Abstract
In this paper, the modular-type operator norm of the general geometric mean operator over spherical cones is investigated. We give two applications of a new limit process, introduced by the present authors, to the establishment of Pólya-Knopp-type inequalities. We not only partially generalize the sufficient parts of Persson-Stepanov’s and Wedestig’s results, but we also provide new proofs to these results.
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1 Introduction
Let E be a spherical cone in \(\Bbb{R}^{n}\). By this, we mean that \(E=\bigcup_{s>0} sA \) for some Borel measurable subset A of the unit sphere \(\Sigma^{n-1}\). Let \(\|\Bbb{K}\|_{D_{\Bbb{K}}\cap L_{\Phi}^{p}(v\,dx)\mapsto L_{\Phi}^{q}(u\,dx)}\) (in brief, \(\|\Bbb{K}\|_{*}\)) denote the smallest constant C in (1.1):
for all \(f\in D_{\Bbb{K}}\cap L_{\Phi}^{p}(v\,dx)\), where \(p, q>0\), \(u(x)\ge0\), \(v(x)>0\), \(\Phi\in CV^{+}(I)\), \(\Phi\circ f(x)=\Phi(f(x))\), and \(\Bbb{K}f(x)\) is of the form
Here \(CV^{+}(I)\) denotes the set of all nonnegative convex functions defined on an open interval I in \(\Bbb{R}\), \(D_{\Bbb{K}}\) is the space of those f such that \(\Bbb{K}f(x)\) is well defined for almost all \(x\in E\), and \(L_{\Phi}^{p}(v\,dx)\) is the set of all real-valued Borel measurable f with
Moreover, \(\tilde{S}_{x}=\bigcup_{0< s\le\|x\|} sA\), \(S_{x}=\tilde{S}_{x}\setminus\| x\|A\), and \(k(x,t)\ge0\) is locally integrable over \(\Bbb{E}\times\Bbb{E}\).
We write \(L^{p}(v\,dx)\) and \(\|f\|_{p,v}\) instead of \(L^{p}_{\Phi}(v\,dx)\) and \(\| f\|_{\Phi,p,v}\), respectively, for the case \(\Phi(s)=|s|\). We also write \(L^{p}(E,v\,dx)\) for \(L^{p}(v\,dx)\), whenever the integral region E is emphasized.
Clearly,
where the supremum is taken over all \(f\in D_{\Bbb{K}}\cap L_{\Phi}^{p}(v\,dx)\) with \(\|\Phi\circ f\|_{p,v}\neq0\). This number reduces to the operator norm of \(\Bbb{K}\) for the case \(\Phi(s)=|s|\). The investigation of the value \(\|\Bbb{K}\|_{*}\) has a long history in the literature. In [1], the present authors introduced a generalized Muckenhoupt constant \(A_{M}(p,q)\) and established the following Muckenhoupt-type estimate for \(\|\Bbb{K}\|_{*}\):
where \(1\le p, q\le\infty\), \(\eta=\max(p,q)\), and \((\cdot)^{*}\) is the conjugate exponent of \((\cdot)\) in the sense that \(1/(\cdot)+1/(\cdot)^{*}=1\). For the particular case that
there are two other types of estimates. They are
and
These two inequalities were proved in [2] and [3], Theorem 3.1 and Lemma 7.4, for the case \(1< p\le q<\infty\) (see also [4], Theorem 2.1). We refer the readers to Section 2 for details.
In this paper, we focus on the evaluation of \(\|\Bbb{K}\|_{*}\) for the following case of (1.1):
where \(f(t)>0\), \(g(t)>0\), and
The corresponding inequality to (1.1) takes the form
which is known as the Pólya-Knopp-type inequality.
In [4], Theorem 3.1, [2, 5], and [3], Theorem 7.3, the particular case \(g(t)=1\) of (1.7) was considered. They obtained the following estimates by means of the formula \((G_{\Bbb{K}}f)(x)=\lim_{\epsilon\to0^{+}} [\Bbb{K}(f^{\epsilon})]^{1/\epsilon}(x)\):
where \(0< p\le q<\infty\). The definitions of \(D^{*}_{PS}\) and \(D^{*}_{OG}(s)\) are given in Section 3.
The purpose of this paper is two-fold. We not only extend the aforementioned sufficient parts of [2, 4, 5], and [3] from \(u(x)>0\) and \(g(t)=1\) to \(u(x)\ge0\) and
but we also provide a new proof of (1.8) from the viewpoint of (1.10):
where \(0< p,q<\infty\), \(\frak{F}_{\Phi}^{+}=\{\epsilon>0: \Phi^{\epsilon}\in CV^{+}(I)\}\), and \(A_{p,q}\) are absolute constants subject to the condition
It is clear that (1.10) is applicable to the case \(\Phi(s)=e^{s}\). In this case, \(\frak{F}_{\Phi}^{+}=\{\epsilon>0\}\) and the second inequality in (1.10) holds. We remark that it may not be an equality (cf. [6]). On the other hand, we have \(p/\epsilon\to\infty\) and \(q/\epsilon\to\infty\) as \(\epsilon\to0^{+}\). This indicates that the infimum in (1.10) can be estimated by evaluating those \(A_{p,q}\) with p, q large enough.
The limit process (1.10) differs from the scheme by means of the formula \((G_{\Bbb{K}}f)(x)=\lim_{\epsilon\to0^{+}} [\Bbb{K}(f^{\epsilon})]^{1/\epsilon}(x)\). It was introduced in [6] to get different types of Pólya-Knopp inequalities, including the n-dimensional extensions of the Levin-Cochran-Lee-type inequalities and Carleson’s result. We showed that the infimum in (1.10) can easily be evaluated by applying the following choice of \(A_{p,q}\) for \(1< p,q<\infty\):
This choice is due to (1.3). We also pointed out that for some cases, the values of \(\|\Bbb{K}\|_{*}\) obtained from (1.10) are better than the known constants in the literature. In this paper, we consider two other choices of \(A_{p,q}\) with \(1< p\le q<\infty\), that is, \(A_{p,q}\le p^{*}\tilde{A}_{PS}(p,q)\) and \(A_{p,q}\le\tilde{A}_{W}(p,q)\), which are general forms of (1.5) and (1.5a). We shall derive them from (1.5) and (1.5a) and relax the conditions on \(u(x)\) and \(g(t)\) from \(u(x)>0\) and \(g(t)=1\) to \(u(x)\ge0\) and \(g(t)>0\) (cf. Section 2). Based on such choices, we prove that (1.8) follows from (1.10). Moreover, (1.8) can be extended from \(u(x)>0\) and \(g(t)=1\) to \(u(x)\ge0\) and \(g(t)\) of the form (1.9). This extension gives Persson-Stepanov-type and Opic-Gurka-tpye estimates of the modular-type operator norm of the general geometric mean operator corresponding to \(g(t)\). We remark that the particular case \(g(t)=|{\tilde{S}}_{t}|^{s-1}\) can lead us to the Levin-Cochran-Lee-type inequality (see Section 3 for details).
2 General forms of (1.5) and (1.5a)
Let \(1< p\le q<\infty\), \(g(t)>0\), \(u(x)\ge0\), and \(v(x)>0\). Consider the inequality:
where \(G(x)\) is defined by (1.6). This corresponds to the case \(\Phi(s)=|s|\) and \(k(x,t)=g(t)/G(x)\) of (1.1). Inequality (2.1) reduces to the form (2.2) for the case \(g(t)=1\):
where \(\tilde{u}(x)=u(x)/G(x)^{q}\). In [4], Theorem 2.1, [2] and [3], Lemma 7.4(a), it was proved that under the conditions \(u(x)>0\) and \(A_{PS}(p,q)<\infty\), (1.5) holds, in other words, (2.2) with \(\tilde{u}(x)\) replaced by \(u(x)\) is true for \(C=p^{*}A_{PS}(p,q)\), where
This result will be extended below from \(g(t)=1\) and \(u(x)>0\) to \(g(t)>0\) and \(u(x)\ge0\). We shall see its application in the proof of Theorem 3.2.
Theorem 2.1
Let \(1< p\le q<\infty\), \(u(x)\ge0\), \(v(x)>0\), \(g(t)>0\), and \(0< G(x)<\infty\), where \(G(x)\) is defined by (1.6). If \(\tilde{A}_{PS}(p,q)<\infty\), then (2.1) holds for \(C\le p^{*}\tilde{A}_{PS}(p,q)\), where
Proof
The case \(u(x)>0\) follows from [4], Theorem 2.1, or [3], Lemma 7.4(a), under the following substitutions:
As for \(u(x)\ge0\), let \(u_{\tau}(x)=u(x)+\rho_{\tau}(x)\), where \(0<\tau<1\) and \(\rho_{\tau}(x)>0\) is subject to the condition
Such \(\rho_{\tau}(x)\) exists. We have \(u_{\tau}(x)>0\) on E. Moreover, the condition \(1/q<1\) implies that \((a+b)^{1/q}\le a^{1/q}+b^{1/q}\) for all \(a,b\ge0\). Putting this together with (2.4) yields
This leads us to
where \(\tilde{A}_{PS}(p,q,\tau)\) is the number obtained from \(\tilde{A}_{PS}(p,q)\) by replacing \(u(t)\) by \(u_{r}(t)\). We have \(u_{\tau}(x)>u(x)\) on E. By the result of the case \(u(x)>0\), the following inequality holds for \(f\ge0\):
It follows from (2.5) that \(\liminf_{\tau\to0^{+}} \tilde{A}_{PS}(p,q,\tau)\le\tilde{A}_{PS}(p,q)\). Putting this together with (2.6) yields the desired inequality. The proof is complete. □
Next, consider (1.5a). The number \(A_{W}(s,p,q)\) in (1.5a) is defined by the formula:
In [3], Lemma 7.4(b), \(A_{W}(s,p,q)\) is replaced by another notation \(A^{*}_{W}(s)\). Like (1.5), (1.5a) can be generalized in the following way, in which \(g(t)=1\) and \(u(x)>0\) are relaxed to \(g(t)>0\) and \(u(x)\ge0\). We shall see its application in the proof of Theorem 3.3.
Theorem 2.2
Let \(1< p\le q<\infty\), \(u(x)\ge0\), \(v(x)>0\), \(g(t)>0\), and \(0< G(x)<\infty\), where \(G(x)\) is defined by (1.6). If \(\tilde{A}_{W}(s,p,q)<\infty\) for some \(1< s< p\), then (2.1) holds for \(C\le\tilde{A}_{W}(p,q)\), where
and
Proof
The case \(u(x)>0\) follows from [3], Lemma 7.4(b), under the substitutions (2.3). For the case \(u(x)\ge0\), we modify the proof of Theorem 2.1 in the following way. Let \(1< s< p\) and \(0<\tau<1\). Set \(u_{\tau}(x,s)=u(x)+\rho_{\tau}(x,s)\), where \(\rho_{\tau}(x,s)>0\) and satisfies the condition
Such \(\rho_{\tau}(x,s)\) exists. We have \(u_{\tau}(x,s)>0\) on \(x\in E\). Moreover,
where \(\tilde{A}^{\tau}_{W}(s,p,q)\) is obtained from \(\tilde{A}_{W}(s,p,q)\) by making the change in (2.8): \(u(t)\longrightarrow u_{\tau}(t,s)\). Obviously, \(u_{\tau}(x,s)>u(x)\). Applying the preceding result of the case \(u(x)>0\) to \(u_{\tau}(x,s)\), we get
Taking ‘\(\inf_{1< s< p}\)’ for both sides of (2.10), we get
Here
From (2.9), we obtain \(\tilde{A}^{\tau}_{W}(p,q)\le \tilde{A}_{W}(p,q)+\tau^{1/q}\). Taking \(\tau\to0^{+}\) for both sides of (2.11), we get the desired inequality. This completes the proof. □
3 Extensions and new proofs of (1.8)
To derive the extensions of (1.8), we need the following lemma.
Lemma 3.1
Let \(0< p<\infty\), \(v(x)>0\), \(g(t)>0\), and \(0< G(x)<\infty\), where \(G(x)\) is defined by (1.6). If \(\sup_{x\in E} \{g(x)/v(x)\}<\infty\), then, for all \(t\in E\),
Proof
Let \(\alpha\ge\sup_{x\in E} \{g(x)/v(x)\}\). Without loss of generality, we may assume \(\alpha>1\). We first consider the case that \(\int_{\tilde{S}_{t}}g(y) |\log (\frac{g(y)}{v(y)} ) |\,dy<\infty\). Let
We have
so \(h(\epsilon)\) is well defined and has a finite value. For \(\epsilon \in[0,p/2)\) and \(0<\tau<\min(p/2-\epsilon,\epsilon)\), it follows from the mean value theorem that
where \(\epsilon_{0}:=\epsilon_{0}(y)\) lies between ϵ and \(\epsilon+\tau\). We know that
By (3.2) and the Lebesgue dominated convergence theorem, h is differentiable on \([0, p/2)\). In addition,
Thus,
We get the desired result for the case \(\int_{\tilde{S}_{t}} g(y) |\log (\frac{g(y)}{v(y)} ) |\,dy<\infty\). Next, consider the case \(\int_{\tilde{S}_{t}} g(y) |\log (\frac {g(y)}{v(y)} ) |\,dy=\infty\). This implies
where \(\Omega_{1}=\{y\in\tilde{S}_{t}: g(y)/v(y)\le1\}\) and \(\Omega_{2}=\{y\in \tilde{S}_{t}: g(y)/v(y)> 1\}\). We have
Combining this with (3.3), we find that \(\int_{\Omega_{1}}g(y) |\log (\frac{g(y)}{v(y)} ) |\,dy=\infty\). This leads us to
We shall show
If so, the desired equality follows. Let \(0<\epsilon<p/2\) and \(y\in \tilde{S}_{t}\). By the mean value theorem, we get
for some \(\epsilon_{0}\in(0, \epsilon)\). This implies
By Fatou’s lemma, we get
Like (3.3), decompose the integral \(\int_{\tilde{S}_{t}} (\cdots)\) as the sum \(\int_{\Omega_{1}} (\cdots) +\int_{\Omega_{2}} (\cdots)\). For the \(\Omega_{2}\) term, we have
which implies
From (3.4) and the fact that \(\lim_{\epsilon\to 0}(1+\epsilon\theta)^{1/\epsilon}=e^{\theta}\) for any \(\theta\in\Bbb{R}\), we get
for any \(\theta<0\). Letting \(\theta\to-\infty\), we get the desired result. □
Lemma 3.1 may be false for the case that \(\sup_{x\in E} g(x)/v(x)=\infty \). A counterexample is given as follows. Consider \(n=1\), \(t=1\), \(g(t)=1\), and \(v(x)=\sum_{m=2}^{\infty}e^{-m}\chi_{(\frac{1}{m}-\frac{1}{m^{3}},\frac{1}{m}]}(x)+\chi_{\Bbb{R}\setminus\bigcup_{m\ge2} (\frac{1}{m}-\frac{1}{m^{3}}, \frac{1}{m}]}(x)\). We have
and
From these, we know that (3.1) is false for this example.
Now, we go back to the investigation of the first part of (1.8). Set
where \(G(x)\) is defined by (1.6). The case \(g(t)=1\) of \(\tilde{D}_{PS}\) reduces to \(D^{*}_{PS}\) mentioned in (1.8). We shall establish the following result, which extends the first inequality in (1.8) from \(u(x)>0\) and \(g(t)=1\) to \(u(x)\ge0\) and those \(g(t)\) subject to the condition (1.9). This extension gives the Persson-Stepanov-type estimate of the modular-type operator norm of the general geometric mean operator corresponding to \(g(t)\). In particular, \(g(t)\) can be of the form \(g(t)=|{\tilde{S}}_{t}|^{s-1}\). An elementary calculation of this case will lead us to the Levin-Cochran-Lee-type inequality. We leave such a calculation to the readers. Our result partially generalizes the sufficient parts of [4], Theorem 3.1, [2], and [3], Theorem 7.3(a).
Theorem 3.2
Let \(0< p\le q<\infty\), \(u(x)\ge0\), \(v(x)>0\), \(g(t)>0\), and \(0< G(x)<\infty\), where \(G(x)\) is defined by (1.6). If (1.9) is true and \(\tilde{D}_{PS}<\infty\), then (1.7) holds for \(C\le e^{1/p}\tilde{D}_{PS}\).
Proof
Let \(\Phi(s)=e^{s}\), \(k(x,t)=g(t)/G(x)\), and \(f(t)\longrightarrow\log f(t)\). The proof is the same as to prove that \(\|\mathbb{K}\|_{*}\le e^{1/p}\tilde{D}_{PS}\). We first assume that \(\sup_{x\in E} \{g(x)/v(x)\}<\infty\). Consider the case that u is bounded on \(\tilde{\Omega}_{r}\) and \(u(x)=0\) on \(E\setminus\tilde{\Omega}_{r}\), where \(r\ge1\) and \(\tilde{\Omega}_{r}=\{x\in E: 1/r\le\|x\|\le r\}\). By (1.10)-(1.11) and Theorem 2.1, we know that
provided that the term \((\cdots)^{1/\epsilon}\) in (3.5) is finite for all sufficiently small \(\epsilon>0\). By an elementary calculation, we obtain \(\lim_{\epsilon\to0^{+}} ((p/\epsilon)^{*} )^{1/\epsilon}=\lim_{\epsilon\to0^{+}} (\frac{p}{p-\epsilon } )^{1/\epsilon}=e^{1/p}\). On the other hand, let \(0<\epsilon<p\). Then \(p/\epsilon>1\) and \(q/\epsilon>1\). Moreover, we have \((p/\epsilon)^{*}=p/(p-\epsilon)\), so
It follows from the definition of \(\tilde{A}_{PS}(p/\epsilon,q/\epsilon)\) that
We have assumed that \(u(x)=0\) on \(E\setminus\tilde{\Omega}_{r}\). Moreover, for \(t\in\tilde{S}_{x}\), we have
These imply
where \(\tilde{B}_{\rho}=\{x\in E: \|x\|\le\rho\}\). The above argument guarantees the validity of (3.5). Now, we try to estimate the limit infimum given in (3.5). It suffices to show that
Clearly, the term \((\int_{\tilde{S}_{x}} (\cdots) )^{-1/p}\) in (3.6) becomes bigger whenever x with \(\|x\|>r\) is replaced by \(rx/\|x\|\). Moreover, the term \((\int_{\tilde{S}_{x}} \{\cdots\} ^{q/\epsilon}u(t)\,dt )^{1/q}\) in (3.6) is zero for \(\|x\|<1/r\) and it keeps the same value for the change: x with \(\|x\|>r\longrightarrow rx/\|x\|\). Hence, the term ‘\(\sup_{x\in E}\)’ in (3.6) can be replaced by ‘\(\sup_{x\in\tilde{\Omega}_{r}}\)’. By the Heine-Borel theorem, we can choose \(0<\epsilon_{m}<p/2\), \(\alpha_{m}>0\), and \(x_{0}, x_{m}\in\tilde{\Omega}_{r}\), such that \(\epsilon_{m}\to0\), \(\alpha_{m}\to0\), \(x_{m}\to x_{0}\), and the following inequality holds for all m:
We have
By the Lebesgue dominated convergence theorem, we infer that
Similarly, the hypotheses on \(u(t)\) and \(g(t)/v(t)\) imply
Applying the Lebesgue dominated convergence theorem again, it follows from Lemma 3.1 that
Putting (3.9)-(3.11) together yields (3.8). This finishes the proof for those u and v with the restrictions stated above. Now, we come back to the proof of the case \(u\ge0\) and \(\sup_{x\in E} \{ g(x)/v(x)\}<\infty\). Let \(u_{r}(x)=\min\{u(x),r\}\chi_{\tilde{\Omega}_{r}}(x)\), where \(r=1,2,\ldots \) . By the preceding result,
where
We have \(u_{r}(t)\le u(t)\), so \(\tilde{D}_{PS}(r)\le\tilde{D}_{PS}\). Replacing \(\tilde{D}_{PS}(r)\) in (3.12) by \(\tilde{D}_{PS}\) first and then applying the monotone convergence theorem to (3.12), we get the desired inequality for this case.
Next, we deal with the case \(\sup_{x\in E} g(x)<\infty\). Let \(v_{\ell}(x)=v(x)+1/\ell\), where \(\ell=1,2,\ldots\) . Then \(\sup_{x\in E} \{g(x)/v_{\ell}(x)\}<\infty\) for each ℓ. By the preceding result,
where
We have \(v_{\ell}(x)\ge v(x)\), so \(\tilde{D}^{\ell}_{PS}\le\tilde{D}_{PS}\). This says that (3.13) can be replaced by (3.14):
We shall claim that \(v_{\ell}(x)\) in (3.14) can be replaced by \(v(x)\). Without loss of generality, we may assume \(\int_{E} (f(x))^{p}v(x)\,dx <\infty\). Set
where \(\tilde{B}_{\rho}\) is defined before and \(h:E\mapsto(0,\infty)\) is chosen so that
Replacing f in (3.14) by \(f_{r}\), we get
For each r, we have
and \(|f_{r}(x)|^{p}v_{\ell}(x)\le(f_{r}(x))^{p}v_{1}(x)\) for \(\ell=1, 2,\ldots\) . Applying the Lebesgue dominated convergence theorem to the right hand side of (3.15), we get
By definition, \(f_{r}(x)\uparrow f(x)\) as \(r\to\infty\). Applying the monotone convergence theorem to both sides of (3.16), the right hand side tends to
and the left hand side has the limit
Let \(x\in E\). Since \(\int_{\tilde{S}_{x}}g(t)\log f(t)\,dt \) is well defined, the following equality makes sense:
where \(\xi^{+} =\max(\xi,0)\) and \(\xi^{-}=\min(-\xi,0)\). Consider \(r\ge \max(\|x\|,1)\). By the monotone convergence theorem,
Inserting this limit in (3.17) yields the desired inequality. This finishes the proof. □
Theorem 3.2 gives a new proof of [3], Theorem 7.3(a). In the following, we shall display another example to show how (1.10) works well for the estimate of Opic-Gurka type. Set
where \(G(x)\) is defined by (1.6). The number \(D^{*}_{OG}(s)\) in (1.8) is just the case \(g(t)=1\) of \(\tilde{D}_{OG}(s)\). In the following, we shall extend the second inequality in (1.8) from \(u(x)>0\) and \(g(t)=1\) to \(u(x)\ge0\) and those \(g(t)\) subject to the condition (1.9). This extension gives the Opic-Gurka-type estimate of the modular-type operator norm of the general geometric mean operator corresponding to \(g(t)\). In particular, \(g(t)\) can be of the form \(g(t)=|{\tilde{S}}_{t}|^{s-1}\), which leads us to the Levin-Cochran-Lee-type inequality. Our result partially generalizes the sufficient parts of [5] and [3], Theorem 7.3(b).
Theorem 3.3
Let \(0< p\le q<\infty\), \(u(x)\ge0\), \(v(x)>0\), \(g(t)>0\), and \(0< G(x)<\infty\), where \(G(x)\) is defined by (1.6). If (1.9) is true and \(\tilde{D}_{OG}(s)<\infty\) for some \(s>1\), then (1.7) holds for \(C\le\inf_{s>1}e^{(s-1)/p}\tilde{D}_{OG}(s)\).
Proof
Let \(\Phi(s)=e^{s}\), \(k(x,t)=g(t)/G(x)\), and \(f(t)\longrightarrow\log f(t)\). The proof is similar to Theorem 3.2. We shall show that \(\|\mathbb{K}\|_{*}\le \inf_{s>1}e^{(s-1)/p}\tilde{D}_{OG}(s)\). To observe the proof of Theorem 3.2, we find that it suffices to prove this inequality for the case: u is bounded on \(\tilde{\Omega}_{r}\), \(u(x)=0\) on \(E\setminus\tilde{\Omega}_{r}\), and \(\sup_{x\in E}\{g(x)/v(x)\}<\infty\), where \(\tilde{\Omega}_{r}\) is defined in the proof of Theorem 3.2. It follows from (1.10)-(1.11) and Theorem 2.2 that
For \(s>1\), we have \(\lim_{\epsilon\to0^{+}} (\frac{p-\epsilon}{p-\epsilon s} )^{1/\epsilon-1/p}=e^{(s-1)/p}\). We shall prove
If so, the desired inequality follows from (3.18). Let \(0<\epsilon<p/s\). We have
The term ‘\((\int_{\tilde{S}_{x}} (\cdots) )^{\frac{s-1}{p}}\)’ in (3.19) increases in \(\|x\|\). On the other hand, the term ‘\((\int_{E\setminus S_{x}} \{\cdots \}^{q(p-\epsilon s)/(\epsilon p)}\frac{u(t)\,dt}{(G(t))^{q/\epsilon}} )^{1/q}\)’ in (3.19) is zero for \(\|x\|>r\) and it keeps the same value for the change: x with \(\|x\|<1/r\longrightarrow(1/r)x/\|x\|\). These imply that the term ‘\(\sup_{x\in E}\)’ in (3.19) can be replaced by ‘\(\sup_{x\in\tilde{\Omega}_{r}}\)’. By the Heine-Borel theorem, we can choose \(0<\epsilon_{m}<p/s\), \(\alpha_{m}>0\), and \(x_{0}, x_{m}\in\tilde{\Omega}_{r}\) such that \(\epsilon_{m}\to0\), \(\alpha_{m}\to0\), \(x_{m}\to x_{0}\), and the following inequality holds for all m:
For the first integral in (3.20), we have
As for the second integral, it follows from Lemma 3.1 that
Moreover, for m large enough,
Integrating the left hand side of (3.22) with respect to \(u(t)\,dt\) first and then applying the Lebesgue dominated convergence theorem, we obtain
Putting (3.20), (3.21), and (3.23) together yields the desired inequality. This finishes the proof. □
For other estimates of Hardy-type inequalities, we may use a similar limit process to Theorems 3.2 and 3.3 to get the corresponding Pólya-Knopp inequalities.
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Acknowledgements
The first author was supported in part by the Ministry of Science and Technology, Taipei, ROC, under Grants Most103-2115-M-364-001 and Most104-2115-M-364-001. We express our gratitude to Professor Lars-Erik Persson and the reviewers for their valued comments in developing the final version of the article.
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Chen, CP., Lan, JW. Estimates of the modular-type operator norm of the general geometric mean operator. J Inequal Appl 2015, 347 (2015). https://doi.org/10.1186/s13660-015-0865-3
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DOI: https://doi.org/10.1186/s13660-015-0865-3