Abstract
In this paper we describe the Euler semigroup \(\{e^{-t\mathbb {E}^{*}\mathbb {E}}\}_{t>0}\) on homogeneous Lie groups, which allows us to obtain various types of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities for the Euler operator \(\mathbb {E}\). Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and \(|\cdot |\)-radial weighted Hardy–Sobolev type inequality are established.
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1 Introduction
In this paper we continue the research from [20] devoted to properties of Euler operators on homogeneous groups, their consequences, and related analysis. In turn, this continues the research direction initiated in [16] devoted to Hardy and other functional inequalities in the setting of Folland and Stein’s [9] homogeneous groups.
Recall the following Sobolev type (improved Hardy) inequality: Let \(1<p<\infty\). Then we have
for all \(f\in C_{0}^{\infty }({\mathbb {R}}^{n})\), where \(\nabla\) is the standard gradient in \({\mathbb {R}}^{n}\).
In [15, Proposition 1.4] the authors showed that the Sobolev type inequality (1.1) is equivalent to the following Hardy inequality when \(p=2\) and \(n\ge 3\):
for all \(f,g\in C_{0}^{\infty }({\mathbb {R}}^{n}\backslash \{0\})\), where \(\Vert x\Vert =\sqrt{x_{1}^{2}+\cdots +x_{n}^{2}}\).
We note that the analysis of the remainder terms in such inequalities has a long history, for example, we refer to [4] for Sobolev inequalities, to [6, Section 4] and [5] for Hardy inequalities and for many others, and a more recent literature review to [10].
In this paper we are interested, among other things, in obtaining (1.1) with remainder terms, that is, for example, in [1, Corollary 4.4] the authors obtained the following: Let \(n\ge 3\) and \(0\le \delta <n^{2}/4\). Then there exists a positive constant C such that
holds for all \(f\in C_{0}^{\infty }({\mathbb {R}}^{n})\), where \(F(x):={\mathscr {M}}(f)=\frac{1}{{\mathbb {S}}^{n-1}}\int _{{\mathbb {S}}^{n-1}}f(r,y)dy\) is the integral mean of f over the unit sphere \({\mathbb {S}}^{n-1}\), and \(2^{*}=2n/(n-2)\), \(r=|x|\). They derived (1.3) from the following Gagliardo–Nirenberg inequality [1, Theorem 4.1] using the explicit representation of the semigroup \(e^{-t(x\cdot \nabla )^{*}(x\cdot \nabla )}\): Let \(1\le p<q<\infty\) and let \(-i(\partial /\partial r) f\in L^{p}({\mathbb {R}}\times {\mathbb {S}}^{n-1})\) and \(f\in B^{p/(p-q)}({\mathbb {R}}^{n})\). Then there exists a positive constant \(C=C(p,q)\) such that
where \({\mathscr {R}}:=\frac{d}{d|x|}\) is the radial derivative, and \(\Vert \cdot \Vert _{B^{\alpha }}\) is the Besov type norm associated with the semigroup.
We show analogues of (1.3) and (1.4), and calculate the semigroup \(e^{-t\mathbb {E}^{*}\mathbb {E}}\) on homogeneous (Lie) groups, where \(\mathbb {E}\) is the Euler operator (see Sect. 2 for more information). We can also obtain (1.3) with sharp constant without using (1.4) and semigroup \(e^{-t\mathbb {E}^{*}\mathbb {E}}\). We will also explain that the obtained homogeneous group results are not only analogues of the known Euclidean results, but also they give new inequalities even in Abelian cases with arbitrary quasi-norms (see Remark 4.13). Since we do not have sub-Laplacian and “horizontal” gradients on general homogeneous groups, we will work with radial derivative (\(\mathscr {R}:=\frac{d}{dr}\)) and Euler (\(\mathbb {E}:=|x|\mathscr {R}\)) operators. Some other discussions on this idea will be provided in the final section. We refer to [19] for horizontal versions of these inequalities.
For the convenience of the reader we briefly summarise the obtained results. Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q and let us fix any homogeneous quasi-norm \(|\cdot |\). Then we have the following results:
-
Let \(x=ry\) with \(r=|x|\) and \(|y|=1\). Then the semigroup \(e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}\) is given by
$$\begin{aligned} \begin{aligned} (e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}f)(x)&=\frac{e^{-tQ^{2}/4}}{\sqrt{4\pi t}}r^{-Q/2}\int ^{\infty }_{0}e^{-\frac{(\ln r - \ln s )^{2}}{4t}}s^{-Q/2}f(sy)s^{Q-1}ds\\&= \frac{e^{-tQ^{2}/4}}{\sqrt{4\pi t}}|x|^{-Q/2}\int ^{\infty }_{0}e^{-\frac{(\ln |x| - \ln s )^{2}}{4t}}s^{-Q/2}f(sy)s^{Q-1}ds, \end{aligned} \end{aligned}$$(1.5)where \({\mathbb {E}}=|x|\mathscr {R}\) is the Euler operator and \({\mathscr {R}}:=\frac{d}{d|x|}\) is the radial derivative.
-
Let \(1\le p<q<\infty\) and let f be such that \(\mathscr {R}f\in L^{p}({\mathbb {R}}\times \mathfrak {S})\) and \(f\in B^{p/(p-q)}({\mathbb {R}}\times \mathfrak {S})\). Then there exists a positive constant \(C=C(p,q)\) such that
$$\begin{aligned} \Vert f \Vert _{L^{q}({\mathbb {R}}\times \mathfrak {S})}\le C \Vert \mathscr {R}f\Vert ^{p/q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})}\Vert f\Vert ^{1-p/q}_{B^{p/(p-q)}({\mathbb {R}}\times \mathfrak {S})}, \end{aligned}$$(1.6)where \({\mathscr {R}}:=\frac{d}{d|x|}\) is the radial derivative, \(\mathfrak {S}\) is the unit sphere in \(\mathbb {G}\) and \(\Vert \cdot \Vert _{B^{\alpha }}\) is the Besov type norm defined by (4.1).
-
Let \(Q>2\). Then we have for all \(f\in C^{\infty }_{0}(\mathbb {G})\) and \(0\le \delta <\frac{Q^{2}}{4}\) the inequality
$$\begin{aligned} \begin{aligned}&\int _{\mathbb {G}}|\mathbb {E}f(x)|^{2}dx-\delta \int _{\mathbb {G}} |f(x)|^{2}dx\\&\quad \ge \frac{\left( \frac{Q^{2}}{4}-\delta \right) ^{\frac{Q-1}{Q}}}{\left( \frac{(Q-2)^{2}}{4}\right) ^{\frac{Q-1}{Q}}}S_{Q} \left( \int _{\mathbb {G}}|x|^{2^{*}}|g(|x|)|^{2^{*}}dx\right) ^{\frac{2}{2^{*}}} \end{aligned} \end{aligned}$$(1.7)with sharp constant, where \(g(|x|)={\mathscr {M}}(f)(|x|):=\frac{1}{|\mathfrak {S}|}\int _{\mathfrak {S}}f(|x|,y)d\sigma (y)\), \(2^{*}=2Q/(Q-2)\) and \(S_{Q}\) is the constant defined in (4.20).
-
Let \(\phi\) and \(\psi\) be positive functions defined on \(\mathbb {G}\). Then there exists a positive constant C such that
$$\begin{aligned} \int _{\mathbb {G}}\phi (x)\exp ({\mathscr {M}} \log f)(x)dx\le C\int _{\mathbb {G}} \psi (x)f(x)dx \end{aligned}$$(1.8)holds for all positive f if and only if \(A(\phi ,\psi )<\infty\), where A is given in (5.2) and \(({\mathscr {M}}f)(x)=\frac{1}{|B(0,|x|)|}\int _{B(0,|x|)}f(z)dz\).
-
Let \(Q\ge 2\). For any quasi-norm \(|\cdot |\), all differentiable \(|\cdot |\)-radial functions \(\phi\), all \(p>1\) and all \(f \in C_{0}^{1}({\mathbb {G}})\) we have
$$\begin{aligned} \int _{{\mathbb {G}}} \frac{\phi ^{\prime }(|x|)}{|x|^{Q-1}}|f(x)|^{p}dx \le \int _{{\mathbb {G}}} \left| \mathscr {R}f(x) \right| ^{p} dx + (p-1)\int _{{\mathbb {G}}} \frac{|\phi (|x|)|^{\frac{p}{p-1}}}{|x|^{\frac{p(Q-1)}{p-1}}} |f(x)|^{p}dx. \end{aligned}$$In the Abelian case with the standard Euclidean distance \(\Vert x\Vert =\sqrt{x^{2}_{1}+\ldots +x^{2}_{n}}\) it implies the “critical” Hardy inequality
$$\begin{aligned} \int _{{\mathbb {R}}^{n}} \frac{|f(x)|^{n}}{\Vert x\Vert ^{n}} dx \le \int _{{\mathbb {R}}^{n}} \left| \nabla f(x) \right| ^{n} dx + (n-1)\int _{{\mathbb {R}}^{n}} \frac{\left| \log \frac{1}{\Vert x\Vert }\right| ^{\frac{n}{n-1}}}{\Vert x\Vert ^{n}} \left| f(x) \right| ^{n} dx, \end{aligned}$$(1.9)where \(\nabla\) is the standard gradient on \({\mathbb {R}}^{n}\).
-
Let \(Q\ge 2\). For any quasi-norm \(|\cdot |\), all differentiable \(|\cdot |\)-radial functions \(\phi\), all \(p>1\) and all \(f \in C_{0}^{1}({\mathbb {G}})\) we have
$$\begin{aligned} \int _{{\mathbb {G}}} \frac{\phi ^{\prime }(|x|)}{|x|^{Q-1}}|f(x)|^{p}dx \le p \left( \int _{{\mathbb {G}}} \left| \mathscr {R}f(x) \right| ^{p} dx \right) ^{\frac{1}{p}} \left( \int _{{\mathbb {G}}} \frac{|\phi (|x|)|^{\frac{p}{p-1}}}{|x|^{\frac{p(Q-1)}{p-1}}} |f(x)|^{p}dx\right) ^{\frac{p-1}{p}}. \end{aligned}$$In the Abelian case with the standard Euclidean distance \(\Vert x\Vert =\sqrt{x^{2}_{1}+\ldots +x^{2}_{n}}\) it implies the “improved” Heisenberg–Pauli–Weyl uncertainty principle
$$\begin{aligned} \left( \int _{{\mathbb {R}}^{n}}|f(x)|^{2}dx\right) ^{2} \le \left( \frac{2}{n}\right) ^{2} \int _{{\mathbb {R}}^{n}} \left| \nabla f(x) \right| ^{2} dx \int _{{\mathbb {R}}^{n}} \Vert x\Vert ^{2} |f(x)|^{2}dx, \quad n\ge 2, \end{aligned}$$(1.10)where \(\nabla\) is the standard gradient on \({\mathbb {R}}^{n}\).
Indeed the same inequality with the constant \(\left( \frac{2}{n-2}\right) ^{2}, n\ge 3,\) (instead of \(\left( \frac{2}{n}\right) ^{2}\)) is known as the Heisenberg–Pauli–Weyl uncertainty principle (see, e.g. [17, Remark 2.10]), that is,
$$\begin{aligned} \left( \int _{{\mathbb {R}}^{n}}|f|^{2}dx\right) ^{2} \le \left( \frac{2}{n-2}\right) ^{2} \int _{{\mathbb {R}}^{n}} \left| \nabla f \right| ^{2} dx \int _{{\mathbb {R}}^{n}} \Vert x\Vert ^{2} |f|^{2}dx, \quad n\ge 3, \end{aligned}$$(1.11)for all \(f\in C^{1}_{0}({\mathbb {R}}^{n}).\) Obviously, since \(\frac{2}{n-2}\ge \frac{2}{n},\, n\ge 3,\) inequality (1.10) is an improved version of (1.11). Note that equality case in (1.10) holds for the function \(f=C\exp (-b \Vert x\Vert ),\,b>0.\)
The organisation of the paper is as follows. In Sect. 2 we briefly recall the necessary concepts of homogeneous Lie groups and fix the notation. The operator semigroup \(\{e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}\}_{t>0}\) is determined in Sect. 3. In Sect. 4 we establish Gagliardo–Nirenberg type inequalities, and obtain sharp remainder terms of the Sobolev type inequality. The maximal Hardy inequality is established in Sect. 5. In Sect. 6 we give some further discussions on (critical) Hardy–Sobolev type inequalities on homogeneous groups.
2 Preliminaries
In this section we briefly recall the necessary notions and fix the notation for homogeneous groups. We refer to [8] for a detailed discussion of the appearing objects.
Let us recall the notion of a family of dilations of a Lie algebra \({\mathfrak {g}}\), that is, a family of linear mappings of the following form
where A is a diagonalisable linear operator on \({\mathfrak {g}}\) with positive eigenvalues. We also recall that \(D_{\lambda }\) is a morphism of \({\mathfrak {g}}\) if it is a linear mapping from \({\mathfrak {g}}\) to itself satisfying the property
where \([X,Y]:=XY-YX\) is the Lie bracket. Then, a homogeneous group\(\mathbb {G}\) is a connected simply connected Lie group whose Lie algebra is equipped with a morphism family of dilations. It induces the dilation structure on \({\mathbb {G}}\) which we denote by \(D_{\lambda }x\) or just by \(\lambda x\).
Let dx be the Haar measure on \({\mathbb {G}}\) and let |S| denote the volume of a measurable subset S of \(\mathbb {G}\). The homogeneous dimension of \({\mathbb {G}}\) is defined by
Then we have
A homogeneous quasi-norm on \({\mathbb {G}}\) is a continuous non-negative function
satisfying the properties
-
\(|x^{-1}| = |x|\) for all \(x\in {\mathbb {G}}\),
-
\(|\lambda x|=\lambda |x|\) for all \(x\in {\mathbb {G}}\) and \(\lambda >0\),
-
\(|x|= 0\) if and only if \(x=0\).
The quasi-ball centred at \(x\in {\mathbb {G}}\) with radius \(R > 0\) can be defined by
The polar decomposition on homogeneous groups can be formulated as follows: there is a (unique) positive Borel measure \(\sigma\) on the unit sphere
such that
holds for all \(f\in L^{1}({\mathbb {G}})\).
We refer to Folland and Stein [9] for more information (see also [8, Section 3.1.7] for a detailed discussion).
If we fix a basis \(\{X_{1},\ldots ,X_{n}\}\) of \({\mathfrak {g}}\) such that we have
for each k, then the matrix A can be taken to be \(A=\mathrm{diag} (\nu _{1},\ldots ,\nu _{n})\), where each \(X_{k}\) is homogeneous of degree \(\nu _{k}\) and
The decomposition of \({\exp }_{{\mathbb {G}}}^{-1}(x)\) in \({\mathfrak {g}}\) defines the vector
by the formula
On the other hand, we also have the following equality
Using homogeneity and denoting \(x=ry,\,y\in \mathfrak {S},\) we obtain
By a direct calculation, we get
which implies the following equality
for each homogeneous quasi-norm |x| on a homogeneous group \({\mathbb {G}}\), where we have denoted
We will also use the Euler operator
Let us recall the following property of the Euler operator \({\mathbb {E}}\).
Lemma 2.1
([20, Lemma 2.2]) We have
where \({\mathbb {I}}\) and \(\mathbb {E^{*}}\) are the identity operator and the formal adjoint operator of \({\mathbb {E}}\), respectively.
Now we give useful inequality and identity for Euler operator.
Lemma 2.2
Let \(1<p<\infty\) and let \(\{\phi _{i}\}_{i=1}^{3}\in L_{loc}^{1}(\mathbb {G})\) be any radial functions. If we denote
then we have
for any \(f\in L_{loc}^{p}(\mathbb {G})\). Moreover, we have
and
for any \(k\in {\mathbb {N}}\) and all \(f\in L_{loc}^{p}(\mathbb {G})\) such that \(\mathbb {E}^{k} f\in L_{loc}^{p}(\mathbb {G})\) with sharp constants. The constants are attained when \(f={\widetilde{f}}\).
Remark 2.3
In most cases, this lemma allows that instead of proving an inequality for non-radial functions, it is enough to prove it for radial functions (see e.g. in the proof of Theorem 4.12 and [21, in the proof of Theorem 3.1]).
Proof
Using representation (2.8), we obtain
which is (2.9).
Now we prove (2.10). First let us prove the following by induction
for any \(k\in {\mathbb {N}}\). So, we need to check this for \(k=1\). Using Hölder’s inequality, we get
Now assuming that
holds for \(\ell \in {\mathbb {N}}\), we prove
Then, using (2.14) we calculate
where we have used Hölder’s inequality in the last line. Thus, we have proved (2.13).
Now using (2.13), we obtain
Thus, we have proved (2.10) for any \(k\in {\mathbb {N}}\). Similarly, one can obtain (2.11). \(\square\)
3 Euler semigroup \(e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}\)
In this section we introduce the operator semigroup \(\{e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}\}_{t>0}\) associated with the Euler operator on homogeneous groups.
First, let us prepare some preliminary results. We define the map \(F : L^{2}(\mathbb {G})\rightarrow L^{2}({\mathbb {R}}\times \mathfrak {S})\) as
for \(y \in \mathfrak {S}\) and \(s\in {\mathbb {R}}\), and its inverse \(F^{-1}: L^{2}({\mathbb {R}}\times \mathfrak {S})\rightarrow L^{2}(\mathbb {G})\) as
We note that F preserves the \(L^{2}\) norm.
We will also use the dilations \(U(t):L^{2}(\mathbb {G})\rightarrow L^{2}(\mathbb {G})\) given by
These form a group of unitary operators with generator \(U(t)=e^{iAt}\), where A is given by
where we have used that
as follows from the following equality with \(x=ry\) and \(\rho :=e^{t}r\),
where \(y\in \mathfrak {S}\). Using Lemma 2.1 we obtain from (3.4) that
which implies
By (3.1) and (3.3), we have \((Ff)(s,y)=(U(s)f)(y)\) for \(y\in \mathfrak {S}\) and \(s\in {\mathbb {R}}\), then this with the group property of the dilations \(U(\cdot )\) gives that
If we define \(M :L^{2}(\mathbb {G})\rightarrow L^{2}({\mathbb {R}}\times \mathfrak {S})\) as
then it follows from (3.7) and the change of variables that
We note that the map \(M={{\mathscr {F}}}\circ F\) is the Mellin transformation, where \({\mathscr {F}}\) is the Fourier transform on \({\mathbb {R}}\). The map M has an explicit representation using the group structure of \({\mathbb {R}}^{+}\) under multiplication: it is the Fourier transform on this group.
Now we are ready to give the representation of the operator semigroup \(\{e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}\}_{t>0}\) on homogeneous groups.
Theorem 3.1
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(x=ry\) and \(r=|x|\) with \(y\in \mathfrak {S}\). Then the semigroup \(e^{-t{\mathbb {E}}^{*}{\mathbb {E}}}\) is given by
Proof
Let us first show that
for f in the domain \({\mathscr {D}}(A)\), which implies the fact that \(f\in {\mathscr {D}}(A)\Leftrightarrow (\tau ,y)\mapsto \tau (Mf)(\tau ,y)\in L^{2}({\mathbb {R}}\times \mathfrak {S})\). Noting \(iAe^{itA}=\partial _{t}U(t)\) we calculate
Now using (3.9) we get from above that
which implies (3.11) after setting \(t=0\).
Now we prove that
To obtain (3.13), we write
On the other hand, by iteration we have from (3.11) that
Putting this in (3.14), we obtain
Thus, we have obtained (3.13). Then, it implies that
Here using \(M={\mathscr {F}}\circ F\), one has
Then, a direct calculation gives that
where we have used the property of the Gaussian integral in the last line
From this and (3.15), using (3.1), (3.2) and \(M={\mathscr {F}}\circ F\) with \(x=ry\) we compute
where we have used the change of variables \(z=e^{s}\) in the last line.
Since we have \(e^{-t\mathbb {E}^{*}\mathbb {E}}=e^{-tQ^{2}/4}e^{-tA^{2}}\) by (3.6), we obtain from above that
yielding (3.10). \(\square\)
Now let us give the following representation for \(e^{-tA^{2}}\), which is useful to obtain Gagliardo–Nirenberg type inequalities (see Sect. 4):
Corollary 3.2
Let F and \(F^{-1}\) be mappings as in (3.1) and (3.2), respectively. Then we have
Proof
Plugging \(e^{-tA^{2}}=e^{tQ^{2}/4}e^{-t\mathbb {E}^{*}\mathbb {E}}\), (3.1) and (3.2) into the left hand side of (3.16), and using (3.10) we obtain
which is (3.16), where we have used the change of variables \(s=e^{s_{1}}\) in the last line. \(\square\)
4 Hardy–Sobolev and Gagliardo–Nirenberg type inequalities
In this section we establish a class of the Hardy–Sobolev and Gagliardo–Nirenberg type inequalities on homogeneous groups. Moreover, sharp remainder terms of the Sobolev type inequality are also obtained.
We define the Besov type space \(B^{\alpha }({\mathbb {R}}\times \mathfrak {S})\) as the space of all tempered distributions f on \({\mathbb {R}}\times \mathfrak {S}\) with the norm
We will also use the one-dimensional case of the following result:
Theorem 4.1
([12, Theorem 1]) Let \(1\le p<q<\infty\). Then for every function \(f\in L^{p}_{1}({\mathbb {R}}^{n})\) there exists a positive constant \(C=C(p,q,n)\) such that
where
Now we state the Gagliardo–Nirenberg type inequalities:
Theorem 4.2
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(1\le p<q<\infty\) and let f be such that \(\mathscr {R}f\in L^{p}({\mathbb {R}}\times \mathfrak {S})\) and \(f\in B^{p/(p-q)}({\mathbb {R}}\times \mathfrak {S})\). Then there exists a positive constant \(C=C(p,q)\) such that
Proof
Using Theorem 4.1 with \(n=1\) and Corollary 3.2, we obtain
for any \(y\in \mathfrak {S}\), in view of (4.1). One obtains (4.3) after integrating the above inequality with respect to y over \(\mathfrak {S}\). \(\square\)
Once Theorems 3.1 and 4.2, and Corollary 3.2 are established, we obtain the following corollaries in exactly the same way as in [1, Section 4]:
Corollary 4.3
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(1\le p\le Q-1\) and \(\mathscr {R}f \in L^{p}({\mathbb {R}}\times \mathfrak {S})\). Let \(p^{*} :=Qp/(Q-p)\). Then we have
-
If \(\;\mathop {\sup }\limits_{{y \in {\mathfrak{S}}}}\Vert f(\cdot ,y)\Vert _{L^{p}({\mathbb {R}})}<\infty\), then there exists a positive constant C such that
$$\begin{aligned} \Vert f\Vert _{L^{p^{*}}({\mathbb {R}}\times \mathfrak {S})}\le C\Vert \mathscr {R}f\Vert ^{1/Q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})} \sup _{y \in \mathfrak {S}}\Vert f(\cdot ,y)\Vert _{L^{p}({\mathbb {R}})}^{(Q-1)/Q}. \end{aligned}$$(4.4) -
If \(f\in L^{p}({\mathbb {R}}\times \mathfrak {S})\) and \(g(r)={\mathscr {M}}(f)(r):=\frac{1}{|\mathfrak {S}|}\int _{\mathfrak {S}}f(r,y)d\sigma (y)\), then there exists a positive constant C such that
$$\begin{aligned} \Vert g\Vert _{L^{p^{*}}({\mathbb {R}})}\le C\Vert \mathscr {R}f\Vert ^{1/Q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})} \Vert f\Vert ^{(Q-1)/Q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})}. \end{aligned}$$(4.5)If f is supported in \([-\varLambda ,\varLambda ]\times \mathfrak {S}\), then there exist positive constants \(C_{1}\) and \(C_{2}\) such that
$$\begin{aligned} \Vert f\Vert _{L^{p^{*}}({\mathbb {R}}\times \mathfrak {S})}\le C_{1} \varLambda ^{(Q-1)/Q^{2}}\Vert \mathscr {R}f\Vert ^{1/Q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})} \sup _{y \in \mathfrak {S}}\Vert f(\cdot ,y)\Vert _{L^{p^{*}}({\mathbb {R}})}^{(Q-1)/Q} \end{aligned}$$(4.6)and
$$\begin{aligned} \Vert g\Vert _{L^{p^{*}}({\mathbb {R}})}\le C_{2}\varLambda ^{(Q-1)/Q}\Vert \mathscr {R}f\Vert _{L^{p}({\mathbb {R}}\times \mathfrak {S})}. \end{aligned}$$(4.7)
Corollary 4.4
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(1\le p<q<\infty\) and \(\mathscr {R}f \in L^{p}({\mathbb {R}}\times \mathfrak {S})\). Then we have
-
If \(\;\mathop {\sup }\limits_{{y \in {\mathfrak{S}}}}\Vert f(\cdot ,y)\Vert _{L^{(q/p)-1}({\mathbb {R}})}<\infty\), then there exists a positive constant C such that
$$\begin{aligned} \Vert f\Vert _{L^{q}({\mathbb {R}}\times \mathfrak {S})}\le C\Vert \mathscr {R}f\Vert ^{p/q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})} \sup _{y \in \mathfrak {S}}\Vert f(\cdot ,y)\Vert _{L^{(q/p)-1}({\mathbb {R}})}^{1-p/q}. \end{aligned}$$(4.8) -
If \(f\in L^{(q/p)-1}({\mathbb {R}}\times \mathfrak {S})\) and \(g:={\mathscr {M}}(f)\), then there exists a positive constant C such that
$$\begin{aligned} \Vert g\Vert _{L^{q}({\mathbb {R}})}\le C\Vert \mathscr {R}f\Vert ^{p/q}_{L^{p}({\mathbb {R}}\times \mathfrak {S})}\Vert f\Vert ^{1-p/q}_{L^{(q/p)-1} ({\mathbb {R}}\times \mathfrak {S})}. \end{aligned}$$(4.9)
Corollary 4.5
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension \(Q\ge 3\). Let \(|\cdot |\) be a homogeneous quasi-norm. Let \(\mathbb {E}h\in L^{2}(\mathbb {G})\) and \(2^{*}=2Q/(Q-2)\). Let \(d\mu =r^{Q-1}dr\). Then we have
-
If \(\;\mathop {\sup }\limits_{{y \in {\mathfrak{S}}}}\Vert h(\cdot , y)\Vert _{L^{2}({\mathbb {R}}^{+};d\mu )}<\infty\), then there exists a positive constant C such that
$$\begin{aligned} \begin{aligned} \Vert |x|h\Vert ^{2}_{L^{2^{*}}(\mathbb {G})}&\le C \left( \Vert \mathbb {E}h\Vert ^{2}_{L^{2}(\mathbb {G})} -\frac{Q^{2}}{4}\Vert h\Vert ^{2}_{L^{2}(\mathbb {G})}\right) ^{1/Q}\\&\quad \times \sup _{y\in \mathfrak {S}} \Vert h(\cdot ,y)\Vert ^{2(1-1/Q)}_{L^{2}({\mathbb {R}}^{+};d\mu )}. \end{aligned} \end{aligned}$$(4.10) -
If \(h\in L^{2}(\mathbb {G})\) and \(g:={\mathscr {M}}(h)\), then there exists a positive constant C such that
$$\begin{aligned} \Vert |x|g\Vert ^{2}_{L^{2^{*}}(\mathbb {G})} \le C \left( \Vert \mathbb {E}h\Vert ^{2}_{L^{2}(\mathbb {G})}-\frac{Q^{2}}{4}\Vert h\Vert ^{2}_{L^{2}(\mathbb {G})}\right) ^{1/Q} \Vert h\Vert ^{2(1-1/Q)}_{L^{2}(\mathbb {G})}. \end{aligned}$$(4.11)For \(0\le \delta <Q^{2}/4\), we have
$$\begin{aligned} \Vert |x|h\Vert ^{2}_{L^{2^{*}}(\mathbb {G})} \le C (Q^{2}/4-\delta )^{-(Q-1)/Q}\left( \Vert \mathbb {E}h\Vert ^{2}_{L^{2}(\mathbb {G})}-\delta \Vert h\Vert ^{2}_{L^{2}(\mathbb {G})}\right) . \end{aligned}$$(4.12)
Corollary 4.6
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension \(Q\ge 3\). Let \(|\cdot |\) be a homogeneous quasi-norm. Let \(f\in C_{0}^{\infty }(\mathbb {G}\backslash \{0\})\) and \(d\mu =r^{Q-1}dr\). Then we have
-
If \(\;\mathop {\sup }\limits_{{y \in {\mathfrak{S}}}}\Vert f(\cdot ,y)/ |\cdot |\Vert ^{2}_{L^{2}({\mathbb {R}}^{+};d\mu )}<\infty\), then there exists a positive constant C such that
$$\begin{aligned} \begin{aligned} \Vert f\Vert ^{2}_{L^{2^{*}}(\mathbb {G})}&\le C\left( \Vert \mathscr {R}f\Vert ^{2}_{L^{2}(\mathbb {G})}-\left( \frac{Q-2}{2}\right) ^{2}\left\| \frac{f}{|x|}\right\| ^{2}_{L^{2}(\mathbb {G})}\right) ^{1/Q}\\&\quad \times \sup _{y \in \mathfrak {S}}\left\{ \left\| \frac{f(|\cdot |,y)}{|\cdot |}\right\| ^{2}_{L^{2}({\mathbb {R}}^{+};d\mu )}\right\} ^{1-1/Q}. \end{aligned} \end{aligned}$$(4.13) -
If \(g :={\mathscr {M}}(f)\), then there exists a positive constant C such that
$$\begin{aligned} \Vert g\Vert ^{2}_{L^{2^{*}}(\mathbb {G})}\le C\left( \Vert \mathscr {R}f\Vert ^{2}_{L^{2}(\mathbb {G})}-\left( \frac{Q-2}{2}\right) ^{2}\left\| \frac{f}{|x|}\right\| ^{2}_{L^{2}(\mathbb {G})}\right) ^{1/Q} \left\| \frac{f}{|x|}\right\| ^{(2Q-2)/Q}_{L^{2}(\mathbb {G})}. \end{aligned}$$(4.14)For \(0 \le \delta <(Q-2)^{2}/4\), we have
$$\begin{aligned} \Vert g\Vert ^{2}_{L^{2^{*}}(\mathbb {G})}\le C\left( \frac{(Q-2)^{2}}{4}-\delta \right) ^{-(Q-1)/Q}\left( \Vert \mathscr {R}f\Vert ^{2}_{L^{2}(\mathbb {G})}-\delta \left\| \frac{f}{|x|}\right\| ^{2}_{L^{2}(\mathbb {G})}\right) . \end{aligned}$$(4.15)
Remark 4.7
In the Abelian case \({\mathbb {G}}=({\mathbb {R}}^{n},+)\) and \(Q=n\), replacing \(|\mathscr {R}f|\) by \(|\nabla f|\) in Corollary 4.6, we see that this corollary implies [1, Corollary 4.5], since \(|\mathscr {R}f|=|\partial _{r}f|\le |\nabla f|\).
Let us give the proof of Corollary 4.6.
Proof
If \(h\in C_{0}^{\infty }(\mathbb {G}\backslash \{0\})\), then using the integration by parts and polar coordinates on \(\mathbb {G}\) we obtain
If we put (4.16) and \(h=f/|x|\) in (4.10), (4.11) and (4.12), then they imply (4.13), (4.14) and (4.15), respectively. \(\square\)
Corollary 4.8
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(|\cdot |\) be a homogeneous quasi-norm. Let \(A_{R}:=\{x\in \mathbb {G}: 1/R\le |x|\le R\}\). Let \(f\in C^{\infty }_{0}(A_{R})\) and \(g :={\mathscr {M}}(h)\). Then there exists a positive constant C such that
By substituting (4.16) and \(h=f/|x|\) in (4.17), we obtain
Corollary 4.9
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(|\cdot |\) be a homogeneous quasi-norm. Let \(A_{R}:=\{x\in \mathbb {G}: 1/R\le |x|\le R\}\). Let \(f\in C^{\infty }_{0}(A_{R})\) and \(g :={\mathscr {M}}(f)\). Then there exists a positive constant C such that
Remark 4.10
In the Abelian case \({\mathbb {G}}=({\mathbb {R}}^{n},+)\) and \(Q=n\), as in Remark 4.7, replacing \(|\mathscr {R}f|\) by \(|\nabla f|\) in (4.18), we note that this corollary implies [1, Corollary 4.7].
Corollary 4.11
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension Q. Let \(|\cdot |\) be a homogeneous quasi-norm. Let \(2<q<\infty\). Let \(f, \mathbb {E}f \in L^{2}(\mathbb {G})\) and
Then there exists a positive constant C such that
where
Now we prove (4.12) with sharp constant, which can be viewed as an analogue of Stubbe’s inequality [23] on homogeneous groups.
Theorem 4.12
Let \({\mathbb {G}}\) be a homogeneous group of homogeneous dimension \(Q>2\). Let \(|\cdot |\) be a homogeneous quasi-norm. Then we have for all \(f\in C^{\infty }_{0}(\mathbb {G})\) and \(0\le \delta <\frac{Q^{2}}{4}\), the inequality
with sharp constant, where
and
Remark 4.13
In the Abelian case \({\mathbb {G}}=({\mathbb {R}}^{n},+)\) and \(Q=n\), the inequality (4.19) gives that
An interesting observation is that the constant in the above inequality is sharp for any quasi-norm \(|\cdot |\), that is, it does not depend on the quasi-norm \(|\cdot |\). Therefore, this inequality is new already in the Euclidean setting of \({\mathbb {R}}^{n}\). When \(\Vert x\Vert =\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}}\) is the Euclidean distance, the inequality (4.21) was investigated in \({\mathbb {R}}^{n}\) in [1, Corollary 4.4] and in [24, Theorem 1.1].
Before starting the proof of this result, let us recall the following result [3]:
Lemma 4.14
([3]) Let f be a non-negative function. Then for \(s\ge 0\) and \(q>p>1\) we have
where
is sharp. Moreover, the equality in (4.22) is attained for functions of the form
Let us now prove Theorem 4.12.
Proof
Let us first prove this theorem for radial functions \(f(x)={\widetilde{f}}(|x|)\). Then we have \(g(r) = {\widetilde{f}}(r)\) since \(g(|x|)={\mathscr {M}}(f)(|x|):=\frac{1}{|\mathfrak {S}|}\int _{\mathfrak {S}}f(|x|,y)d\sigma (y)\), and \(\mathbb {E}f = |x|{\widetilde{f}}'(|x|)\). By a direct calculation we have
where \(0\le \beta <Q/2\). If we set \(h(|x|):=|x|^{\beta }{\widetilde{f}}(|x|)\), then the integration by parts gives that
On the other hand, we have by changing the variables \(s=r^{Q-2\beta }\) that
Taking into account (4.25) and (4.26), we rewrite (4.24) as
Now denoting \(\phi (s):=h'(s)\) and \(\psi (s):=s^{-2}\phi (s^{-1})\), and using Lemma 4.14 with \(p=2\) and \(q=\frac{2Q}{Q-2}=:2^{*}\) we have
where we have used \(s=r^{Q-2\beta }\) and \(h(r)=r^{\beta }{\widetilde{f}}(r)\) in the last line. Combining this with (4.27), we arrive at
Here, if we set \(\beta =(Q-\sqrt{Q^{2}-4\delta })/2\) for \(0\le \delta <Q^{2}/4\), then recalling \(g(|x|)={\widetilde{f}}(|x|)\) we see that (4.28) implies that
since (4.20). Thus, we have obtained (4.19) with sharp constant for all radial functions \(f\in C^{\infty }_{0}(\mathbb {G})\).
Finally, using Lemma 2.2, and taking into account \(g(|x|)={\widetilde{f}}(|x|)\) in (4.29), we obtain (4.19) for non-radial functions. The constant in (4.19) is sharp, since this constant is sharp for radial functions by Lemma 4.14. \(\square\)
5 Maximal Hardy inequality
In this section we discuss a weighted exponential inequality.
Theorem 5.1
Let \(\phi\) and \(\psi\) be positive functions defined on \(\mathbb {G}\). Then there exists a positive constant C such that
holds for all positive f if and only if
where \(({\mathscr {M}}f)(x)=\frac{1}{|B(0,|x|)|}\int _{B(0,|x|)}f(z)dz\).
Remark 5.2
In the Abelian case \({\mathbb {G}}=({\mathbb {R}}^{n},+)\) and \(Q=n\), the inequality (5.1) was studied in [11] for \(n=1\), and in [7] for \(n\ge 1\).
Proof
First, we show (5.2)\(\Rightarrow\)(5.1). Denoting
and \(u(x):=f(x)\psi (x)\), and changing the variables \(z=|x|\xi\), we obtain
Now taking into account \(\int _{B(0,1)}\log (|\xi |^{Q})d\xi =Q\int _{\mathfrak {S}}\int _{0}^{1}r^{Q-1}\log rdrd\sigma (y) =-|B(0,1)|\) and using Jensen’s inequality, we obtain from (5.4) that
where \(|\xi |=s\) and \(|x|=r\). Here, we continue our calculation by changing the variables \(rs=t\) to get
yielding (5.1), where we have used (5.2) in the last line.
Now let us show (5.1)\(\Rightarrow\)(5.2). We note, using (5.4), that (5.1) is equivalent to
Here, choosing the following radial function for a fixed \(R>0\),
we compute
since \(\chi\) is the cut-off function. Now, taking into account this and plugging (5.8) into the left hand side of (5.7) we get
which implies (5.2), where we have used \(\frac{|\mathfrak {S}|}{|B(0,1)|}=Q\), \(\frac{2R^{Q}}{s^{Q}}-\frac{2R^{Q}}{Qs^{Q}}>0\) and (5.3) in the last two lines. \(\square\)
6 Further inequalities
In this section we discuss a number of related inequalities, also interesting on their own.
Theorem 6.1
For any quasi-norm \(|\cdot |\), all differentiable \(|\cdot |\)-radial functions \(\phi\), all \(p>1\), \(Q\ge 2\), and all \(f \in C_{0}^{1}({\mathbb {G}})\) we have
and
In (6.1) taking \(\phi =\log |x|\) in the Euclidean (Abelian) case \({{\mathbb {G}}}=({\mathbb {R}}^{n},+)\), \(n\ge 2\), we have \(Q=n\), and taking \(p=n\ge 2\), so for any quasi-norm \(|\cdot |\) on \({\mathbb {R}}^{n}\) it implies the new inequality:
which in turn, by using Schwarz’s inequality with the standard Euclidean distance \(\Vert x\Vert =\sqrt{x^{2}_{1}+\ldots +x^{2}_{n}}\), implies the ‘critical’ Hardy inequality
where \(\nabla\) is the standard gradient on \({\mathbb {R}}^{n}\). It is known (see, e.g. [2, Section 1.2.5]) that there is no positive constant C such that
for all \(f\in C^{1}_{0}({\mathbb {R}}^{n}).\) Therefore, the appearance of a positive additional term (the second term) on the right hand side of (6.3) seems essential. We refer to [18] and references therein for different versions of critical Hardy–Sobolev type inequalities. Note that this type of inequalities (Hardy–Sobolev type inequalities with an additional term on the right hand side) can be applied, for example in the Euclidean case, to establish the existence and nonexistence of positive exponentially bounded weak solutions to a parabolic type operator perturbed by a critical singular potential (see [22]).
In (6.2) taking \(\phi =|x|^{n}\) in the Euclidean (Abelian) case \({{\mathbb {G}}}=({\mathbb {R}}^{n},+)\), \(n\ge 2\), we have \(Q=n\), so for any quasi-norm \(|\cdot |\) on \({\mathbb {R}}^{n}\) it implies the following uncertainty principle:
which in turn, by using Schwarz’s inequality with the standard Euclidean distance \(\Vert x\Vert =\sqrt{x^{2}_{1}+\ldots +x^{2}_{n}}\), implies that
where \(\nabla\) is the standard gradient on \({\mathbb {R}}^{n}\). In the case when \(p=2\) we have
for all \(f\in C^{1}_{0}({\mathbb {R}}^{n}).\) The same inequality with the constant \(\left( \frac{2}{n-2}\right) ^{2}, n\ge 3,\) (instead of \(\left( \frac{2}{n}\right) ^{2}\)) is known as the Heisenberg-Pauli-Weyl uncertainty principle (see, e.g. [17, Remark 2.10]), that is,
for all \(f\in C^{1}_{0}({\mathbb {R}}^{n}).\) Thus, when \(n=2\) inequality (6.6) gives the critical case of the Heisenberg-Pauli-Weyl uncertainty principle. Moreover, since \(\frac{2}{n-2}\ge \frac{2}{n},\, n\ge 3,\) inequality (6.6) is an improved version of (6.7). Note that equality case in (6.6) holds for the function \(f=C\exp (-b \Vert x\Vert ),\,b>0.\)
Now we prove Theorem 6.1.
Proof
By applying the polarization formula and integration by parts we obtain
Furthermore, using the Young inequality for \(p>1\), we arrive at
This proves inequality (6.1). On the other hand, from (6.8) using the Hölder inequality for \(p>1\), we establish
This completes the proof. \(\square\)
Note that a similar proof technique gives
Theorem 6.2
For any quasi-norm \(|\cdot |\), all differentiable \(|\cdot |\)-radial functions \(\phi\), all \(p>1\), \(Q\ge 2\) with \(\frac{1}{p}+\frac{1}{q}=1\), and all \(f \in C_{0}^{1}({\mathbb {G}})\) we have
and
Usually, classical Hardy and Sobolev type inequalities are stated with a gradient. As a reader noticed througout this paper by using the Euler or/and Radial operators we have obtained different type of gradient free functional inequalities. On the one hand, such analysis is important since, as we mentioned in the introduction, in general there is no homogeneous gradient on homogeneous (Lie) groups. On the other hand, these inequalities give new inequalities even in Euclidean cases as well as cover classical inequalities with gradients. However, in addition to these methods there are other techniques to obtain gradient free Hardy–Sobolev type inequalities. To conclude these discussions let us introduce the following functional
where \(\circ\) is the group operation on \({\mathbb {G}}\).
In the Euclidean (Abelian) case by using this functional, H.-M. Nguyen and M. Squassina (see, e.g. [13] and [14]) obtained nonlocal verions of the classical Hardy–Sobolev type inequality. Note that from their inequalities in the singular limit \(\delta \searrow 0\) one recovers the classical Hardy–Sobolev type results since in the Euclidean case the functional \(I_{\delta }\) converges to the Dirichlet energy up to a normalisation constant. Their main inequalities are gradient free ones. Therefore, those are extendable to the homogeneous (Lie) groups. We believe that such ideas of proofs of (nonlocal) gradient free inequalities can be generalised to the homogeneous groups. Below we demonstrate this idea in a special case.
Proposition 6.3
Let \(Q\ge 3\), \(I_{\delta }(f)<\infty\) and
for some positive constants \(C_{Q}\) and \(\lambda _{Q}\). Then for all \(\delta >0\) there exists a positive constant \(C_{Q}\) such that
Proof
The proof of Proposition 6.3 relies on the same technique as the proof of [14, Theorem 1.1] with the difference that now the quasi-norm (instead of the Euclidean distance) is used. For the proof we only need to recall the fact that the Lebesque measure on \({\mathbb {R}}^{n}\) gives the Haar measure for \({\mathbb {G}}\). The rest of the proof is exactly the same as in the proof of [14, Theorem 1.1]. \(\square\)
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Acknowledgements
The first author was supported in parts by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations, the EPSRC Grant EP/R003025/1 and by the Leverhulme Grant RPG-2017-151. The second author was supported by the Nazarbayev University program 091019CRP2120 and the Nazarbayev University Grant 240919FD3901. The third author was supported by the FWO Odysseus 1 Grant G.0H94.18N: Analysis and Partial Differential Equations, by the LMS Early Career Fellowship Grant ECF-1819-12 and by the MESRK Grant AP05133271.
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Ruzhansky, M., Suragan, D. & Yessirkegenov, N. Euler semigroup, Hardy–Sobolev and Gagliardo–Nirenberg type inequalities on homogeneous groups. Semigroup Forum 101, 162–191 (2020). https://doi.org/10.1007/s00233-020-10110-9
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DOI: https://doi.org/10.1007/s00233-020-10110-9