In this section, we derive three results as applications of Theorem A: If the compact set K is specified in detail, we can reduce \(K_{(\delta )}\) into \(K_{\delta }\), as appears in Theorem A. Let \(\alpha \in \mathbb{Z}_{\ast }^{n}\) and let \(0\leq \lambda _{\alpha } \leq \infty \). We define the set generated by the number sequence \(\{ \lambda _{\alpha }\}\) as \(G\{\lambda _{\alpha }\}\) consisting of all points \(\xi \in \mathbb{R}^{n}\) such that
$$ \bigl\vert \xi ^{\alpha } \bigr\vert \leq \lambda _{\alpha } \quad \text{for all } \alpha \in \mathbb{Z}_{\ast }^{n}. $$
Now for any set \(E\subset \mathbb{R}^{n}\), put the g-hull of E by
$$ g(E)= G\Bigl\{ \sup_{E} \bigl\vert \xi ^{\alpha } \bigr\vert \Bigr\} . $$
Then \(E\subset g(E)\) readily. We say that E satisfies the g-property if \(E=g(E)\). We note two facts that every set generated by a number sequence \(G\{\lambda _{\alpha }\}\) has the g-property and vice versa and every symmetric compact convex set also has the g-property. For more information, refer to [4, 5].
Theorem 5.1
Let
\(0< p<1\)
and let
\(K\subset \mathbb{R}^{n}\)
be compact satisfying the
g-property. Then
\(\operatorname{supp}\hat{f} \subset K\)
if and only if for any
\(\delta >0\)
there exists a constant
\(C_{p,K,\delta }\)
independent of
f, α
such that
$$\begin{aligned} \bigl\Vert D^{\alpha }f \bigr\Vert _{p} \leq C_{p,K,\delta } \Bigl(\sup_{x\in K_{\delta }} \bigl\vert x^{\alpha } \bigr\vert \Bigr) \Vert f \Vert _{p} \end{aligned}$$
(29)
for any
\(\alpha \in \mathbb{Z}_{\ast }^{n}\).
For any polynomial P and for \(r>0\), define an r-neighborhood with respect to P by
$$ N_{P}(r)=\bigl\{ x \in \mathbb{R}^{n}: \bigl\vert P(x ) \bigr\vert \leq r\bigr\} . $$
Theorem 5.2
Let
\(0< p <1\)
and let
\(K= N_{P}(r)\). Then
\(\operatorname{supp}\hat{f} \subset K\)
if and only if for any
\(\delta >0\)
there exists a constant
\(C_{p,K,\delta }\)
independent of
f, m
such that
$$\begin{aligned} \bigl\Vert P^{m}(D)f \bigr\Vert _{p} \leq C_{p,K,\delta } (r+ \delta )^{m} \Vert f \Vert _{p} \end{aligned}$$
(30)
for any
\(m\in \mathbb{Z}_{\ast }\).
Theorem 5.3
Let
\(0< p <1\)
and let
\(K\subset \mathbb{R}^{n}\)
be convex and compact. Then
\(\operatorname{supp}\hat{f} \subset K\)
if and only if for any
\(\delta >0\)
there exists a constant
\(C_{p,K,\delta }\)
independent of
f, P, m
such that
$$\begin{aligned} \bigl\Vert P^{m}(D)f \bigr\Vert _{p} \leq C_{p,K,\delta } \Bigl(\sup_{x\in K_{ \delta }} \bigl\vert P(x) \bigr\vert ^{m} \Bigr) \Vert f \Vert _{p} \end{aligned}$$
(31)
for any real polynomial
P
of degree 1 and for any
\(m\in \mathbb{Z}_{\ast }\).
To prove Theorem 5.1, we need a lemma: The inclusion of \(K_{\delta } \subset K_{(\delta )}\), implies
$$ \sup_{K_{(\delta )}} \bigl\vert z^{\alpha } \bigr\vert \ge \sup_{K_{\delta }} \bigl\vert x^{\alpha } \bigr\vert \quad \text{for all } \alpha \in \mathbb{Z}_{\ast }^{n}. $$
Moreover, for \(z\in K_{(\delta )}\) there exist \(\xi \in K\) and \(\eta \in \mathbb{C}^{n}\) such that \(z= \xi + \eta \), \(|\eta |\le \delta \) and so \(|z_{j}|\le |\xi _{j}|+|\eta _{j}|\)
\((1\le j\le n)\). Put \(x=(\xi _{1} + |\eta _{1}| \operatorname{sign} (\xi _{1}), \dots , \xi _{n} + | \eta _{n}| \operatorname{sign} (\xi _{n}))\). Clearly, \(x \in K_{\delta }\) and \(|x_{j}|= |\xi _{j}| + |\eta _{j}| \geq |z_{j}|\) for all \(1\le j\le n\). Thus for each \(z\in K_{(\delta )}\), there exists \(x\in K_{\delta }\) such that
$$ \bigl\vert z^{\alpha } \bigr\vert \le \bigl\vert x^{\alpha } \bigr\vert \quad \text{for all } \alpha \in \mathbb{Z}_{\ast }^{n}. $$
Therefore, we conclude the following.
Lemma 5.4
If
K
is compact on
\(\mathbb{R}^{n}\), then, for any
\(\delta >0\),
$$ \sup_{K_{(\delta )}} \bigl\vert z^{\alpha } \bigr\vert = \sup_{K_{\delta }} \bigl\vert x^{\alpha } \bigr\vert \quad \textit{for all } \alpha \in \mathbb{Z}_{\ast }^{n}. $$
Proof of Theorem 5.1
Fix \(\delta >0\). By Theorem A, there exists a constant \(C_{p,K,\delta } <\infty \) such that
$$ \begin{aligned}[b] \bigl\Vert D^{\alpha }f \bigr\Vert _{p} &\leq C_{p,K,\delta } \sup_{z\in K_{(\delta )}} \bigl\vert z ^{\alpha } \bigr\vert \Vert f \Vert _{p} \\ &\leq C_{p,K,\delta } \sup_{x\in K_{\delta }} \bigl\vert x^{\alpha } \bigr\vert \Vert f \Vert _{p} \end{aligned} $$
(32)
for all \(\alpha \in \mathbb{Z}_{\ast }^{n}\), where the second inequality follows from Lemma 5.4. This proves the necessity.
To see the sufficiency, on the contrary, assume that there exists \(x_{0} \in \operatorname{supp}\hat{f}\) with \(x_{0} \notin K\). Since K has the g-property, we find \(\alpha \in \mathbb{Z}_{\ast }^{n}\) such that
$$ \bigl\vert x_{0}^{\alpha } \bigr\vert \gneq \sup_{x\in K} \bigl\vert x^{\alpha } \bigr\vert . $$
(33)
By the hypothesis of (29) with mα
\((m=1,2,\ldots )\),
$$ \bigl\Vert D^{m\alpha }f \bigr\Vert _{p} \leq C_{p,K,\delta } \Bigl(\sup_{x\in K_{\delta }} \bigl\vert x^{m\alpha } \bigr\vert \Bigr) \Vert f \Vert _{p}. $$
According to Proposition 2.1, applying limsup, we have
$$ \limsup_{m\to \infty } \bigl\Vert D^{m\alpha }f \bigr\Vert _{1}^{1/m}\le \sup_{x\in K_{\delta }} \bigl\vert x^{\alpha } \bigr\vert , $$
and taking \(\delta \searrow 0\), we get
$$ \limsup_{m\to \infty } \bigl\Vert D^{m\alpha }f \bigr\Vert _{p}^{1/m}\le \sup _{x\in K} \bigl\vert x ^{\alpha } \bigr\vert . $$
(34)
Also, by Lemma 2.2 with \(P(x)=x^{\alpha }\),
$$\begin{aligned} \liminf_{m\to \infty } \bigl\Vert D^{m\alpha } f \bigr\Vert _{p}^{1/m}\ge \bigl\vert x_{0} ^{\alpha } \bigr\vert . \end{aligned}$$
(35)
Thus, the equations of (34) and (35) yield
$$ \bigl\vert x_{0}^{\alpha } \bigr\vert \le \sup _{x\in K} \bigl\vert x^{\alpha } \bigr\vert . $$
This contradicts (33). Therefore, the proof is complete. □
Since any symmetric convex compact set satisfies the g-property ([4, 5]), Theorem 5.1 produces the corollary:
Corollary 5.5
Assume
K
is a symmetric convex compact set in
\(\mathbb{R}^{n}\), \(0< p <1\). Then, for any
\(\delta >0\), there exists a constant
\(C_{p,K,\delta } < \infty \)
such that
$$ \bigl\Vert D^{\alpha }f \bigr\Vert _{p} \leq C_{p,K,\delta } \Bigl(\sup_{x\in K_{\delta }} \bigl\vert x ^{\alpha } \bigr\vert \Bigr) \Vert f \Vert _{p} $$
for all
\(\alpha \in \mathbb{Z}_{\ast }^{n}\).
Let us note that a symmetric convex compact set is a typical example for a compact set that has the g-property. Since \(D^{\alpha }\) is simpler than \(P(D)\), in Corollary 5.5 the supremum runs over \(K_{\delta }\) instead of \(K_{(\delta )}\).
Proof of Theorem 5.2
We first prove the necessity. For any \(\delta >0\), by continuity, there is \(\delta '>0\) so that
$$ \sup_{K_{(\delta ')}} \bigl\vert P(z) \bigr\vert \le r+\delta , $$
since \(\sup_{K}|P(x)|= r\). By Theorem A, there exists a constant \(C_{p,K,\delta '}\) such that
$$ \begin{aligned} \bigl\Vert P^{m}(D)f \bigr\Vert _{p} &\le C_{p,K,\delta '} \Bigl(\sup_{K_{(\delta ')}} \bigl\vert P ^{m}(z) \bigr\vert \Bigr) \Vert f \Vert _{p} \\ &\le C_{p,K,\delta '}(r+\delta )^{m} \Vert f \Vert _{p}. \end{aligned} $$
To prove the sufficiency, suppose that there exists \(x_{0} \in \operatorname{supp}\hat{f}\) and \(x_{0} \notin K=N_{P}(r)\). Then \(|P(x_{0})|>r\). From hypothesis (30) and by Proposition 2.1,
$$ \limsup_{m\to \infty } \bigl\Vert P^{m}(D)f \bigr\Vert _{1}^{1/m}\le r+\delta . $$
(36)
Also, by Lemma 2.2,
$$ \liminf_{m\to \infty } \bigl\Vert P^{m}(D)f \bigr\Vert _{p}^{1/m}\ge \bigl\vert P(x_{0}) \bigr\vert . $$
(37)
Hence, by (36), by (37), and by assumption,
$$ r< \bigl\vert P(x_{0}) \bigr\vert \le r+\delta $$
for any \(\delta >0\). Letting \(\delta \searrow 0\), we reach a contradiction of the inequality. This gives the proof. □
By Theorems 5.1 and 5.2, we have a corollary.
Corollary 5.6
Let
\(r>0\)
and let
\(P_{j}\)
be polynomial
\((1\le j\le q)\). Put
\(K= H\bigcap_{j=1}^{q} N_{P_{j}}(r)\), where
H, \(N_{P_{j}}(r)\)
are compact. Then
\(\operatorname{supp}\hat{f}\subset K\)
if and only if for any
\(\delta >0\)
there exists a constant
\(C_{p,K,\delta }\)
such that
$$ \Biggl\Vert D^{\alpha }\prod_{j=1}^{q}P_{j}^{m_{j}}(D)f \Biggr\Vert _{p} \le C_{p,K,\delta }(r+\delta )^{\sum _{j=1}^{q}m_{j}} \Bigl( \sup_{z\in K_{(\delta )}} \bigl\vert z^{\alpha } \bigr\vert \Bigr) \Vert f \Vert _{p} $$
for all
\(m_{j}\in \mathbb{Z}_{\ast }\)
\((1\le j\le q)\), for all
\(\alpha \in \mathbb{Z}_{\ast }^{n}\).
We are ready to derive Theorem 5.3 with a lemma.
Lemma 5.7
Let
P
be a real polynomial of degree 1. If
\(E\subset \mathbb{R}^{n}\)
is any set, then
$$ \sup_{z\in E_{(\delta )}} \bigl\vert P(z) \bigr\vert = \sup _{x\in E_{\delta }} \bigl\vert P(x) \bigr\vert . $$
Proof
From \(E_{\delta }\subset E_{(\delta )}\), we have \(\sup_{z\in E_{(\delta )}}|P(z)|\ge \sup_{x\in E_{\delta }}|P(x)|\). To complete the proof, we need prove that \(\sup_{z\in E_{(\delta )}}|P(z)| \le \sup_{x\in E_{\delta }}|P(x)|\). Indeed, let \(z\in E_{(\delta )}\). Then there are \(x\in E\) and \(r\eta \in \mathbb{C}^{n}\)
\((0\le r\le \delta , |\eta |=1)\) such that \(z=x+r\eta \). Since we can replace P with −P, we may assume that \(P(x)\ge 0\). Taking \(y=x+r( \operatorname{sign} (\partial _{x_{1}}P)|\eta _{1}|,\ldots ,\operatorname{sign} ( \partial _{x_{n}}P)|\eta _{n}|)\in E_{r}\subset E_{\delta }\) so that
$$ \bigl\vert P(y) \bigr\vert = P(x) + r\sum_{j=1}^{n} \vert \partial _{x_{j}} P \vert \vert \eta _{j} \vert . $$
By the triangle inequality,
$$ \bigl\vert P(z) \bigr\vert \le P(x) + r\sum_{j=1}^{n} \vert \partial _{x_{j}} P \vert \vert \eta _{j} \vert . $$
The previous two inequalities show that for each \(z\in E_{(\delta )}\), there is \(y\in E_{\delta }\) such that \(|P(z)|\le |P(y)|\). Therefore, taking supremums successively, we justify the lemma. □
Proof of Theorem 5.3
The necessity follows readily. Indeed, let \(\delta >0\). By Theorem A, for some \(C_{p,K,\delta }\), we have
$$ \begin{aligned} \bigl\Vert P^{m}(D)f \bigr\Vert _{p} &\le C_{p,K,\delta } \Bigl(\sup_{z\in K_{(\delta )}} \bigl\vert P(z) \bigr\vert ^{m} \Bigr) \Vert f \Vert _{p} \\ &\le C_{p,K,\delta } \Bigl(\sup_{z\in K_{\delta }} \bigl\vert P(z) \bigr\vert ^{m} \Bigr) \Vert f \Vert _{p} \end{aligned} $$
for a real polynomial P of degree 1 and for all \(m\in \mathbb{Z} _{\ast }\), where the second inequality comes from Lemma 5.7.
It remains to prove the sufficiency. On the contrary, assume that there exists \(x_{0}\in H\) with \(x_{0} \notin K\), where \(H= \operatorname{supp}\hat{f}\). Since K is convex and compact, we easily find a linear P such that
$$ \bigl\vert P(x_{0}) \bigr\vert > \sup _{x\in K} \bigl\vert P(x) \bigr\vert . $$
(38)
By (31) and Proposition 2.1,
$$ \bigl\Vert P^{m}(D)f \bigr\Vert _{1} \le C_{p,H} C_{p,K,\delta } \Bigl(\sup_{x\in K_{ \delta }} \bigl\vert P^{m}(x) \bigr\vert \Bigr) \Vert f \Vert _{p} $$
for all \(m\in \mathbb{Z}_{\ast }\). So,
$$ \limsup_{m\to \infty } \bigl\Vert P^{m}(D)f \bigr\Vert _{1}^{1/m} \le \sup _{x\in K_{ \delta }} \bigl\vert P(x) \bigr\vert . $$
(39)
By (39) and by Lemma 2.2,
$$ \bigl\vert P(x_{0}) \bigr\vert \le \sup_{x\in K_{\delta }} \bigl\vert P(x) \bigr\vert , $$
and, putting \(\delta \searrow 0\), we get
$$ \bigl\vert P(x_{0}) \bigr\vert \le \sup_{x\in K} \bigl\vert P(x) \bigr\vert . $$
This contradicts (38), therefore, the proof is complete. □
From Theorems 5.1 and 5.3, we have the following.
Corollary 5.8
Let
K
be convex compact in
\(\mathbb{R}^{n}\), \(0< p <1\). Then, for every
\(\delta >0\), there exists a constant
\(C_{p,K,\delta }\)
independent of
f, P, m
such that
$$\begin{aligned} \bigl\Vert P^{m}(D) f \bigr\Vert _{p} \le C_{p,K,\delta } \Bigl(\sup_{x\in K_{\delta }} \bigl\vert P(x) \bigr\vert ^{m} \Bigr) \Vert f \Vert _{p} \end{aligned}$$
for all
\(P(x)\)
having degree 1 and for all
\(m\in \mathbb{Z}_{\ast }\).
Corollary 5.9
Let
\(0< p<1\). Suppose that
\(K_{1}\)
is convex and compact and that
\(K_{2}\)
compact satisfying the
g-property. Then
\(\operatorname{supp}\hat{f} \subset K_{1}\cap K_{2}=K\), say, if and only if for any
\(\delta >0\)
there exists a constant
\(C_{p,K,\delta }\)
such that
$$ \bigl\Vert P^{m}(D)D^{\alpha }f \bigr\Vert _{p} \le C_{p,K,\delta } \Bigl( \sup_{x\in K_{\delta }} \bigl\vert P^{m}(x) \bigr\vert \Bigr) \Vert f \Vert _{p} $$
for any real polynomials
P
of degree 1 and for any
\(m\in \mathbb{Z} _{\ast }\).
Remark 3
According to the Nikolskii inequality all \(L^{p}\)–\(L^{p}\) inequalities for differential operators in this paper can be extended to the \(L^{p}\)–\(L^{q}\) inequalities for \(0< p< q<\infty \).