Abstract
In this paper, we derive some new Gagliardo–Nirenberg type inequalities in Lorentz type spaces without restrictions on the second index of Lorentz norms, which generalize almost all known corresponding results. Our proof mainly relies on the Bernstein inequalities in Lorentz spaces, the embedding relation among various Lorentz type spaces, and Littlewood–Paley decomposition techniques.
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1 Introduction
An important way to study well-posedness of partial differential equations is via the research of their weak solutions. It is well known that Gagliardo-Nirenberg inequality is a fundamental tool to improve the regularity of weak solutions, which has gained widespread applications such as in Hilbert’s 19th problem [10], Caffarelli–Kohn–Nirenberg theorem [6] for the 3D Navier–Stokes equations and the critical quasi-geostrophic equations [5]. The classical integer version of Gagliardo–Nirenberg inequality is the generalization of Sobolev embedding theorem and was discovered independently by Gagliardo [13] and Nirenberg [23] as follows: for all smooth functions u in \(\mathbb {R}^{n}\) with compact support, there holds
where j, m are any integers satisfying \(0\le j<m\), \(1\le q,r\le \infty ,\) and
for all \(\theta \in [\frac{j}{m},\,1]\), unless \(1<r<\infty \) and \(m-j-\frac{n}{r}\) is a nonnegative integer.
Up to now, there have been extensive investigations on Gagliardo-Nirenberg inequalities involving fractional derivatives and various function spaces (see [4, 8, 9, 16, 19, 21, 22, 25, 27, 28, 30, 31]). On one hand, the most general form of fractional Gagliardo-Nirenberg inequality in Lebesgue spaces was recently presented by Hajaiej–Molinet–Ozawa–Wang [15] and by Chikami [8]: for \(0 \le \sigma<s<\infty \) and \(1< q, r \le \infty ,\) there holds
with
where \(0 \le \theta \le 1-\frac{\sigma }{s}\,\left( \theta \ne 0\right. \) if \(\left. s-\sigma \ge \frac{n}{r}\right) \) and \(\Lambda ^{s}=(-\Delta )^{\frac{s}{2}} \) is defined via \(\widehat{\Lambda ^{s} f}(\xi )=|\xi |^{s}\hat{f}(\xi ).\) For the fractional Gagliardo-Nirenberg inequality in the framework of nonhomogeneous Sobolev spaces, Besov spaces and Fourier–Herz spaces, we refer the reader to [4, 8, 15] and references therein. On the other hand, recent progress on Gagliardo–Nirenberg inequality in Lorentz spaces for some special cases was made in [9, 16, 22]. In particular, Hajaiej–Yu–Zhai [16] established the Gagliardo–Nirenberg inequality for Lorentz spaces below: for \(1 \le p, p_{2}, q, q_{1}, q_{2}<\infty , 0<\alpha<q, 0<s<n\) and \(1<p_{1}<n/s\),
with
In [22], McCormick–Robinson–Rodrigo proved the following inequality
where
Subsequently, this result was improved by Dao–Díaz–Nguyen [9] as follows: for any \(\alpha >0\),
with
One of main purposes of this paper is to establish the most general form of Gagliardo–Nirenberg inequality in Lorentz spaces, parallel to the classical version (1.1) and the fractional case (1.2).
Theorem 1.1
Suppose that \(u\in L^{q,\infty }(\mathbb {R}^{n})\) and \(\Lambda ^{s}u\in L^{r,\infty }(\mathbb {R}^{n})\). Let \(0 \le \sigma<s<\infty \) and \(1< q, r \le \infty . \) Then there exists a positive constant \(C=C(n,q,p,r, s,\sigma )\) such that
with
where \(0<\theta < 1-\frac{\sigma }{s}\,\) and \(\,s-\frac{n}{r} \ne \sigma -\frac{n}{p}.\)
Remark 1.1
The motivation of this theorem is twofold. On one hand, it is worth noting that Gagliardo–Nirenberg inequalities (1.3)–(1.5) in Lorentz spaces mentioned above are only concerned with the subcritical case or the case that \(q<p\), and the left-hand side of estimates (1.3)–(1.5) is without higher derivatives. On the other hand, since there are some extra restrictions on the second index of Lorentz norms in (1.3)–(1.5) and that in the results of [19], it seems that their versions of Gagliardo–Nirenberg inequalities for Lorentz spaces are not easily applicable in the field of nonlinear partial differential equations. We emphasize here that Theorem 1.1 removes all the aforementioned restrictions and covers most of the possible range of exponents.
Remark 1.2
It is essential to make an exception that \(\,s-\frac{n}{r} \ne \sigma -\frac{n}{p}\,\) in this theorem, otherwise the generalized Gagliardo-Nirenberg inequality (1.6) is invalid for \(u(x)=|x|^{s-\frac{n}{r}}\). We also remark that the restriction that \(q, r >1\) results from an application of Lemma 3.1, and the homogeneity of (1.6) dictates the relation (1.7).
Remark 1.3
A special case of (1.6) is that
where
This result still extends the aforementioned inequalities (1.3)–(1.5) for Lorentz spaces.
Inspired by the work [8], we shall divide the proof of Theorem 1.1 into three cases by the relationship between \(s-\sigma \) and n/r. Firstly, we focus on the subcritical case \(s-\sigma <n/r\). Roughly speaking, we observe that there exist at least four kinds of equivalent definitions of Lorentz norms (see (2.3) for details). This together with the pointwise interpolation estimate for derivatives (3.5) given in [1] enables us to prove that
Then, making use of the interpolation characteristic (2.4) and Sobolev inequality in Lorentz spaces, we can obtain the desired estimates in this case. The second case is devoted to dealing with the critical case \(s-\sigma =n/r\). To this end, in the spirit of [8, 30, 31], we establish the following estimate in the critical Besov–Lorentz spaces
To achieve this, various Bernstein inequalities (2.12)–(2.15) in Lorentz spaces are derived. The new Bernstein inequality (2.15) allows us to get the fact that
Combining (1.8), (1.9) and (1.10), we may prove the critical case. For the third case, we proceed with the case \(s-\sigma > n/r\) by setting up the following key estimate
By a similar argument used in the previous two cases, this yields the generalized Gagliardo-Nirenberg inequality (1.6) under the supercritical case, which concludes Theorem 1.1. Moreover, it is worth remarking that critical inequality (1.9) is an extension of corresponding results in [29, Theorem 4.1].
Furthermore, it should be stated that the Bernstein inequalities (2.13)–(2.14) together with low-high frequency techniques as in [8] also guarantee the following generalized Gagliardo–Nirenberg inequality in the framework of Besov–Lorentz spaces, which extends the corresponding results in [8, 15]. Moreover, our proof of this main result is self-contained.
Theorem 1.2
Assume that \(u \in \dot{B}_{r,\infty , \infty }^{s}\left( \mathbb {R}^{n}\right) \cap \dot{B}_{q,\infty , \infty }^{0}\left( \mathbb {R}^{n}\right) \) with \(1 < q, r \le \infty \) and \(0 \le \sigma<s<\infty .\) Then there exists a positive constant \(C=C(n,q,p,r,s,\sigma )\) such that
with
Remark 1.4
This theorem implies several versions of Gagliardo–Nirenberg inequalities, such as Theorem 1.1 for Lorentz spaces, [8, Theorem 5.3] and [29, Theorem 3.2] for Besov spaces. Due to [15], it is necessary to make an assumption that \(\,s-\frac{n}{r} \ne \sigma -\frac{n}{p}\,\) in Theorem 1.2.
It is worth pointing out that there has existed an extensive study on Besov–Lorentz spaces (see [18, 26, 31, 32] and references therein). In [31], Wadade presented the critical Besov–Lorentz inequality (1.9) under the case that \(q>\max \{p,r\}\), and he also proved that
which improves the classical result below due to Ozawa [25]
Combining the Besov–Lorentz inequality (1.9) and the embedding relation in Lemma 2.5, we have the following corollary.
Corollary 1.3
For \(1<p<q<\infty ,\,1<r<\infty \) and \(1\le l\le \infty \), there exists a positive constant \(C=C(n,q,p,r,l)\) such that
Remark 1.5
This corollary generalizes the critical interpolation inequalities (1.12) and (1.13). The results obtained here can be applied to deduce the Trudinger–Moser type inequality as in [25, 30, 31].
Remark 1.6
In a forthcoming paper, we will present an application of the above inequalities to establish several novel criteria in terms of the velocity or the gradient of the velocity in Lorentz spaces for energy conservation of the 3D Navier–Stokes equations.
The rest of this paper is organized as follows. In Sect. 2, we recall some basic materials of various Lorentz type spaces and present embedding relation among these spaces. The generalized Young inequality and Bernstein inequalities for Lorentz spaces are also established in this section. Section 3 is devoted to the proof of Theorem 1.1 and Theorem 1.2.
2 Notations and Key Auxiliary Lemmas
2.1 Lorentz Spaces and Generalized Bernstein Inequality
Throughout this paper, we will use the summation convention on repeated indices. C will denote positive absolute constants which may be different from line to line unless otherwise stated in this paper. \(a\approx b\) means that \(C^{-1}b\le a\le Cb\) for some constant \(C>1\). \(\chi _{\Omega }\) stands for the characteristic function of a set \(\Omega \subset \mathbb {R}^{n}\). |E| represents the n-dimensional Lebesgue measure of a set \(E\subset \mathbb {R}^{n}\). Let \(\mathcal {M}\) be the Hardy–Littlewood maximal operator and its definition is given by
where f is any locally integrable function on \(\mathbb {R}^{n}\), and B(r) is the open ball centered at the origin with radius \(r>0\).
Next, we present some basic facts on Lorentz spaces. Recall that the distribution function of a measurable function f on \(\Omega \) is the function \(f_{*}\) defined on \([0,\infty )\) by
The decreasing rearrangement of f is the function \(f^{*}\) defined on \([0,\infty )\) by
For \(p,q\in (0,\infty ]\), we define
Furthermore,
which implies that \(L^{\infty ,\infty }=L^{\infty }\), \(L^{q,q}=L^{q}\) and \(L^{\infty ,q}=\{0\}\) for \(0<q<\infty \).
Notice that identity definition of Lorentz norm can be found in [14, 20]. Indeed, for \(0<p\le \infty \) and \(0<q\le \infty \), there holds
Similarly, one can define Lorentz spaces \(L^{p,q}(0,T;X)\) in time for \(0<p, q\le \infty \). \(f\in L^{p, q}(0,T;X)\) means that \(\Vert f\Vert _{L^{p,q}(0,T;X)}<\infty \), where
Note that the triangle inequality is not valid for \(\Vert \cdot \Vert _{L^{p, q}(\mathbb {R}^{n})}.\) Another equivalent norm in Lorentz spaces is defined as
where
In addition, Lorentz spaces endowed with the norm \(\Vert \cdot \Vert ^{*}_{L^{p, q} }\) are Banach spaces, and there holds
Most of the above statement is borrowed from [3, 7, 14].
Subsequently, we present norm-equivalence concerning Lorentz spaces.
Lemma 2.1
Let f be in \(L^{p,q}(\mathbb {R}^{n})\) with \(1<p\le \infty \) and \(1\le q\le \infty \). Then there holds
where \(C_{1},C_{2},C_{3}\) and \(C_{4}\) are positive constants depending only on p, q and n.
Proof
Since \(f\in L^{p,q}(\mathbb {R}^{n})\) with \(1<p\le \infty \) and \(1\le q\le \infty \), it follows from (2.7) that f is a locally integrable function on \(\mathbb {R}^{n}\). Recall the pointwise inequality involving Hardy–Littlewood maximal operator (see [11, p. 41] and [2, Chapter 3]) below,
where \(c_{1}\) and \(c_{2}\) are positive constants depending only on n. This together with (2.1) means
The conclusion is a straightforward consequence of the latter and (2.2). \(\square \)
Remark 2.1
Even if \(0<q<1\), the equivalent relation \(\Vert \mathcal {M}f\Vert _{L^{p,q}(\mathbb {R}^{n})}\thickapprox \Vert f\Vert _{L^{p,q}(\mathbb {R}^{n})}\) still holds for all functions \(f\in L^{p,q}(\mathbb {R}^{n})\) with \(1<p\le \infty \). Indeed, thanks to Marcinkiewicz’s interpolation theorem for Lorentz spaces [14, Theorem 1.4.19], it follows from the fact that Hardy-Littlewood maximal operator \(\mathcal {M}\) is a sublinear operator of both weak type (1, 1) and strong type \((\infty ,\infty )\) that \(\mathcal {M}\) is also bounded on \(L^{p,q}(\mathbb {R}^{n})\) for any \(p\in (1,\infty ]\) and \(q\in (0,\infty ]\), which yields that \(\Vert \mathcal {M}f\Vert _{L^{p,q}(\mathbb {R}^{n})}\le C(n,p,q)\Vert f\Vert _{L^{p,q}(\mathbb {R}^{n})}\) for all functions \(f\in L^{p,q}(\mathbb {R}^{n})\). On the other hand, Lebesgue’s differentiation theorem implies that \(|f(x)|\le \mathcal {M}f(x)\) for almost all \(x\in \mathbb {R}^{n}\), which yields that \(f_{*}\le (\mathcal {M}f)_{*}\) and \(\Vert f\Vert _{L^{p,q}(\mathbb {R}^{n})}\le \Vert \mathcal {M}f\Vert _{L^{p,q}(\mathbb {R}^{n})}\) for \(1<p\le \infty \) and \(0<q\le \infty \).
We list the properties of Lorentz spaces as follows.
-
Interpolation characteristic of Lorentz spaces [3, 9]
$$\begin{aligned} \begin{aligned}&\Vert f \Vert _{L^{p,p_{1}}(\mathbb {R}^{n})} \le \left[ \frac{(r-q)p^{2}}{(r-p)(p-q)p_{1}}\right] ^{\frac{1}{p_{1}}}\Vert f \Vert _{L^{q,\infty }(\mathbb {R}^{n})}^{\alpha } \Vert f \Vert _{L^{r,\infty }(\mathbb {R}^{n})}^{1-\alpha },\\&\text {with}~~ \frac{1}{p}=\frac{\alpha }{q} +\frac{1-\alpha }{r},~0<\alpha<1,~0<q<p<r\le \infty ~~ \text {and}~~ 0<p_{1}\le \infty .\end{aligned} \end{aligned}$$(2.4) -
$$\begin{aligned} \Vert |f|^{\lambda } \Vert _{L^{p,q}(\mathbb {R}^{n})}= \Vert f \Vert ^{\lambda }_{L^{\lambda p,\lambda q}(\mathbb {R}^{n})},~~\text {with}~~ 0< \lambda<\infty ~~\text {and}~~0<p,q\le \infty . \end{aligned}$$
-
Hölder’s inequality in Lorentz spaces [24]
$$\begin{aligned} \begin{aligned}&\Vert fg\Vert _{L^{r,s}(\Omega )}\le C(r_{1},r_{2},s_{1},s_{2})\,\Vert f\Vert _{L^{r_{1},s_{1}}(\Omega )}\Vert g\Vert _{L^{r_{2},s_{2}}(\Omega )}, \\&\text {with}~~\frac{1}{r}=\frac{1}{r_{1}}+\frac{1}{r_{2}},~~\frac{1}{s}=\frac{1}{s_{1}}+\frac{1}{s_{2}},~~0<r_{1},r_{2},s_{1},s_{2}\le \infty .\end{aligned}\end{aligned}$$(2.5) -
The Lorentz spaces increase as the exponent q increases [14, 20] For \(0< p\le \infty \) and \(0< q_{1}<q_{2}\le \infty ,\)
$$\begin{aligned} \Vert f\Vert _{L^{p,q_{2}}(\Omega )}\le \Big (\frac{q_{1}}{p}\Big )^{\frac{1}{q_{1}}-\frac{1}{q_{2}}}\Vert f\Vert _{L^{p,q_{1}}(\Omega )}. \end{aligned}$$(2.6) -
Inclusion in Lorentz spaces on bounded domains [14, 20] For any \(1\le m<M\le \infty \), \(~~1\le r,q\le \infty \),
$$\begin{aligned} \Vert f\Vert _{L^{m,r}(\Omega )}\le \Big (\frac{1}{m}\Big )^{\frac{r-1}{r}} \Big (\frac{q}{M}\Big )^{\frac{1}{q}}\frac{|\Omega | ^{\frac{1}{m}-\frac{1}{M}}}{\frac{1}{m}-\frac{1}{M}} \Vert f\Vert _{L^{M,q}(\Omega )}. \end{aligned}$$(2.7) -
Sobolev inequality in Lorentz spaces [24, 28]
$$\begin{aligned} \Vert f\Vert _{L^{\frac{np}{n-p},p}(\mathbb {R}^{n})}\le C(n,p)\,\Vert \nabla f\Vert _{L^{p}(\mathbb {R}^{ n })}~~\text {with}~~1\le p <n.\end{aligned}$$(2.8) -
Young inequality in Lorentz spaces [24] Let \(1<p,q,r<\infty \), \(0<s_{1},s_{2}\le \infty \),\(\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1\), and \( \frac{1}{s}=\frac{1}{s_{1}}+\frac{1}{s_{2}}\). Then there holds
$$\begin{aligned} \Vert f*g\Vert _{L^{r,s}(\mathbb {R}^{n})}\le C(p,q,s_{1},s_{2})\,\Vert f \Vert _{L^{p,s_{1}}(\mathbb {R}^{n})}\Vert g \Vert _{L^{q,s_{2}}(\mathbb {R}^{n})}. \end{aligned}$$(2.9)
It should be mentioned that the classical Young inequality (2.9) due to O’Neil requires the first index of every Lorentz norm is larger than 1. Grafakos [14] improved O’Neil’s result and showed that
The following lemma extends it to a more general version, which we shall give a different proof from that of [14, Theorem 1.2.13].
Lemma 2.2
Suppose that \(0<l\le s \le \infty \), \(1 \le r<\infty \) and \(1<p, q<\infty \). If \(f \in L^{q, l}(\mathbb {R}^{n})\) and \(g \in L^{r}(\mathbb {R}^{n})\) with
then \(f *g \in L^{p, s}(\mathbb {R}^{n})\), and there exists a positive constant C depending only on r, q, s and l such that
Remark 2.2
In general, (2.11) fails when \(l>s\). Indeed, here is a counterexample as follows: for any \(f \in L^{q, l}(\mathbb {R}^{n})\) with \(0<s<l\le \infty \) and \(1<q<\infty \), it follows from (2.7) and (2.6) that f is a locally integrable function on \(\mathbb {R}^{n}\). Take \(g=\chi _{Q}\), where \(Q=\left\{ (y_{1},y_{2},\ldots ,y_{n})\in \mathbb {R}^{n}: \max _{i}|y_{i}|\le 1/2 \right\} \) is a cube in \(\mathbb {R}^{n}\). Let \((\chi _{Q})_{r}(x)=r^{-n}\chi _{Q}(x/r)\) for all \(x\in \mathbb {R}^{n}\) and \(r>0\). Then Lebesgue’s differentiation theorem guarantees that \(\lim \limits _{m\rightarrow \infty } f *(\chi _{Q})_{\frac{1}{m}}(x)=f(x)\) for almost all \(x\in \mathbb {R}^{n}\), which together with [14, Proposition 1.4.5] implies that \(f^{*}\le \liminf \limits _{m\rightarrow \infty } \left( f *(\chi _{Q})_{\frac{1}{m}}\right) ^{*}\). Hence we may apply Fatou’s lemma and (2.11) with \(r=1\) to derive that for \(0<s<l\le \infty \) and \(1<q<\infty \),
This contradicts the fact that \(L^{q, s}(\mathbb {R}^{n})\subsetneqq L^{q, l}(\mathbb {R}^{n})\). Additionally, we remark that necessity of the condition (2.10) results from dilation structure of the convolution \(f *g\) in (2.11), and the exclusion of endpoint cases for the three indices \(q,\,r\) and p is due to [14, Example 1.2.14].
The proof of this lemma relies on Marcinkiewicz’s interpolation theorem for Lorentz spaces as follows, which is given in [14, Theorem 1.4.19].
Lemma 2.3
Let \(0<r \le \infty , 0<p_{0} \ne p_{1} \le \infty ,\) and \(0<q_{0} \ne q_{1} \le \infty \) and let \((X, \mu )\) and (Y, v) be two measure spaces. Let T be either a quasilinear operator with some constant \(K>0\) defined on \(L^{p_{0}}(X)+L^{p_{1}}(X)\) and taking values in the set of measurable functions on Y or a linear operator defined on the set of simple functions on X and taking values as before. Assume that for some \(M_{0}, M_{1}<\infty \) the following (restricted) weak type estimates hold:
for all measurable subsets A of X with \(\mu (A)<\infty .\) Fix \(0<\theta <1\) and let
Then there exists a positive constant C, which depends only on \(K, p_{0}, p_{1}, q_{0}, q_{1}, r\) and \(\theta ,\) such that for all functions f in the domain of T and in \(L^{p, r}(X)\) we have
Now we continue with the proof of Lemma 2.2.
Proof of Lemma 2.2
Since (2.6) implies \(L^{p, l}(\mathbb {R}^{n}) \hookrightarrow L^{p, s}(\mathbb {R}^{n})\), it suffices to prove (2.11) for the case when \(s=l\).
To this end, fix \(g\in L^{r}(\mathbb {R}^{n})\). Let \(T(f)=f*g\), then T is a linear operator defined on the set of simple functions on \(\mathbb {R}^{n}\). For all measurable subsets A of \(\mathbb {R}^{n}\) with \(|A|<\infty \), it follows from (2.6) and Young’s inequality for Lebesgue spaces that
where \(1<r'\le \infty \) and \(1 /r + 1 /r'=1.\)
Take \(\theta =1-r/p=r(1-1/q)\in (0,1)\). Then the hypotheses on the indices imply that
With the help of Lemma 2.3, we obtain that for all functions \(f\in L^{q,l}(\mathbb {R}^{n})\) with \(l\in (0,\infty ]\) there holds
Here \(C>0\) depends only on r, q and l. This completes the proof. \(\square \)
As an application of this lemma, we may derive the generalized Bernstein inequality for Lorentz spaces as follows.
Lemma 2.4
Let a ball \(B=\left\{ \xi \in \mathbb {R}^{n}: |\xi | \le R\right\} \) with \(0<R<\infty \) and an annulus \(\mathcal {C}=\left\{ \xi \in \mathbb {R}^{n}: r_{1} \le |\xi | \le r_{2}\right\} \) with \(0<r_{1}<r_{2}<\infty \). Then a positive constant C exists such that for any nonnegative integer k, any couple (p, q) with \(1< p<q<\infty ,\) any \(\lambda \in (0, \infty )\), and any function u in \(L^{p,\infty }(\mathbb {R}^{n})\) or in \(L^{q,l}(\mathbb {R}^{n})\) with \(0< l\le \infty \), there hold
Here \(\lambda B=\left\{ \xi \in \mathbb {R}^{n}: |\xi | \le \lambda R\right\} \) and \(\lambda \mathcal {C}=\left\{ \xi \in \mathbb {R}^{n}: \lambda r_{1} \le |\xi | \le \lambda r_{2}\right\} \).
Remark 2.3
This lemma extends the following result due to McCormick–Robinson–Rodrigo in [22], for \({\text {supp}} \widehat{f} \subset \lambda B\),
Remark 2.4
In this lemma, the necessity of the assumption that \(p,q>1\) results from Young inequality (2.11) and (2.9) for Lorentz spaces.
Proof
(1) Let \(\psi \) be a Schwartz function on \(\mathbb {R}^{n}\) such that \(\chi _{B}\le \hat{\psi }\le \chi _{2B}\). Since \(\widehat{u}(\xi )=\hat{\psi }(\xi /\lambda ) \widehat{u}(\xi )\) when \({\text {supp}} \widehat{u} \subset \lambda B\), we have
Here \((\partial ^{\alpha } \psi )_{\lambda }(x)=\lambda ^{n}\,\partial ^{\alpha } \psi (\lambda x)\) for all \(x\in \mathbb {R}^{n}\).
Fix \(|\alpha |=k\). From the Hölder inequality (2.5) for Lorentz spaces, we infer that
where \(C=C(p)>0\). Note that \(\partial ^{\alpha } \psi \in \mathcal {S}(\mathbb {R}^{n})\), then for any \(\varepsilon >0\), there exists a constant \(C=C(\varepsilon ,n,\partial ^{\alpha } \psi )>0\) such that \(0\le (\partial ^{\alpha } \psi )_{*}(s)\le Cs^{-\varepsilon }\) for all \(s>0\). This yields that
which implies (2.12).
(2) Take \(1/r=1+1/q-1/p\), then the hypotheses on the indices imply that \(1<r<q<\infty \). In light of (2.9), we see that there exists a positive constant \(C=C(p,q)\) such that
By a similar argument used in the proof of (2.12), we may derive that \(\left\| \partial ^{\alpha } \psi \right\| _{L^{r,1}(\mathbb {R}^{n})}<\infty \). This implies (2.13).
(3) As \(\partial ^{\alpha } \psi \in \mathcal {S}(\mathbb {R}^{n})\subset L^{1}(\mathbb {R}^{n})\), Lemma 2.2 enable us to deduce that
where \(C=C(q,l)>0\). This implies (2.14).
(4) Observe that (2.14) implies the second inequality of (2.15), it suffices to show the first inequality in (2.15). Let \(\eta \) be a Schwartz function on \(\mathbb {R}^{n}\) such that \(\chi _{\mathcal {C}}\le \hat{\eta }\le \chi _{\tilde{\mathcal {C}}}\), where \(\tilde{\mathcal {C}}=\left\{ \xi \in \mathbb {R}^{n}: r_{1}/2 \le |\xi | \le 2 r_{2}\right\} \). It follows from \( {\text {supp}} \hat{u} \subset \lambda \mathcal {C}\) that for all \(\xi \in \mathbb {R}^{n}\),
Therefore, we may write
where \((g_{\alpha })_{\lambda }(x)=\lambda ^{n} g_{\alpha }(\lambda x)\) for all \(x\in \mathbb {R}^{n}\), and
This together with Lemma 2.2 yields that a constant \(C=C(n,q,l,k)>0\) exists such that
This concludes (2.15) and the proof is complete. \(\square \)
2.2 Besov–Lorentz Spaces, Sobolev–Lorentz Spaces and Triebel–Lizorkin–Lorentz Spaces
\(\mathcal {S}\) denotes the Schwartz class of rapidly decreasing functions, \(\mathcal {S}'\) the space of tempered distributions, \(\mathcal {S}'/\mathcal {P}\) the quotient space of tempered distributions which modulo polynomials. We use \(\mathcal {F}f\) or \(\widehat{f}\) to denote the Fourier transform of a tempered distribution f. To define Besov–Lorentz spaces, we need the following dyadic unity partition (see e.g. [1]). Choose two nonnegative radial functions \(\varrho \), \(\varphi \in C^{\infty }(\mathbb {R}^{n})\) be supported respectively in the ball \(\{\xi \in \mathbb {R}^{n}:|\xi |\le \frac{4}{3} \}\) and the shell \(\{\xi \in \mathbb {R}^{n}: \frac{3}{4}\le |\xi |\le \frac{8}{3} \}\) such that
In particular, let \(K(r)\in C^{\infty }(\mathbb {R}^{+})\) be a decreasing function on \(\mathbb {R}^{+}\) supported in the interval [0, 16/9], and \(K(r)=1\) for any \(r\in [0,9/16].\) Then for every \(\xi \in \mathbb {R}^{n},\) we may choose \(\varrho (\xi )=K(|\xi |^{2})\) and \(\varphi (\xi )=\varrho (\xi /2)-\varrho (\xi )\) here.
The nonhomogeneous dyadic blocks \(\Delta _{j}\) are defined by
The homogeneous Littlewood–Paley operators are defined as follows
The homogeneous Besov–Lorentz space \( \dot{B}^{s}_{p,q,r}(\mathbb {R}^{n})\) is the set of \(f\in \mathcal {S}'(\mathbb {R}^{n})/\mathcal {P}(\mathbb {R}^{n})\) such that
Here \(\ell ^{r}(\mathbb {Z})\) represents the set of sequences with summable r-th powers. The homogeneous Sobolev–Lorentz norm \(\Vert \cdot \Vert _{\dot{H}^{s}_{p,p_{1}}(\mathbb {R}^{n})}\) is defined as \(\Vert f\Vert _{\dot{H}^{s}_{p,p_{1}}(\mathbb {R}^{n})}=\Vert \Lambda ^{s}f\Vert _{L^{p,p_{1}}(\mathbb {R}^{n})}.\) When \(p_{1}=p\), the Sobolev–Lorentz spaces reduce to the classical Sobolev spaces \(\dot{H}^{s}_{p}(\mathbb {R}^{n})\).
For \(p,q,r\in (0,\infty ]\) and \(s \in \mathbb {R},\) the homogeneous Triebel–Lizorkin–Lorentz space \(\dot{F}_{p, q,r}^{s}(\mathbb {R}^{n})\) is defined by
Here
Lemma 2.5
Suppose that \(f\in \dot{H}^{s}_{p,\infty }(\mathbb {R}^{n})\) with \(s\in \mathbb {R}\) and \(p\in (1,\infty ]\). Then there holds
where \(C>0\) depends only on n, s, p and \(\varphi \).
Remark 2.5
It is worth remarking that the nonhomogeneous embedding \(H^{s}_{p,\infty }\hookrightarrow F^{s}_{p,\infty ,\infty }\) was recently proved by Ko and Lee in [18] and they mentioned that the results hold for the homogeneous case. One can modify the argument in [18] to get the homogeneous case \(\dot{H}^{s}_{p,\infty }\hookrightarrow \dot{F}^{s}_{p,\infty ,\infty }\). Here, we shall present a different proof via the Hardy–Littlewood maximal function.
Proof
Since it is obvious that
we focus on the proof
It suffices to show that for all functions \(g\in L^{p,\infty }(\mathbb {R}^{n})\),
where \(\phi =\left( |\xi |^{-s}\varphi (\xi )\right) ^{\vee }\) and \(\phi _{j}(x)=2^{jn}\phi (2^{j}x)\) for all \(x\in \mathbb {R}^{n}\).
To this end, observe that \(\phi \) is a Schwartz function on \(\mathbb {R}^{n}\) and there exists a positive constant C such that for all \(x\in \mathbb {R}^{n}\), \(|\phi (x)|\le C (1+|x|)^{-n-1}.\) This yields that for all \(j \in \mathbb {Z}\) and \(x\in \mathbb {R}^{n}\),
which implies that there exists a positive constant \(C=C(n,\phi )\) such that for all \(x\in \mathbb {R}^{n}\),
Then it follows from (2.3) that
where \(C>0\) depends only on n, p and \(\phi \). This concludes the proof. \(\square \)
3 Proof of Theorem 1.1 and 1.2
3.1 Gagliardo–Nirenberg Inequalities in Lorentz Spaces
The goal of this subsection is to prove Theorem 1.1 involving Gagliardo–Nirenberg inequalities in Lorentz spaces. Firstly, we establish three key inequalities, which play an important role in the proof of three cases in Theorem 1.1. Secondly, in view of a pointwise interpolation estimate for derivatives found in [1] and equivalent norms of Lorentz spaces, we get Proposition 3.2. Finally, we are in a position to show Theorem 1.1.
Lemma 3.1
-
(1)
Suppose that \(u \in \dot{H}_{r,p_{1}}^{s}\left( \mathbb {R}^{n}\right) \) with \(1< r<\infty \), \(0 \le s<n / r\) and \(0< p_{1}\le \infty \). Then there exists a positive constant \(C=C(n,s,p,r,p_{1})\) such that
$$\begin{aligned} \Vert u\Vert _{L^{p,p_{1}}(\mathbb {R}^{n})} \le C\Vert \Lambda ^{s}u\Vert _{L^{r,p_{1}}(\mathbb {R}^{n})} ~~~\text {with}~~~ \frac{n}{p}=\frac{n}{r}-s. \end{aligned}$$(3.1) -
(2)
Let \(1< q,p,r<\infty \), \(\,1 \le l \le \infty \) and \( s= {n}/{r}.\) Then there exists a positive constant C such that for all functions \(u \in L^{p,\infty }\left( \mathbb {R}^{n}\right) \cap \dot{B}^{s}_{r,\infty ,\infty }\left( \mathbb {R}^{n}\right) \), there holds
$$\begin{aligned} \begin{aligned} \Vert u\Vert _{L^{q,l}(\mathbb {R}^{n})} \le C \Vert u\Vert _{L^{p,\infty }(\mathbb {R}^{n})}^{\frac{p}{q}} \Vert u\Vert ^{1-\frac{p}{q}}_{\dot{B}^{s}_{r,\infty ,\infty }}~~~\text {with}~~~p<q. \end{aligned} \end{aligned}$$ -
(3)
Let \(u \in L^{p,\infty }\left( \mathbb {R}^{n}\right) \cap \dot{B}^{s}_{r,\infty ,\infty }\left( \mathbb {R}^{n}\right) \) with \(s>n / r \) and \(1 < p, r\le \infty .\) Then there exists a positive constant \(C=C(n,s,p,r)\) such that
$$\begin{aligned} \Vert u\Vert _{L^{\infty }(\mathbb {R}^{n})} \le C \Vert u\Vert _{L^{p,\infty }(\mathbb {R}^{n})}^{\theta }\Vert u\Vert _{\dot{B}^{s}_{r,\infty ,\infty }}^{1-\theta }, \end{aligned}$$where
$$\begin{aligned} 0=\theta \frac{n}{p}+(1-\theta )\left( \frac{n}{r}-s\right) ,~~0<\theta \le 1. \end{aligned}$$
Remark 3.1
As a corollary of this lemma and imbedding relation in Lemma 2.5, we have
with \(1<p<q<\infty ,\,1<r<\infty \) and \(1 \le l \le \infty \), and
with \(0=\theta n/p+(1-\theta )\left( n/r-s+\sigma \right) \), \(0<\theta \le 1\), \(s-\sigma >n/r\) and \(1 < p, r\le \infty \).
Remark 3.2
In this lemma, the necessity of the condition that \(p,r>1\) results from Lemma 2.2, Young inequality (2.9) and Bernstein inequality (2.13) for Lorentz spaces. Due to (2.2), it is also essential to make the assumption that \(q>1\) and \(l \ge 1\) to ensure that the space \(L^{q,l}(\mathbb {R}^{n})\) is normable and the corresponding triangular inequality is applicable.
Proof of Lemma 3.1
(1) Thanks to Fourier transform, there exists a positive constant \(C=C(n,s)\) such that
With the help of the Young inequality (2.9) in Lorentz spaces and the fact that \(|x|^{-1}\in L^{n,\infty }\), we see that
(2) By means of the low and high frequencies, it follows from (2.2) that
Here \(j_{0}\) is an integer to be chosen later. Observe that
where \(h_{j_{0}}(x)=2^{j_{0}n} h(2^{j_{0}} x)\) for all \(x\in \mathbb {R}^{n}\), and
Owing to Bernstein’s inequality (2.13) and (2.6), we may apply Lemma 2.2 to infer that there exists a positive constant C independent of \(j_{0}\) such that
For the high-frequency part, it follows from (2.4), (2.6), (2.13) and Lemma 2.2 that
Here \(s=n/r\), \(1/q=(1-\alpha )/p+\alpha /(q+r)\) with \(0<\alpha <1\), and \(\varphi _{j}(x)=2^{jn}\varphi (2^{j}x)\) for all \(x\in \mathbb {R}^{n}\). It turns out that
where the positive constant C is independent of \(j_{0}\). Since \(1/p-1/q+\alpha /(q+r)=\alpha /p\), by choosing \(j_{0}\) such that \(2^{j_{0}n(\frac{1}{p}-\frac{1}{q})}\Vert u\Vert _{L^{p,\infty } (\mathbb {R}^{n})}\approx 2^{-\frac{j_{0}n\alpha }{q+r}} \Vert u\Vert _{L^{p,\infty }(\mathbb {R}^{n})}^{1-\alpha }\Vert u\Vert _{\dot{B}^{s}_{r,\infty ,\infty }}^{\alpha }\), we may derive that
(3) If \(s> {n}/{r}\), we take \(q= l=\infty \) in (3.4). Using Bernstein’s inequality (2.12) and Young inequality for Lorentz spaces, we find that
where the positive constant C is independent of \(j_{0}\). As the derivation of the above, we may choose \(j_{0}\) appropriately to conclude that
This completes the proof of this lemma. \(\square \)
Proposition 3.2
Assume that \(u \in L^{q,q_{1}}\left( \mathbb {R}^{n}\right) \cap \dot{H}_{r,r_{1}}^{s}\left( \mathbb {R}^{n}\right) \) with \(1<q, r \le \infty \) and \(1\le q_{1}, r_{1} \le \infty \). Then there holds for \(0< \sigma<s<\infty \),
with
Here C is a positive constant depending only on \(q,r,q_{1},r_{1},s,\sigma \) and n.
Remark 3.3
As a special case of this proposition, there holds the following estimate
where q, r, p, s and \(\sigma \) satisfy the same conditions as in Proposition 3.2. This inequality will be used in the proof of Theorem 1.1.
Remark 3.4
Very recently, by means of Maz’ya–Shaposhnikova pointwise estimates in [21], Fiorenza–Formica–Roskovec–Soudský [12] showed the following estimate
where \(j, k \in \mathbb {N}, 1 \le j<k, 1<q, r\le \infty ,\) \(1\le q_{1}, r_{1 }\le \infty \) and
Proposition 3.2 extends the aforementioned integer cases of Gagliardo-Nirenberg inequalities in [12] to the fractional cases.
Proof
Thanks to the decomposition of low and high frequencies, one can derive the following pointwise estimate
whose proof can be found in [1, p. 84].
As a consequence, it follows from the Hölder’s inequality for Lorentz spaces that
According to (2.3), we conclude the desired estimate. \(\square \)
Now, at this stage, we can prove Theorem 1.1.
Proof of Theorem 1.1
(I) First, we consider (1.6) under the hypothesis that \(0< s-\sigma <\frac{n}{r}\).
(\(I_{1}\)) If \(\sigma =0\), it follows from the interpolation characteristic (2.4) of Lorentz spaces that
where we have used the fact that \(\,\frac{1}{\tilde{p}}=\frac{1}{r}-\frac{s}{n}\ne \frac{1}{q}\). This together with the Sobolev inequality (3.1) ensures that
with
(\(I_{2}\)) If \(\sigma >0\), the Sobolev embedding (3.1) yields
with
It follows from Proposition 3.2 that for \(1<q, r \le \infty \),
with
Then the hypothesis on the indices that
imply the following relation
Observe that the condition \(\,s-\frac{n}{r} \ne \sigma -\frac{n}{p}\,\) guarantees \(\,\widetilde{p}\ne r^{*}\). From the interpolation characteristic (2.4) of Lorentz spaces, we see that
Plugging (3.7) and (3.8) into (3.9), we get
(II) Next, we turn our attention to the case that \(s-\sigma = {n}/{r}\) in (1.6). By virtue of (3.2), we have proved (1.6) with \(s= {n}/{r}\) and \(\sigma =0\). In the following, we assume that \(\sigma >0\). According to Proposition 3.2, we arrive at that for \(\frac{1}{\tilde{p}}=\left( 1-\frac{\sigma }{s}\right) \frac{1}{q}+\frac{\sigma }{sr}=\left( 1-\frac{\sigma }{s}\right) \left( \frac{1}{q}+\frac{\sigma }{n}\right) \),
Since \(\frac{1}{p}=\theta \left( \frac{1}{q}+\frac{\sigma }{n}\right) <\left( 1-\frac{\sigma }{s}\right) \left( \frac{1}{q}+\frac{\sigma }{n}\right) =\frac{1}{\tilde{p}}\), it follows from (3.2) that
Substituting (3.10) into (3.11), we obtain
(III) Finally, it is enough to show (1.6) under the hypothesis that \(s-\sigma > {n}/{r}\).
(\(III_{1}\)) If \(\sigma >0\), we conclude from Proposition 3.2 that for \(1<q, r \le \infty \),
where
Observe that
In view of the interpolation characteristic (2.4) of Lorentz spaces, we see that
with
Furthermore (3.3) ensures that
where
Inserting (3.14) into (3.13) and using (3.12), we have
(\(III_{2}\)) If \(\sigma =0\), we note that
Then it follows from the interpolation characteristic (2.4) of Lorentz spaces that
In view of (3.3), we obtain
This together with (3.15) yields that
We complete the proof of Theorem 1.1. \(\square \)
3.2 Gagliardo–Nirenberg Inequalities in Besov–Lorentz Spaces
In this subsection, by means of the Littlewood–Paley decomposition and generalized Bernstein inequalities in Lemma 2.4, we shall prove the Gagliardo–Nirenberg inequality (1.11) in Besov–Lorentz spaces.
Proof of Theorem 1.2
First, we assert that \(q\le p\), \(\sigma >0\) and \(\frac{n}{p}-\sigma =\theta \frac{n}{q}+(1-\theta )(\frac{n}{r}-s)\) imply that
which will be frequently used later. Indeed, thanks to \(q\le p\) and \(\sigma >0\), we see that
that is,
The assertion follows. Similarly, we also assert that \(q< p\) and \(\sigma =0\) yield (3.16).
(I) We next consider the case \(q=r\) of (1.11). Note that
Therefore, we see that \(r=q<p\) and (3.16) are valid. With the help of the Bernstein inequality (2.13) in Lorentz spaces, we infer that
where we have used \(\sigma +n(\frac{1}{r}-\frac{1}{p})>0\) and (3.16). Observe that the positive constant C in (3.17) is independent of k. As a consequence, by choosing the integer k appropriately such that \( 2^{k[\sigma +n(\frac{1}{r}-\frac{1}{p})]} \Vert u\Vert _{\dot{B}^{0}_{r,\infty ,\infty }}\approx 2^{-k[s-\sigma -n(\frac{1}{r}-\frac{1}{p})]} \Vert u\Vert _{\dot{B}^{s}_{r,\infty ,\infty }}\), we further get
(II) We turn our attention to the case \(q<r\). In order to get (3.16), we check that \(q<p\) via the following straightforward calculation
We also see that
The following discussion will be divided into three subcases that \(q<r<p\), \( q<p< r\) and \( q<p= r\).
(\(II_{1}\)) We examine the case \(q<r<p\). As the derivation of (3.17), we find that
where we have used \(\sigma +n(\frac{1}{q}-\frac{1}{p})>0\) and (3.16).
Hence, we conclude that
(\(II_{2}\)) We deal with the case \( q<p< r\). By means of the interpolation characteristic (2.4) of Lorentz spaces, we observe that
Combining this, the Bernstein inequality (2.13) in Lorentz spaces and (3.16), we know that
where we have used \(\sigma +n(\frac{1}{q}-\frac{1}{p})>0\) and \(s(1-\alpha )-\sigma >0\) which is derived from (3.18). Choosing the integer k such that \( 2^{k[\sigma +n(\frac{1}{q}-\frac{1}{p})]}\Vert u\Vert _{\dot{B}^{0}_{q,\infty ,\infty }}\approx 2^{-k[s(1-\alpha )-\sigma ]}\Vert u\Vert _{\dot{B}^{0}_{q,\infty ,\infty }}^{\alpha } \Vert u\Vert _{\dot{B}^{s}_{r,\infty ,\infty }}^{1-\alpha }\), we also have
(\(II_{3}\)) We treat the case \( q<p= r\). It follows from the interpolation characteristic (2.4) of Lorentz spaces that
where \(\varepsilon >0\) will be determined later. We derive from this, the Bernstein inequality (2.13) in Lorentz spaces and (3.16) that
where we have used the fact that \(\sigma +n(\frac{1}{q}-\frac{1}{p})>0\).
Denote \(\delta (\varepsilon )=s\alpha - \frac{n\varepsilon \alpha }{(1+\varepsilon )p}-\sigma \). From (3.20), we see that
and \(\delta (\varepsilon )\) is a continuous function on a neighborhood of 0. Since \(\delta (0)>0\), there exists a sufficiently small \(\varepsilon >0\) such that \(\delta (\varepsilon )>0\). It follows from (3.21) that
which also yields that
(III) Finally, it remains to show (1.11) under the case that \(q>r\).
(\(III_{1}\)) We first consider (1.11) under the hypothesis that \(r<q\le p\). We divide this case into two subcases that \(r<q <p\) and \(r<q =p\).
(\(III_{11}\)) We handle with the case \(r<q <p\). In view of the Bernstein inequality (2.13), we see that
where we have used \(\sigma +n(\frac{1}{q}-\frac{1}{p})>0\) and (3.16).
Therefore, we conclude that
(\(III_{12}\)) We need to show (1.11) under the hypothesis that \(r<q=p\). Observe that the condition \(\,s-\frac{n}{r} \ne \sigma -\frac{n}{p}\,\) implies that \(\sigma >0 \) in this case. In the same manner as (3.20), we see that
where \(\varepsilon >0\) will be determined later. Then we obtain
As the arguments in (\(II_{3}\)), we can choose \(\varepsilon >0\) sufficiently small to ensure that \(\sigma +\frac{n\varepsilon \alpha }{p(1+\varepsilon )}-s(1-\alpha )>0\). This yields the desired inequality (1.11). We omit the details.
(\(III_{2}\)) Let \(r<q\) and \( p<q\). It is clear that
which yields that \(r<p\) in this case. Additionally, the condition \(\,s-\frac{n}{r} \ne \sigma -\frac{n}{p}\,\) guarantees that \(\frac{ 1}{p} \ne (1-\frac{\sigma }{s})\frac{1}{q}+\frac{\sigma }{s}\frac{1}{r}\).
(\(III_{21}\)) We assume that \(r<q,\) \(p<q \) and \(\frac{ 1}{p} < (1-\frac{\sigma }{s})\frac{1}{q}+\frac{\sigma }{s}\frac{1}{r}\). Then (3.16) follows from (3.24). Note that \(r<p<q\), a slight modification of the proof of (3.19) together with (3.16) implies that
Here we need the fact that \(\sigma -s(1-\alpha )>0\) with \(1/p=\alpha /q+(1-\alpha )/r\), which is derived from the hypothesis that \(\frac{ 1}{p} < (1-\frac{\sigma }{s})\frac{1}{q}+\frac{\sigma }{s}\frac{1}{r}\). Therefore, we obtain the desired estimate
(\(III_{22}\)) Now, it remains to show (1.11) with \(r<q \) and \(\frac{ 1}{p} > (1-\frac{\sigma }{s})\frac{1}{q}+\frac{\sigma }{s}\frac{1}{r}\), which imply that
It follows from (3.24) and \(\frac{ 1}{p} > (1-\frac{\sigma }{s})\frac{1}{q}+\frac{\sigma }{s}\frac{1}{r}\) that
which together with \(p<q\) enables us to derive that
Making use of the Bernstein inequality (2.13) for Lorentz spaces, (2.4), (3.25) and (3.26), we arrive at
We thereby deduce the inequality
The proof of this theorem is completed. \(\square \)
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Acknowledgements
The authors thank the anonymous referees and the associated editor for the invaluable comments and suggestions which helped to improve the paper greatly. The authors would like to express their sincere gratitude to Dr. Xiaoxin Zheng at Beihang University, for a lot of useful discussion on this topic. Wei was partially supported by the National Natural Science Foundation of China under grant (Nos. 11601423, 12271433, 11771352, 11871057). Wang was partially supported by the National Natural Science Foundation of China under grant (Nos. 11971446, 12071113 and 11601492) and sponsored by Natural Science Foundation of Henan (No. 232300421077). Ye was partially supported by the National Natural Science Foundation of China under grant (No. 11701145) and sponsored by Natural Science Foundation of Henan (No. 232300420111).
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Wei, W., Wang, Y. & Ye, Y. Gagliardo–Nirenberg Inequalities in Lorentz Type Spaces. J Fourier Anal Appl 29, 35 (2023). https://doi.org/10.1007/s00041-023-10018-2
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DOI: https://doi.org/10.1007/s00041-023-10018-2