Let Γ be the gamma function, that is, \(\Gamma(x)=\int _{0}^{\infty}t^{x-1}e^{-t}\,dt\) for \(x>0\). For \(f\in L^{r}(\mathbb{R}^{N})\) and \(g\in L^{s}(\mathbb{R}^{N})\) (\(1\leq r,s\leq\infty\)), let \(f*g: \mathbb{R}^{N}\to\mathbb{R}\) be the convolution of f and g defined by
$$\begin{aligned} (f*g) (x):= \int_{\mathbb{R}^{N}}f(x-y)g(y)\,dy \biggl(= \int_{\mathbb {R}^{N}}f(x)g(x-y)\,dy \biggr). \end{aligned}$$
In the following three lemmas, we recall some known results required to obtain explicit values of \(D_{p}(\Omega_{i})\) in (5) for bounded convex domains \(\Omega_{i}\).
Lemma 3.1
(see, e.g., [30, 31])
Let
\(\Omega\subset\mathbb{R}^{N}\) (\(N\in\mathbb{N}\)) be a bounded convex domain. For
\(u\in W^{1,1}(\Omega)\)
and any point
\(x\in\Omega \), we have
$$\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert \leq\frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-y \vert ^{1-N} \bigl\vert \nabla u(y) \bigr\vert \,dy. \end{aligned}$$
A proof of Lemma 3.1 is provided in Appendix 2 because Lemma 3.1 plays an especially important role in obtaining the explicit values of \(D_{p}(\Omega_{i})\).
Lemma 3.2
(Hardy-Littlewood-Sobolev’s inequality [32])
For
\(\lambda>0\), we put
\(h_{\lambda}(x):= \vert x \vert ^{-\lambda}\). If
\(0<\lambda<N\),
$$\begin{aligned} \Vert h_{\lambda}*g \Vert _{L^{\frac{2N}{\lambda }}(\mathbb{R}^{N})}\leq C_{\lambda, N} \Vert g \Vert _{L^{\frac{2N}{2N-\lambda}}(\mathbb{R}^{N})}\quad \textit{for all } g\in L^{\frac{2N}{2N-\lambda}}\bigl(\mathbb{R}^{N}\bigr) \end{aligned}$$
(8)
holds valid for
$$\begin{aligned} C_{\lambda, N}=\pi^{\frac{\lambda}{2}}\frac{\Gamma(\frac {N}{2}-\frac{\lambda}{2})}{\Gamma(N-\frac{\lambda}{2})} \biggl( \frac{\Gamma(\frac{N}{2})}{\Gamma(N)} \biggr)^{-1+\frac{\lambda}{N}}, \end{aligned}$$
(9)
where this is the best constant in (8).
Moreover, if
\(N<2\lambda<2N\),
$$\begin{aligned} \Vert h_{\lambda}*g \Vert _{L^{\frac{2N}{2\lambda -N}}(\mathbb{R}^{N})}\leq\tilde{C}_{\lambda, N} \Vert g \Vert _{L^{2}(\mathbb{R}^{N})}\quad \textit{for all } g\in L^{2}\bigl(\mathbb{R}^{N}\bigr) \end{aligned}$$
(10)
holds valid for
$$\begin{aligned} \tilde{C}_{\lambda, N}=\pi^{\frac{\lambda}{2}}\frac{\Gamma(\frac {N}{2}-\frac{\lambda}{2})}{\Gamma(\frac{\lambda}{2})} \sqrt{\frac {\Gamma(\lambda-\frac{N}{2})}{\Gamma(\frac{3N}{2}-\lambda)}} \biggl(\frac{\Gamma(\frac{N}{2})}{\Gamma(N)} \biggr)^{-1+\frac{\lambda}{N}}, \end{aligned}$$
(11)
where this is the best constant in (10).
Lemma 3.3
(Young’s inequality [33])
Suppose that
\(1\leq t,r,s\leq\infty\)
and
\(1/t=1/r+1/s-1\geq0\). For
\(f\in L^{r}(\mathbb{R}^{N})\)
and
\(g\in L^{s}(\mathbb{R}^{N})\), we have
$$\begin{aligned} \Vert f*g \Vert _{L^{t}(\mathbb{R}^{N})}\leq (A_{r}A_{s}A_{t'})^{N} \Vert f \Vert _{L^{r}(\mathbb {R}^{N})} \Vert g \Vert _{L^{s}(\mathbb{R}^{N})} \end{aligned}$$
(12)
with
$$\begin{aligned} A_{m}= \textstyle\begin{cases} \sqrt{m^{\frac{2}{m}-1}(m-1)^{1-\frac{1}{m}}}&(1< m< \infty),\\ 1&(m=1, \infty). \end{cases}\displaystyle \end{aligned}$$
The constant
\((A_{r}A_{s}A_{t'})^{N}\)
is the best constant in (12).
The following Theorems 3.1, 3.2, 3.3, and 3.4 provide estimations of \(D_{p}(\Omega)\) for a bounded convex domain Ω, where p, q, and N are imposed on the assumptions listed in Table 1.
Theorem 3.1
Let
\(\Omega\subset\mathbb{R}^{N}\) (\(N\in\mathbb{N}\)) be a bounded convex domain. Assume that
\(p\in\mathbb{R}\)
satisfies
\(2< p\leq 2N/(N-1)\)
if
\(N\geq2\)
and
\(2< p<\infty\)
if
\(N=1\). For
\(q\in\mathbb{R}\)
such that
\(q\geq p/(p-1)\), we have
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}\leq D_{p}(\Omega ) \Vert \nabla u \Vert _{L^{q}(\Omega)} \quad \textit{for all } u\in W^{1,q}(\Omega) \end{aligned}$$
with
$$\begin{aligned} D_{p}(\Omega)=\frac{d_{\Omega}^{1+\frac{2N}{p}}\pi^{\frac {N}{p}}}{N \vert \Omega \vert ^{\frac{1}{p}+\frac {1}{q}}}\frac{\Gamma(\frac{p-2}{2p}N)}{\Gamma(\frac {p-1}{p}N)} \biggl( \frac{\Gamma(N)}{\Gamma(\frac{N}{2})} \biggr)^{\frac{p-2}{p}}. \end{aligned}$$
Proof
Let \(u\in W^{1,q}(\Omega)\). Since \(p\leq2N/(N-1)\) and \(1-N+(2N/p)\geq 0\), it follows that \(\vert x-z \vert ^{1-N+\frac {2N}{p}}\leq d_{\Omega}^{1-N+\frac{2N}{p}}\) for \(x, z\in\Omega\). Lemma 3.1 implies that, for a fixed \(x\in\Omega\),
$$\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq \frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{1-N+\frac{2N}{p}} \vert x-z \vert ^{-\frac{2N}{p}} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{-\frac {2N}{p}} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac {2N}{p}} \bigl(E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz. \end{aligned} $$
Therefore,
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \biggl( \int_{\Omega} \biggl( \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac{2N}{p}} \bigl(E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}} \\ &\leq\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert } \biggl( \int_{\mathbb{R}^{N}} \biggl( \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac{2N}{p}} \bigl(E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}$$
Since \(q\geq p/(p-1)\) and Ω is bounded, we have \(\vert \nabla u \vert \in L^{p/(p-1)}(\Omega)\). Therefore, Lemma 3.2 ensures
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert }C_{\frac{2N}{p}, N} \bigl\Vert E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr\Vert _{L^{\frac{p}{p-1}}(\mathbb{R}^{N})} \\ &=\frac{d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert }C_{\frac{2N}{p}, N} \Vert \nabla u \Vert _{L^{\frac {p}{p-1}}(\Omega)}, \end{aligned}$$
where \(C_{\frac{2N}{p}, N}\) is defined in (9) with \(\lambda=2N/p\). Since \(q\geq p/(p-1)\), Hölder’s inequality moreover implies
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{2N}{p}}}{N \vert \Omega \vert ^{\frac{1}{p}+\frac{1}{q}}}C_{\frac{2N}{p}, N} \Vert \nabla u \Vert _{L^{q}(\Omega)}. \end{aligned}$$
□
Theorem 3.2
Let
\(\Omega\subset\mathbb{R}^{N}\) (\(N\geq2\)) be a bounded convex domain. Assume that
\(2< p\leq2N/(N-2)\)
if
\(N\geq3\)
and
\(2< p<\infty\)
if
\(N=2\). For all
\(u\in W^{1,2}(\Omega)\), we have
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}\leq D_{p}(\Omega ) \Vert \nabla u \Vert _{L^{2}(\Omega)} \end{aligned}$$
with
$$\begin{aligned} D_{p}(\Omega)=\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}\pi^{\frac {p+2}{4p}N}}{N \vert \Omega \vert }\frac{\Gamma(\frac {p-2}{4p}N)}{\Gamma(\frac{p+2}{4p}N)}\sqrt{ \frac{\Gamma(\frac {N}{p})}{\Gamma(\frac{p-1}{p}N)}} \biggl(\frac{\Gamma(N)}{\Gamma (\frac{N}{2})} \biggr)^{\frac{p-2}{2p}}. \end{aligned}$$
Proof
Let \(u\in W^{1,2}(\Omega)\). Since \(p\leq2N/(N-2)\), it follows that \(\vert x-z \vert ^{1-N+(p+2)N/(2p)}\leq d_{\Omega }^{1-N+(p+2)N/(2p)}\) for \(x, z\in\Omega\). Lemma 3.1 leads to
$$\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq \frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{1-N+\frac{p+2}{2p}N} \vert x-z \vert ^{-\frac{p+2}{2p}N} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-z \vert ^{-\frac {p+2}{2p}N} \bigl\vert \nabla u(z) \bigr\vert \,dz \\ &\leq\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac {p+2}{2p}N} \bigl(E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz. \end{aligned} $$
Therefore,
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \biggl( \int_{\Omega} \biggl( \int_{\mathbb{R}^{N}} \vert x-z \vert ^{-\frac{p+2}{2p}N} \bigl(E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}} \\ &\leq\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert } \biggl( \int_{\mathbb{R}^{N}} \biggl( \int_{\mathbb {R}^{N}} \vert x-z \vert ^{-\frac{p+2}{2p}N} \bigl(E_{\Omega ,\mathbb{R}^{N}} \vert \nabla u \vert \bigr) (z)\,dz \biggr)^{p}\,dx \biggr)^{\frac{1}{p}}. \end{aligned}$$
From (10), it follows that
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert }\tilde{C}_{\frac{p+2}{2p}N, N} \bigl\Vert E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr\Vert _{L^{2}(\mathbb {R}^{N})} \\ &=\frac{d_{\Omega}^{1+\frac{p+2}{2p}N}}{N \vert \Omega \vert }\tilde{C}_{\frac{p+2}{2p}N, N} \Vert \nabla u \Vert _{L^{2}(\Omega)}, \end{aligned}$$
where \(\tilde{C}_{\frac{p+2}{2p}N, N}\) is defined in (11) with \(\lambda=(p+2)N/(2p)\). □
Theorem 3.3
Let
\(\Omega\subset\mathbb{R}^{N}\) (\(N\in\mathbb{N}\)) be a bounded convex domain. Suppose that
\(1\leq q\leq p< qN/(N-q)\)
if
\(N>q\), and
\(1\leq q\leq p<\infty\)
if
\(N=q\). Then we have
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}\leq D_{p}(\Omega ) \Vert \nabla u \Vert _{L^{q}(\Omega)}\quad \textit{for all } u\in W^{1,q}(\Omega) \end{aligned}$$
(13)
with
$$\begin{aligned} D_{p}(\Omega)=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }(A_{r}A_{q}A_{p'})^{N} \bigl\Vert \vert x \vert ^{1-N} \bigr\Vert _{L^{r}(V)}, \end{aligned}$$
where
\(\Omega_{x}:=\{x-y\mid y\in\Omega\}\)
for
\(x\in\Omega\), \(V:=\bigcup_{x\in\Omega}\Omega_{x}\), and
\(r=qp/((q-1)p+q)\).
Proof
First, we prove \(I:= \Vert \vert x \vert ^{1-N} \Vert _{L^{r}(V)}^{r}<\infty\). Let \(\rho=2d_{\Omega}\) so that \(V\subset B(0,\rho)\). We have
$$\begin{aligned} \frac{pq(1-N)}{(q-1)p+q}+N-1&=\frac{pq(1-N)+Np(q-1)+Nq}{(q-1)p+q}-1 \\ &=\frac{Nq-(N-q)p}{(q-1)p+q}-1>-1. \end{aligned}$$
Therefore,
$$\begin{aligned} I&= \int_{V} \vert x \vert ^{\frac{pq(1-N)}{(q-1)p+q}}\,dx \leq \int_{B(0,\rho)} \vert x \vert ^{\frac{pq(1-N)}{(q-1)p+q}}\,dx =J \int_{0}^{\rho}\rho^{\frac{pq(1-N)}{(q-1)p+q}+N-1}\,d\rho < \infty, \end{aligned}$$
where J is defined by
$$J= \textstyle\begin{cases} 2&(N=1),\\ 2\pi&(N=2),\\ 2\pi\int_{[0,\pi]^{N-2}}\prod_{i=1}^{N-2}(\sin\theta_{i})^{N-i-1} \,d\theta_{1}\cdots \,d\theta_{N-2}&(N\geq3). \end{cases} $$
Next, we show (13). For \(x\in\Omega\), it follows from Lemma 3.1 that
$$\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq\frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\Omega} \vert x-y \vert ^{1-N} \bigl\vert \nabla u(y) \bigr\vert \,dy \\ &=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int_{\Omega _{x}} \vert y \vert ^{1-N} \bigl\vert \nabla u(x-y) \bigr\vert \,dy \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \int _{V} \vert y \vert ^{1-N} \bigl(E_{\Omega,V} \vert \nabla u \vert \bigr) (x-y)\,dy. \end{aligned}$$
Since \(E_{V,\mathbb{R}^{N}}E_{\Omega,V}=E_{\Omega,\mathbb{R}^{N}}\),
$$\begin{aligned} \bigl\vert u(x)-u_{\Omega}(x) \bigr\vert &\leq \frac{d_{\Omega }^{N}}{N \vert \Omega \vert } \int_{\mathbb{R}^{N}} (E_{V,\mathbb{R}^{N}}\psi ) (y) \bigl(E_{\Omega,\mathbb {R}^{N}} \vert \nabla u \vert \bigr) (x-y)\,dy, \end{aligned}$$
(14)
where \(\psi(y)= \vert y \vert ^{1-N}\) for \(y\in V\). We denote \(f(x)= (E_{V,\mathbb{R}^{N}}\psi )(x)\) and \(g(x)= (E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert )(x)\). Lemma 3.3 and (14) give
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{p}(\Omega)}&\leq\frac {d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{p}(\Omega)} \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{p}(\mathbb{R}^{N})} \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }(A_{r}A_{q}A_{p'})^{N} \Vert f \Vert _{L^{r}(\mathbb {R}^{N})} \Vert g \Vert _{L^{q}(\mathbb{R}^{N})} \\ &=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }(A_{r}A_{q}A_{p'})^{N} I^{\frac{1}{r}} \Vert \nabla u \Vert _{L^{q}(\Omega)}. \end{aligned}$$
□
Theorem 3.4
Let
\(\Omega\subset\mathbb{R}^{N}\) (\(N\in\mathbb{N}\)) be a bounded convex domain, and let
\(q>N\). Then we have
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{\infty}(\Omega)}\leq D_{\infty}(\Omega) \Vert \nabla u \Vert _{L^{q}(\Omega )}\quad \textit{for all } u\in W^{1,q}(\Omega) \end{aligned}$$
(15)
with
$$\begin{aligned} D_{\infty}(\Omega)=\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \bigl\Vert \vert x \vert ^{1-N} \bigr\Vert _{L^{q'}(V)}, \end{aligned}$$
where
V
is defined in Theorem
3.3.
Proof
First, we show \(I:= \Vert \vert x \vert ^{1-N} \Vert _{L^{q'}(V)}^{q'}<\infty\). Let \(\rho=2d_{\Omega}\) so that \(V\subset B(0,\rho)\). We have
$$\begin{aligned} q'(1-N)+N-1=\frac{q(1-N)+N(q-1)}{q-1}-1=\frac{q-N}{q-1}-1>-1. \end{aligned}$$
Therefore,
$$\begin{aligned} I= \int_{V} \vert x \vert ^{q'(1-N)}\,dx\leq \int_{B(0,\rho )} \vert x \vert ^{q'(1-N)}\,dx =J \int_{0}^{\rho}\rho^{q'(1-N)+N-1}\,d\rho < \infty, \end{aligned}$$
where J is defined in the proof of Theorem 3.3.
Next, we prove (15). Let \(r=\frac{q}{q-1}(\geq1)\), \(f(x)= (E_{V,\mathbb{R}^{N}}\psi )(x)\), and \(g(x)= (E_{\Omega,\mathbb{R}^{N}} \vert \nabla u \vert )(x)\), where ψ is denoted in the proof of Theorem 3.3. From Lemma 3.3 and (14), for \(u\in W^{1,q}(\Omega)\), it follows that
$$\begin{aligned} \Vert u-u_{\Omega} \Vert _{L^{\infty}(\Omega)}&\leq\frac {d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{\infty}(\Omega)} \leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f*g \Vert _{L^{\infty}(\mathbb{R}^{N})} \\ &\leq\frac{d_{\Omega}^{N}}{N \vert \Omega \vert } \Vert f \Vert _{L^{q'}(\mathbb{R}^{N})} \Vert g \Vert _{L^{q}(\mathbb{R}^{N})} =\frac{d_{\Omega}^{N}}{N \vert \Omega \vert }I^{\frac {1}{q'}} \Vert \nabla u \Vert _{L^{q}(\Omega)}. \end{aligned}$$
□