lambda ,q_{1}}^{p_{1},r_{1}}(\mathbb{R}^{n}, m)}. $$

Proof

Let $f \in \dot{HL}_{\lambda ,q_{1}}^{p_{1},r_{1}}(\mathbb{R}^{n}, m)$. Then it is easy to see that

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{HL}_{\lambda ,q_{2}}^{p_{2},r_{2}}( \mathbb{R}^{n}, \mu )} \leq & \biggl[\sum _{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \Vert I_{\gamma}f \cdot \chi _{\Omega _{t}} \Vert _{L^{p_{2},r_{2}}(\mathbb{R}^{n}, \mu )}^{q_{1}} \biggr]^{ \frac{1}{q_{1}}}. \end{aligned}$$

As before, setting $f_{s}=f\chi _{\Omega _{s}}$, $s \in \mathbb{Z}$, and using the triangle inequality of the Lorentz norm, we obtain

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{HL}_{\lambda ,q_{2}}^{p_{2},r_{2}}( \mathbb{R}^{n}, \mu )} \leq & \biggl[\sum _{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl\Vert I_{\gamma} \biggl( \sum_{s \in \mathbb{Z}}f_{s} \biggr) \cdot \chi _{\Omega _{t}} \biggr\Vert _{L^{p_{2},r_{2}}( \mathbb{R}^{n}, \mu )}^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ \leq & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl\Vert \sum_{s \in \mathbb{Z}}I_{\gamma}f_{s} \cdot \chi _{\Omega _{t}} \biggr\Vert _{L^{(p_{2},r_{2})}(\mathbb{R}^{n}, \mu )}^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ \lesssim & \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum_{s \in \mathbb{Z}} \Vert I_{\gamma}f_{s} \cdot \chi _{\Omega _{t}} \Vert _{L^{p_{2},r_{2}}(\mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}}. \end{aligned}$$

The inner sum can be broken into three parts, and then by the application of Minkowski’s inequality we may write

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{HL}_{\lambda ,q_{2}}^{L^{p_{2},r_{2}}( \mathbb{R}^{n}, \mu )}(\mathbb{R}^{n}, \mu )} \lesssim & \biggl[ \sum _{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl(\sum _{s \leq t-2} \Vert I_{\gamma}f_{s} \cdot \chi _{\Omega _{t}} \Vert _{L^{p_{2},r_{2}}( \mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ & {}+\biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl( \sum_{t-1 \leq s \leq t+1} \Vert I_{\gamma}f_{s} \cdot \chi _{ \Omega _{t}} \Vert _{L^{p_{2},r_{2}}(\mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ &{}+ \biggl[\sum_{t \in \mathbb{Z}} 2^{t \lambda q_{1}} \biggl( \sum_{s \geq t+2} \Vert I_{\gamma}f_{s} \cdot \chi _{\Omega _{t}} \Vert _{L^{p_{2},r_{2}}(\mathbb{R}^{n}, \mu )} \biggr)^{q_{1}} \biggr]^{\frac{1}{q_{1}}} \\ := & E_{1}+E_{2}+E_{3}. \end{aligned}$$

Now, we proceed as in Theorem 2.1, except that we use the Hölder inequality for Lorentz spaces and rely on the fact that $\|\chi _{\Omega _{s}}\|_{L^{p^{\prime}_{1}, r^{\prime}_{1}}} \Vert \chi _{\Omega _{t}} \Vert _{L^{p_{2},r_{2}}(\mathbb{R}^{n}, \mu )} \lesssim 2^{[{t(n-\gamma )}+{\frac{n}{p^{\prime}_{1}}(s-t)}]}$ (which follows along lines similar to [6, Lemma 3.1.2.1]). Furthermore, the estimation of $E_{2}$ is based on Corollary 1.2. □

We wrap up this section with the following theorem addressing the limiting case $p_{1}=p_{2}=p$. By employing a simple modification of the proof of either Theorem 2.1 or Theorem 2.3, in combination with [14, Theorem 1.2] (see also [13, Theorem 3.1]), we get

Theorem 2.4

Let $1 < p< \infty$, $1 \leq q_{1} \leq q_{2} < \infty$, $0 < \gamma < \frac{n}{p}$, and $\gamma - \frac{n}{p} < \lambda < n - \frac{n}{p}$. Suppose that for every ball $B \subset \mathbb{R}^{n}$, we have $\mu (B) \lesssim [m(B)]^{ (1 - \frac{\gamma p}{n} )}$. Then

$$ \Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p}( \mathbb{R}^{n},\mu )} \lesssim \Vert f \Vert _{\dot{HL}_{ \lambda ,q_{1}}^{p,1}(\mathbb{R}^{n},m)}. $$

The question whether Theorem 2.4 holds when replacing $\dot{HL}_{\lambda ,q_{1}}^{p,1}$ with a space that is not as narrow as this (e.g., $\dot{HL}_{\lambda ,q_{1}}^{p,r}$ for $1< r< p$) remains open.

3 Optimality conditions

In this section, we present some examples to illustrate the optimality of certain parametric conditions assumed in Theorem 2.1 or Theorem 2.3. To that end, we focus on the case where $n=1$ and $\beta :=p_{2} (\frac{1}{p_{1}}-\gamma ) \leq 1$. Suppose μ is the positive Borel measure generated by $g(x)=x^{\beta -1} \chi _{(0, \infty )}(x)$ on the the Borel sigma algebra $\mathcal{B}$ of $\mathbb{R}$, i.e., $\mu (S)=\int _{S} g dm$ for $S \in \mathcal{B}$. We will proceed by working out a few examples using this choice of μ.

Example 3.1

Consider the function $f=\chi _{\Omega _{1}}$ in $\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R},m)$. For $x \in \Omega _{t}$ and $y \in \Omega _{1}$, it is evident that

$$ \frac{1}{ \vert x-y \vert ^{1-\gamma}} \gtrsim \frac{1}{(2+2^{t})^{1-\gamma}}. $$

Consequently,

$$\begin{aligned} \Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R},\mu )} = & \biggl[ \sum _{t \in \mathbb{Z}} 2^{t \lambda q_{2}} \biggl( \int _{\Omega _{t}} \biggl\vert \int _{\mathbb{R}} \frac{\chi _{\Omega _{1}}(y)}{ \vert x-y \vert ^{1-\gamma}}\,dm(y) \biggr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}} \\ \gtrsim & \biggl[ \sum_{t \in \mathbb{Z}} \frac{2^{t\lambda q_{2}} [\mu (\Omega _{t})]^{\frac{q_{2}}{p_{2}}}}{(2+2^{t})^{q_{2}(1-\gamma )}} \biggr]^{\frac{1}{q_{2}}} \\ \gtrsim & \biggl[ \sum_{t \leq 0} 2^{t q_{2}\lambda} \bigl[\mu (\Omega _{t})\bigr]^{ \frac{q_{2}}{p_{2}}}+\sum _{t \geq 1} 2^{t q_{2}(\lambda -1+\gamma )} \bigl[ \mu (\Omega _{t}) \bigr]^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}}. \end{aligned}$$

Therefore

$$ \Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R},\mu )} \gtrsim \biggl[\sum _{t \leq 0} 2^{t q_{2} ( \lambda +\frac{1}{p_{1}}-\gamma )}+ \sum_{t \geq 1} 2^{t q_{2} (\lambda -1+\frac{1}{p_{1}} )} \biggr]^{\frac{1}{q_{2}}}. $$

Thus, for the estimate $\Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R},\mu )} \lesssim \Vert f \Vert _{\dot{K}_{ \lambda ,q_{1}}^{p_{1}}(\mathbb{R},m)}$, it is necessary that $\gamma - \frac{1}{p_{1}} < \lambda < 1- \frac{1}{p_{1}}$.

The subsequent example demonstrates the necessity of the condition $q_{1} \leq q_{2}$.

Example 3.2

Consider the function $f_{k}(x)=|x|^{- (\lambda +\frac{1}{p_{1}} )} \chi _{\{1<|x|<2^{k} \}}(x)$, $k \in \mathbb{N}$. It is not difficult to see that $\|f_{k}\|_{\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R},m)} \lesssim k^{ \frac{1}{q_{1}}}$. Moreover,

$$\begin{aligned} \Vert I_{\gamma}f_{k} \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}(\mathbb{R},\mu )} = & \biggl[ \sum_{t \in \mathbb{Z}} 2^{t \lambda q_{2}} \biggl( \int _{\Omega _{t}} \biggl\vert \int _{\mathbb{R}} \frac{f_{k}(y)}{ \vert x-y \vert ^{1-\gamma}}\,dm(y) \biggr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}} \\ \gtrsim & \biggl[ \sum_{t \in \mathbb{Z}} 2^{t \lambda q_{2}} \biggl( \int _{\Omega _{t}} \biggl\vert \int _{\Omega _{t}} \frac{f_{k}(y)}{ \vert x-y \vert ^{1-\gamma}}\,dm(y) \biggr\vert ^{p_{2}}\,d\mu (x) \biggr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}}. \end{aligned}$$

Notice that $\frac{1}{|x-y|^{1-\gamma}} \geq 2^{-(t+1)(1-\gamma )}$ and $f_{k}(y) \geq 2^{-t (\lambda +\frac{1}{p_{1}} )}$ for $x,y \in \Omega _{t}$, $1 \leq t \leq k$. Consequently,

$$\begin{aligned} \Vert I_{\gamma}f_{k} \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}(\mathbb{R},\mu )} \gtrsim & \biggl[ \sum_{1 \leq t \leq k} 2^{q_{2} (t\lambda -(t+1)(1- \gamma )-t (\lambda +\frac{1}{p_{1}} ) )} \bigl( \bigl[m( \Omega _{t})\bigr]^{p_{2}}\mu (\Omega _{t}) \bigr)^{\frac{q_{2}}{p_{2}}} \biggr]^{\frac{1}{q_{2}}} \\ \gtrsim& \biggl[ \sum _{1 \leq t \leq k} 1 \biggr]^{\frac{1}{q_{2}}} = k^{\frac{1}{q_{2}}}. \end{aligned}$$

Using $\Vert I_{\gamma}f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R},\mu )} \lesssim \Vert f \Vert _{\dot{K}_{ \lambda ,q_{1}}^{p_{1}}(\mathbb{R},m)}$, we deduce that $k^{\frac{1}{q_{2}}} \lesssim k^{\frac{1}{q_{1}}}$. As $k \in \mathbb{N}$ is arbitrary, we must have $q_{1} \leq q_{2}$.

The remaining conditions, $p_{1} < p_{2}$ and $\gamma < \frac{n}{p_{1}}$, are known to be optimal in Lebesgue spaces, and, consequently, they stand as optimal conditions for Theorem 2.1 (or Theorem 2.3).

4 Sobolev inequalities

The Riesz potential operator on $\mathbb{R}^{n}$ classically arises from the $\frac{\gamma}{2}$th-order fractional Laplace equation $(-\Delta )^{\frac{\gamma}{2}}(u)= f$. For $0<\gamma <n$, the function $\mathcal{G}(\gamma )(2\pi |x|)^{-\gamma}$ is the Fourier transform of the function $|x|^{\gamma - n}$ [20, p. 66]. Here the constant $\mathcal{G}(\gamma )$, known as the normalized constant, is given by

$$ \mathcal{G}(\gamma )= \frac{\pi ^{\frac{n}{2}} 2^{\gamma} \Gamma (\frac{\gamma}{2} )}{\Gamma ( \frac{n-\alpha}{2} )}, $$

where Γ is the Euler gamma function. Based on this, it can be readily inferred that the equation $u=\frac{I_{\gamma}(f)}{\mathcal{G}(\gamma )}$ solves the aforementioned equation. Thus the results in Sect. 2 indicate that under the conditions of Theorem 2.1 (or Theorem 2.3), if f belongs to $\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)$ [or $\dot{HL}_{\lambda ,q}^{p,r}(\mathbb{R}^{n}, m)$], then the solution of fractional-order equation $(-\Delta )^{\frac{\gamma}{2}}(u)= f$ belongs to $\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},\mu )$ [ resp., $\dot{HL}_{\lambda ,q}^{p,r}(\mathbb{R}^{n}, \mu )$].

Another important observation is that if μ is the Lebesgue measure restricted to Borel sets in $\mathbb{R}^{n}$ and $\frac{1}{p_{1}}-\frac{1}{p_{2}} = \frac{\gamma}{n}$, then we can immediately deduce the following Hardy–Littlewood–Sobolev theorem of fractional integration in the context of Lorentz–Herz spaces.

Corollary 4.1

Let $1< p_{1}< p_{2}<\infty$, $1 \leq q_{1} \leq q_{2}<\infty$, and $1 \leq r_{1} < r_{2} \leq \infty$ (or $r_{1},r_{2}=\infty$). Then

$$ \Vert I_{\gamma}f \Vert _{\dot{HL}_{\lambda ,q_{2}}^{p_{2},r_{2}}( \mathbb{R}^{n}, m)} \lesssim \Vert f \Vert _{\dot{HL}_{ \lambda ,q_{1}}^{p_{1},r_{1}}(\mathbb{R}^{n}, m)}, $$

provided that $\frac{1}{p_{1}}-\frac{1}{p_{2}}=\frac{\gamma}{n}$ and $\gamma -\frac{n}{p_{1}}<\lambda <n-\frac{n}{p_{1}}$.

In particular, if $r_{i}=p_{i}$, $i=1,2$, then we get the corresponding theorem for Herz spaces (cf. [17]).

Let us recall the definition of homogeneous Herz-type Sobolev spaces from [18]. For consistency, we make a slight adjustment to the notation.

Definition 4.2

Let $1< p<\infty$, $0<\lambda <n (1-\frac{1}{p} )$, $0< q<\infty$, and $k\in \mathbb{Z}_{+}$. The homogeneous Herz-type Sobolev space $\dot{K}_{\lambda ,q}^{p,k}(\mathbb{R}^{n})$ is defined by

$$ \dot{K}_{\lambda ,q}^{p,k}\bigl(\mathbb{R}^{n}\bigr):= \biggl\{ f \in \dot{K}^{p}_{ \lambda ,q}\bigl(\mathbb{R}^{n} \bigr): \text{for } \vert \beta \vert \leq k, \frac{\partial ^{\beta}f}{\partial f^{\beta}} \text{ exists on } \mathcal{D}^{\prime}\bigl(\mathbb{R}^{n}\bigr), \text{and } \frac{\partial ^{\beta}f}{\partial f^{\beta}} \in \dot{K}^{p}_{ \lambda ,q}\bigl( \mathbb{R}^{n}\bigr) \biggr\} , $$

where $\beta =(\beta _{1},\beta _{2}, \ldots , \beta _{n}) \in \mathbb{Z}_{+}^{n}$, $\frac{\partial ^{0}f}{\partial f^{0}}=f$, and the space is equipped with the functional

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q}^{p,k}(\mathbb{R}^{n})}:= \sum_{ \vert \beta \vert \leq k} \biggl\Vert \frac{\partial ^{\beta}f}{\partial f^{\beta}} \biggr\Vert _{\dot{K}^{p}_{ \lambda ,q}(\mathbb{R}^{n})}. $$

The parameters in this definition are subjected to specific conditions to ensure the reasonableness of definition, as outlined in [18].

Theorem 4.3

Let $1< p_{1}< n$, $p_{2}<\infty$, $1 \leq q_{1} \leq q_{2}<\infty$, and $0<\lambda <n-\frac{n}{p_{1}}$. Suppose that for every ball $B \subset \mathbb{R}^{n}$, we have $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}}-\frac{1}{n} )}$. Then

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n}, \mu )} \lesssim \Vert \nabla f \Vert _{ \dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n}, m)} $$

for every $f \in \dot{K}_{\lambda ,q_{1}}^{p_{1},1}(\mathbb{R}^{n})$.

Proof

First, assume that $g \in \mathcal{D}(\mathbb{R}^{n})$, the space of infinitely differentiable functions on $\mathbb{R}^{n}$ with compact support. Then it is well known that $|g(x)| \lesssim I_{1}(|\nabla g|)(x)$ for every $x \in \mathbb{R}^{n}$. Therefore by Theorem 2.1 and the ideal property of Herz spaces ([6, Proposition 3.6]) it follows that

$$\begin{aligned} \Vert g \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} \lesssim & \bigl\Vert I_{1}\bigl( \vert \nabla g \vert \bigr) \bigr\Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}(\mathbb{R}^{n},\mu )} \\ \lesssim & \Vert \nabla g \Vert _{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n},m)}, \end{aligned}$$

(6)

where $\Vert \nabla g \Vert _{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n},m)}= \Vert (|\nabla g |) \Vert _{\dot{K}_{ \lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)}$ and $|\nabla g |= \sum_{j=1}^{n} |\frac{\partial g}{\partial x_{j}} |$. Now let $f \in \dot{K}_{\lambda ,q_{1}}^{p_{1},1}(\mathbb{R}^{n})$. Then $f \in \dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n})$ and $\frac{\partial f}{\partial x_{j}} \in \dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n})$, $j=1,2, \dots , n$. Moreover, there exists a sequence $\{f_{k}\}$ in $\mathcal{D}(\mathbb{R}^{n})$ such that $f_{k} \to f$ in $\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n})$ and $\frac{\partial f_{k}}{\partial x_{j}} \to \frac{\partial f}{\partial x_{j}}$ in $\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n})$ [18, Proposition 2.1]. Therefore by equation (6) we get

$$ \Vert f_{k}-f_{l} \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n},\mu )} \lesssim \Biggl\Vert \sum_{j=1}^{n} \biggl\vert \frac{\partial f_{k}}{\partial x_{j}} - \frac{\partial f_{l}}{\partial x_{j}} \biggr\vert \Biggr\Vert _{\dot{K}_{ \lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n},m)}, $$

from which it follows that the sequence $\{f_{k}\}$ converges to f in $\dot{K}_{\lambda ,q_{2}}^{p_{2}}(\mathbb{R}^{n},\mu )$. This completes the proof. □

The repeated application of the pointwise estimate $|g(x)| \lesssim I_{1}(|\nabla g|)(x)$, in combination with semigroup property $I_{\alpha}I_{\beta}=I_{\alpha +\beta}$, ensures the above inequality for higher-order Sobolev-type Herz spaces as well.

Theorem 4.4

Let $k \in \mathbb{N}$, $1< p_{1}<\frac{n}{k}$, $p_{2}<\infty$, $1 \leq q_{1} \leq q_{2}<\infty$, and $0<\lambda <n-\frac{n}{p_{1}}$. If $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{1}}-\frac{k}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$, then

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q_{2}}^{p_{2}}( \mathbb{R}^{n}, \mu )} \lesssim \bigl\Vert \nabla ^{k} f \bigr\Vert _{\dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n}, m)} $$

for every $f \in \dot{K}_{\lambda ,q_{1}}^{p_{1},k}(\mathbb{R}^{n})$.

We say that $p^{*}$ is the k-Sobolev conjugate of p if $\frac{1}{p^{*}}=\frac{1}{p}-\frac{k}{n}$, where k is a positive integer. We simply write Sobolev conjugate for 1-Sobolev conjugate. Putting $\mu =m$ in the above theorem, we get the following Sobolev embedding theorem for Herz-type Sobolev spaces.

Corollary 4.5

Let $k \in \mathbb{N}$, $1< p<\frac{n}{k}$, $1 \leq q < \infty$, and $0<\lambda <n-\frac{n}{p}$, and let $p^{*}$ be the k-Sobolev conjuage of p. Then

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q}^{p^{*}}(\mathbb{R}^{n}, m)} \lesssim \bigl\Vert \nabla ^{k} f \bigr\Vert _{\dot{K}_{ \lambda ,q}^{p}(\mathbb{R}^{n}, m)} $$

for every $f \in \dot{K}_{\lambda ,q}^{p,k}(\mathbb{R}^{n})$. In particular, $\dot{K}_{\lambda ,q}^{p,k}(\mathbb{R}^{n}) \hookrightarrow \dot{K}_{ \lambda ,q}^{p^{*}}(\mathbb{R}^{n})$.

Finally, we prove the following Gagliardo–Nirenberg–Sobolev (GNS) inequality in the setting of Herz spaces.

Theorem 4.6

Let $0 \leq \theta \leq 1$, $1< p_{0}< n$, $1 \leq p_{0}$, $p_{1} < p_{2}<\infty$, $1 \leq q_{0}$, $q_{1} < q_{2}<\infty$, and $0<\lambda <n-\frac{n}{p_{0}}$. Suppose that $\mu (B) \lesssim [m(B)]^{p_{2} (\frac{1}{p_{0}}-\frac{1}{n} )}$ for every ball $B \subset \mathbb{R}^{n}$. Then

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q}^{p}(\mathbb{R}^{n}, \mu )} \leq \Vert f \Vert ^{1-\theta}_{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n}, \mu )} \Vert \nabla f \Vert ^{\theta}_{ \dot{K}_{\lambda ,q_{0}}^{p_{0}}(\mathbb{R}^{n}, m)} $$

for every $f \in \dot{K}_{\lambda ,q_{1}}^{p_{1}}(\mathbb{R}^{n}, \mu ) \cap \dot{K}_{\lambda ,q_{0}}^{p_{0},1}(\mathbb{R}^{n},m)$, provided that $\frac{1}{q}=\frac{1-\theta}{q_{1}} + \frac{\theta}{q_{2}}$ and $\frac{1}{p}=\frac{1-\theta}{p_{1}} + \frac{\theta}{p_{2}}$.

Proof

Let $0\leq \theta \leq 1$ and $1\leq p_{1}< p_{2} \leq \infty$. Using the interpolation inequality $\|f\|_{L^{p}(\mathbb{R}^{n}, \nu )} \leq \|f\|^{1-\theta}_{L^{p_{1}}( \mathbb{R}^{n}, \nu )} \|f\|^{\theta}_{L^{p_{2}}(\mathbb{R}^{n}, \nu )}$, which holds for any positive measure ν on $\mathbb{R}^{n}$, provided that $\frac{1}{p}=\frac{1-\theta}{p_{1}} + \frac{\theta}{p_{2}}$, and the Hölder inequality, we obtain

$$\begin{aligned} \sum_{t \in \mathbb{Z}}2^{t \lambda q} \Vert f \chi _{\Omega _{t}} \Vert ^{q}_{L^{p}( \mathbb{R}^{n}, \nu )} \leq & \biggl(\sum _{t \in \mathbb{Z}}2^{t \lambda q_{1}} \Vert f \chi _{\Omega _{t}} \Vert ^{q_{1}}_{L^{p_{1}}(\mathbb{R}^{n}, \nu )} \biggr)^{\frac{q(1-\theta )}{q_{1}}} \biggl(\sum_{t \in \mathbb{Z}}2^{t \lambda q_{2}} \Vert f \chi _{\Omega _{t}} \Vert ^{q_{2}}_{L^{p_{2}}( \mathbb{R}^{n}, \nu )} \biggr)^{\frac{q \theta}{q_{2}}}. \end{aligned}$$

Consequently,

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q}^{p}(\mathbb{R}^{n}, \nu )} \leq \Vert f \Vert ^{1-\theta}_{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n}, \nu )} \Vert f \Vert ^{\theta}_{\dot{K}_{ \lambda ,q_{2}}^{p_{2}}(\mathbb{R}^{n}, \nu )}. $$

Using Theorem 4.3, it follows that

$$ \Vert f \Vert _{\dot{K}_{\lambda ,q}^{p}(\mathbb{R}^{n}, \mu )} \leq \Vert f \Vert ^{1-\theta}_{\dot{K}_{\lambda ,q_{1}}^{p_{1}}( \mathbb{R}^{n}, \mu )} \Vert \nabla f \Vert ^{\theta}_{ \dot{K}_{\lambda ,q_{0}}^{p_{0}}(\mathbb{R}^{n}, m)}. $$

 □

Remark 4.7

• (i) If $q \leq p$, then the definition of $\dot{K}_{\lambda ,q}^{p,k}(\mathbb{R}^{n})$ remains reasonable even when $\lambda =0$. Evidently, all the aforementioned results (Theorem 4.3 onwards) are still true when $\lambda =0$ and $q_{i} \leq p_{i}$ ($i=0,1,2$).

• (ii) If $\lambda =0$, $\mu =m$, $p_{i}=q_{i}$ for $i=0,1,2$, and $p_{2}$ is the Sobolev conjugate of $p_{0}$, we obtain the GNS inequality for the Lebesgue spaces (see [21, Sect. 1]).