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\noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=1.75]{BJMA.jpg}}
\noindent\parbox{4.85in}{\hspace{0.1mm}\\[1.5cm]\noindent
Banach J. Math. Anal. 2 (2008), no. 2, 150--162\\
$\frac{\rule{4.55in}{0.05in}}{{}}$\\
{\footnotesize
\textcolor[rgb]{0.65,0.00,0.95}{\textsc{\textbf{\large{B}}anach
\textbf{\large{J}}ournal of \textbf{\large{M}}athematical
\textbf{\large{A}}nalysis}}\\
ISSN: 1735-8787 (electronic)\\
\textcolor[rgb]{0.00,0.00,0.84}{\textbf{http://www.math-analysis.org }}\\
$\frac{{}}{\rule{4.55in}{0.05in}}$}\\[.5in]}



\title[Mixed means over spheres]
{Mixed means for centered and uncentered  averaging operators over
spheres and related results}


\author{I. Peri\'c}
\address{Faculty of Food Technology and Biotechnology \\ University of Zagreb, Pierottijeva 6, 10000 Zagreb, Croatia.}
\email{\textcolor[rgb]{0.00,0.00,0.84}{iperic@pbf.hr}}

\dedicatory{This paper is dedicated to Professor Josip Pe\v{c}ari\'c\\
\vspace{.5cm} {\rm Submitted by M. S. Moslehian}}

\subjclass[2000]{Primary 26D10; Secondary 26D15.}

\date{Received: 30 April 2008; Accepted: 5 July 2008.}

\keywords{Mixed means, integral power means, power weights, centered
and uncentered spheres, polar coordinates, Hardy's inequality,
Carleman's inequality, spherical maximal functions, lower bounds for
operator norms.}


\begin{abstract}
\noindent Mixed-mean inequalities for integral power means over
centered and uncentered spheres are proved. Therefrom we deduce the
Hardy type inequalities for corresponding averaging operators.
Moreover, we discuss estimates related to the spherical maximal
functions.
\end{abstract}


\maketitle

\section{Introduction}

This paper is a continuation of series of papers \cite{CP1, CPP,
CizPer} which deal with the problem of deriving mixed-mean
inequalities for various averaging operators acting on functions
defined on $\mathbb{R}^n$. The mixed-mean inequalities are of
interest themselves, but they can also produce important
inequalities, of which the most important are the Hardy type
inequalities.

Throughout the paper we assume that all involved functions are
non-negative.

M. Christ and L. Grafakos introduced in \cite{CG} the averaging
operator  particularly suitable for deriving mixed-mean inequalities
$$\left(T_{\delta}f\right)(\x )=\frac{1}{\left|B\left(\x ,\delta |\x
|\right)\right|}\int_{B\left(\x ,\delta |\x |\right)}f(\y
)d\y,\;f\in L_{\mathrm{loc}}^{1}\left(\mathbb{R}^n\right)),$$ where
$\delta>0$, $B(\x ,r)$ is the ball in $\mathbb{R}^n$ centered at
$\x\in \mathbb{R}^n$ and of radius $r>0$, $|\x |$ is the Euclidean
norm of $\x\in \mathbb{R}^n$ and $|A|$ is the Lebesgue measure of a
measurable set $A\subset\mathbb{R}^n$. In the same paper they proved
the Hardy type inequality for the operator $T_{\delta}$ and
henceforth deduced its operator norm on
$L^{p}\left(\mathbb{R}^n\right)$. The basic tool in their proof was
Young's inequality $\|f\ast K\|_p\leq \|f\|_p\|K\|_1$ for the
convolution on the group $\left(\mathbb{R}^{+},\frac{dt}{t}\right)$.
An interesting and important feature of this norm is that it is an
lower bound for the Hardy-Littlewood (centered) maximal function
$$\left(M_{c}f\right)(\x )=\mathrm{sup}_{r>0}\frac{1}{\left|B\left(\x
,r\right)\right|}\int_{B\left(\x ,r\right)}f(\y )d\y.$$ In
\cite{CizPer} by proving the appropriate mixed-mean inequality, we
derived the generalization of this result, in the sense that we
obtained the operator norm on weighted $L^p$ spaces (with power
weights) of the operator
$$\left(T_{\delta ,\alpha}f\right)(\x )=\frac{1}{\left|B\left(\x ,\delta |\x
|\right)\right|_{\alpha}}\int_{B\left(\x ,\delta |\x |\right)}f(\y
)|\y |^{\alpha}d\y,$$ where $|A|_{\alpha }=\int_{A}|\y
|^{\alpha}d\y$.

The second motivation for this paper is that maximal function can be
defined for various collections $\mathcal{C}$ of sets,
$\mathcal{C}=\{C\;:\;C\subset\mathbb{R}^n\}$, by
$$\left(M_{\mathcal{C}}f\right)(\x
)=\sup_{C\in\mathcal{C}}\frac{1}{|C|}\int_{C}f(\x-\y )d\y.$$ This
maximal function is closely related to one of the main problems in
real-variable theory: For what collections $\mathcal{C}$
$$\lim_{\mathrm{diam}(C)\to 0}\frac{1}{|C|}\int_{C}f(\x-\y
)d\y=f(\x)\;a.e.$$ holds for ''all'' $f$ (see \cite{S1}).

In this paper we consider two collection of sets, a collection of
centered spheres and a collection of uncentered spheres and
analogous averaging operators
$$\left(\mathcal{S}_{\delta
,\alpha}^{c}f\right)(\x)=\frac{1}{\left|S^{n-1}\left(\x ,\delta |\x
|\right)\right|_{\alpha}}\int_{S^{n-1}\left(\x ,\delta |\x
|\right)}f(\y )|\y |^{\alpha }ds(\y ),\;\delta >0,$$
$$\left(\mathcal{S}_{\delta
,\alpha}^{unc}f\right)(\x)=\frac{1}{\left|S^{n-1}\left(\delta\x
,|1-\delta | |\x
|\right)\right|_{\alpha}}\int_{S^{n-1}\left(\delta\x ,|1-\delta| |\x
|\right)}f(\y )|\y |^{\alpha }ds(\y ),\;\delta
\in\mathbb{R},\delta\neq 1,$$ defined for suitable $f$ (say
continuous with compact support), where $S^{n-1}(\mathbf{a},r)$ is
the sphere in $\mathbb{R}^n$ centered at $\mathbf{a}\in\mathbb{R}^n$
and of radius $r>0$ and $ds$ is the induced Lebesgue measure. Of
course, in both cases the operator norms of these operators are
lower bounds for operator norms of appropriate maximal functions
defined by
$$\left(\mathcal{M}_{\mathrm{c}}f\right)(\x
)=\sup_{r>0}\frac{1}{\left|S^{n-1}(\x ,r)\right|}\int_{S^{n-1}(\x
,r)}f(\y )ds (\y ),$$
$$\left(\mathcal{M}_{\mathrm{unc}}f\right)(\x
)=\sup_{\mathbf{a}\in \mathbb{R}^n,r>0,\x\in
S^{n-1}(\mathbf{a},r)}\frac{1}{\left|S^{n-1}(\mathbf{a}
,r)\right|}\int_{S^{n-1}(\mathbf{a},r)}f(\y )ds (\y ).$$ The
importance of these lower bounds can be seen by comparing  the
operator norm of an maximal function, when it is known, with the
maximum (with respect to $\delta$) of the operator norms of
operators defined as $\mathcal{S}_{\delta ,\alpha}^{\mathrm{c}}$ and
$\mathcal{S}_{\delta ,\alpha}^{\mathrm{unc}}$. For example, this can
be done using results from \cite{GM-S} and calculating the norms of
an operator defined analogously as $\mathcal{S}_{\delta
,\alpha}^{\mathrm{unc}}$ but for balls instead of spheres.

Our results will be given in a priori forms, in the sense that we
shall not go into  details about existence and integrability of
functions $\mathcal{S}_{\delta ,\alpha}^{\mathrm{c}}f$ and
$\mathcal{S}_{\delta ,\alpha}^{\mathrm{unc}}f$. For further details
in this matter see \cite{S, S1}. In what follows we assume that all
integrals exist on the respective domains of their definitions.

We shall frequently use the obvious identities
\begin{eqnarray*}
|B(r)|_{\alpha}=\frac{n}{n+\alpha}r^{n+\alpha}|B|,\; \left|B(\x
,\delta |\x |)\right|_{\alpha}=|\x
|^{n+\alpha}\left|B(\mathbf{e},\delta
)\right|_{\alpha},\end{eqnarray*}
\begin{eqnarray*}
\left|S^{n-1}(r)\right|_{\alpha}=r^{n+\alpha
-1}\left|S^{n-1}\right|,\; \left|S^{n-1}(\x ,\delta |\x
|)\right|_{\alpha}=|\x |^{n+\alpha
-1}\left|S^{n-1}(\mathbf{e},\delta )\right|_{\alpha},\end{eqnarray*}
where $B=B(\mathbf{0},1)$ and $S^{n-1}=S^{n-1}(\mathbf{0},1)$ are
the unit ball and the unit sphere respectively.

We shall also use the integral representation (see \cite{SW})
\begin{equation}\label{Haar}
\frac{1}{\left|S^{n-1}\right|}\int_{S^{n-1}}f(\theta
)d\theta=\int_{SO(n)}f\left(\sigma\bf{e}\right)d\sigma
,\end{equation} where $d\sigma$ is the normalized Haar measure on
the rotation group $SO(n)$ of $\mathbb{R}^n$ (which is left and
right invariant due to the compactness of $SO(n)$, \cite{HR}),
$d\theta$ is induced Lebesgue measure on unit sphere $S^{n-1}$,
$\mathbf{e}\in \mathbb{R}^n$ is any unit vector. Note that we change
notation of the surface measure $ds$ in the case of unit sphere, in
order to be in accordance with the standard notation of polar
coordinates in integral over domains in $\mathbb{R}^n$.

\section{Mixed-mean inequality}

We begin with a technical lemma, which is especially useful in
calculating the norms of the operators $\mathcal{S}_{\delta
,\alpha}^{\mathrm{c}}$ and
$\mathcal{S}_{\delta,\alpha}^{\mathrm{unc}}$. This lemma is a
generalization of the calculus arc length formula.

\begin{lemma}\label{SurfMeas}Suppose that some hypersurface in $\rn$
is given in polar coordinates with
$y=u\phi=tF\left(\phi\cdot\theta\right)\phi$, where $t>0$,
$\theta\in S^{n-1}$ are fixed, $\phi\in U$, $U$ is an open subset of
$S^{n-1}$, and $F:[-1,1]\to \R$ is an differentiable function. Then
\begin{equation}\label{SurfMeas1}ds
(\y)=t^{n-1}F^{n-2}\left(\phi\cdot\theta\right)\sqrt{F^2\left(\phi\cdot\theta\right)+F'^2
\left(\phi\cdot\theta\right)\left(1-\phi\cdot\theta^2\right)}\;d\phi\end{equation}\end{lemma}
\begin{proof}Using rotational invariance of the induced Lebesgue measure on $S^{n-1}$ it is enough to prove the case
when $\theta =\textbf{e}_n\equiv (0,\ldots ,0,1)$. In that case
$\phi\cdot\theta=\cos{\varphi_{n-1}},\;\varphi_{n-1}\in [0,\pi]$ and
the equation of the hypersurface is
$\y=tF\left(\cos{\varphi_{n-1}}\right)\phi$. The polar coordinates
are used in the sense that
$\phi=\left(\sin{\varphi_{n-1}}\bar{\phi},\cos{\varphi_{n-1}}\right)$,
where $\bar{\phi}\in S^{n-2}$. To prove the formula we should
calculate the Jacobian $\mathrm{J}\Phi$, where $$\y=\left(y_1,\ldots
,y_n\right)=\Phi \left(\varphi_1,\ldots
,\varphi_{n-1}\right)=tF\left(\cos{\varphi_{n-1}}\right)\left(\sin{\varphi_{n-1}}\bar{\phi},\cos{\varphi_{n-1}}\right).$$
Note that $\mathrm{J}\Phi$ is a $n\times (n-1)$ determinant. Using
Pythagorean theorem for non-square determinants (see for example
\cite{GE}) we have
$$\left(\mathrm{J}\Phi\right)^2=\sum_{k=1}^n\left(\frac{\partial\left(y_1,\ldots
,\hat{y_k},\ldots ,y_n\right)}{\partial\left(\varphi_1,\ldots
,\varphi_{n-1}\right)}\right)^2,$$ where $\hat{y_k}$ denotes the
missing variable. A straightforward calculation reveals
\begin{align}\label{Pyth1}
&\lefteqn{\frac{\partial\left(y_1,\ldots
,,y_{n-1}\right)}{\partial\left(\varphi_1,\ldots
,\varphi_{n-1}\right)}}
\\ & =  t^{n-1}F^{n-2}\left(-F'\sin^2{\varphi_{n-1}}+F\cos{\varphi_{n-1}}\right)\sin^{n-2}{\varphi_{n-1}}
\left|\begin{array}{cccc} \frac{\partial \bar{\phi}_1}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_1}{\partial
\varphi_{n-2}} & \bar{\phi}_1 \\ \vdots & \vdots & \vdots & \vdots
\\ \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_{n-2}} & \bar{\phi}_{n-1}\end{array}\right|\nonumber
\end{align}
and for $k=1,\ldots ,n-1$,
\begin{eqnarray}\label{Pyth2}\lefteqn{\frac{\partial\left(y_1,\ldots
,\hat{y_k},\ldots ,y_{n}\right)}{\partial\left(\varphi_1,\ldots
,\varphi_{n-1}\right)}}
\\ &  & = -t^{n-1}F^{n-2}\left(F'\cos{\varphi_{n-1}}+F\right)\sin^{n-1}{\varphi_{n-1}}
\frac{\partial \left(\bar{\phi}_1,\ldots ,\hat{\bar{\phi}_k},\ldots
,\bar{\phi}_{n-1}\right)}{\partial \left(\varphi_1,\ldots
,\varphi_{n-2}\right)}. \nonumber
\end{eqnarray}Using $\sum_{k=1}^{n-1}\bar{\phi}_{k}^2=1$ and
Pythagorean theorem we obtain
\begin{align}\label{Pyth3}
\left|\begin{array}{cccc} \frac{\partial \bar{\phi}_1}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_1}{\partial
\varphi_{n-2}} & \bar{\phi}_1 \\ \vdots & \vdots & \vdots & \vdots
\\ \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_{n-2}} & \bar{\phi}_{n-1}\end{array}\right|^2 &=\mathrm{det}
\left[\begin{array}{ccc} \frac{\partial \bar{\phi}_1}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_1}{\partial
\varphi_{n-2}} \\ \vdots & \vdots & \vdots
\\ \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_{n-2}}\end{array}\right]\left[\begin{array}{ccc}
\frac{\partial \bar{\phi}_1}{\partial \varphi_1} & \cdots &
\frac{\partial \bar{\phi}_1}{\partial \varphi_{n-2}} \\ \vdots &
\vdots & \vdots
\\ \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_1} & \cdots & \frac{\partial \bar{\phi}_{n-1}}{\partial
\varphi_{n-2}}\end{array}\right]^T \nonumber \\ &=
\sum_{k=1}^{n-1}\left(\frac{\partial \left(\bar{\phi}_1,\ldots
,\hat{\bar{\phi}_k},\ldots ,\bar{\phi}_{n-1}\right)}{\partial
\left(\varphi_1,\ldots
,\varphi_{n-2}\right)}\right)^2=\left(\mathrm{J}\bar{\phi}\right)^2.
\end{align}
Using (\ref{Pyth1}), (\ref{Pyth2}) and (\ref{Pyth3})
we have
\begin{eqnarray}\label{Pyth4}\lefteqn{\left(\mathrm{J}\Phi\right)^2=\sum_{k=1}^n\left(\frac{\partial\left(y_1,\ldots
,\hat{y_k},\ldots ,y_n\right)}{\partial\left(\varphi_1,\ldots
,\varphi_{n-1}\right)}\right)^2} \nonumber
\\ & & =t^{2(n-1)}F^{2(n-2)}\left(F'\cos{\varphi_{n-1}}+F\right)^2\sin^{2(n-1)}{\varphi_{n-1}}
\left(\mathrm{J}\bar{\phi}\right)^2 \nonumber \\ & & +
t^{2(n-1)}F^{2(n-2)}\left(-F'\sin^2{\varphi_{n-1}}+F\cos{\varphi_{n-1}}\right)^2\sin^{2(n-2)}{\varphi_{n-1}}
\left(\mathrm{J}\bar{\phi}\right)^2 \nonumber \\
& & =t^{2(n-1)}F^{2(n-2)}\sin^{2(n-2)}{\varphi_{n-1}}\nonumber
\\ & & \left[\sin^2{\varphi_{n-1}}
\left(F'\cos{\varphi_{n-1}}+F\right)^2+\left(-F'\sin^2{\varphi_{n-1}}+F\cos{\varphi_{n-1}}\right)^2\right]
\left(\mathrm{J}\bar{\phi}\right)^2 \nonumber \\
& &
=t^{2(n-1)}F^{2(n-2)}\sin^{2(n-2)}{\varphi_{n-1}}\left[F^2+F'^2\sin^2{\varphi_{n-1}}\right]
\left(\mathrm{J}\bar{\phi}\right)^2.
\end{eqnarray}
Finally, using (\ref{Pyth4}) follows
\begin{eqnarray*}\lefteqn{ds(\y)=\mathrm{J}\Phi\;d\varphi_1\cdots
d\varphi_{n-1}}  \\
& & =t^{n-1}F^{n-2}\sin^{n-2}{\varphi_{n-1}}
\sqrt{F^2+F'^2\sin^2{\varphi_{n-1}}}\;\mathrm{J}\bar{\phi}\;d\varphi_1\cdots
d\varphi_{n-1}  \\
& & =t^{n-1}F^{n-2}
\sqrt{F^2+F'^2\sin^2{\varphi_{n-1}}}\;\sin^{n-2}{\varphi_{n-1}}\;d\varphi_{n-1}d\bar{\phi}
 \\
& & =t^{n-1}F^{n-2}
\sqrt{F^2+F'^2\left(1-\cos^2{\varphi_{n-1}}\right)}\;d\phi,
\end{eqnarray*}
which, jointly with rotational invariance, gives (\ref{SurfMeas1}).
\end{proof}

Our basic inequality reads as follows. When there is no danger of
confusion, we write $S$ instead of $S^{n-1}$.

\begin{theorem}\label{Th1}Let $r, s, b, \delta ,\alpha_1
,\alpha_2\in\mathbb{R}$ be such that $r\leq s$, $r,s\neq 0$, $b>0$,
$\delta>0$, $\alpha_2>-n$ and $\alpha_1>-n+1$ in the case
$\delta=1$. If $f$ is a non-negative function on $B\left((1+\delta
)b\right)$ ($f$ positive in the case $r<0$) and
$\textbf{b}=b\textbf{ e}$, $|\textbf{e}|=1$, then the inequality
\begin{eqnarray}
\label{In1} \lefteqn{\left[\ff{1}{|B(b)|_{\alpha_2}}\int_{B(b)}
\left(\ff{1}{|S(\textbf{x},\delta |\textbf{x}|)|_{\alpha_1}}
\int_{S(\textbf{x},\delta |\textbf{x}|)}f^r(\textbf{
y})|\y|^{\alpha_1}ds(\y)\right)^ {\ff{s}{r}}
|\x|^{\alpha_2}d\x\right]^{\ff{1}{s}}} \nonumber \\
& & \leq\left[\ff{1}{|S(\textbf{b},\delta b)|_{\alpha_1}}
\int_{S(\textbf{b},\delta b)} \left(\ff{1}{|B(|\textbf{
x}|)|_{\alpha_2}} \int_{B(|\textbf{x}|)}f^{s}(\textbf{
y})|\y|^{\alpha_2}d\y\right)^{\ff{r}{s}}
|\x|^{\alpha_1}ds(\x)\right]^{\ff{1}{r}}.    \nonumber \\
\end{eqnarray} holds. Inequality (\ref{In1}) is sharp and equality holds for functions of the form
$f(\x )=C|\x|^{\lambda}$, $C>0$. In the case $r\geq s$ the sign of
inequality in (\ref{In1}) is reversed.\end{theorem}
\begin{proof}To transform the LHS of inequality (\ref{In1}) we use the polar coordinates, so let
$\x =t\theta$ and $\y=u\phi$, $t,u\geq 0$, $\theta ,\phi\in
S^{n-1}$. The relation $|\y-\x|=\delta |\x|$ is now equivalent to
expression $u=t\left(\phi\cdot\theta\pm
\sqrt{\phi\cdot\theta^2+\delta^2-1}\right)$, where $\phi\cdot\theta$
denotes the inner product in $\rn$. In the case $0<\delta<1$, we
have $\phi\cdot\theta\geq \sqrt{1-\delta^2}$ and we must decompose
the sphere into two parts, $S_{+}^{n-1}\left(\x ,\delta |\x|\right)$
and $S_{-}^{n-1}\left(\x ,\delta |\x|\right)$. In the case
$\delta\geq 1$, the minus case has no geometrical meaning.

We continue by considering the case $0<\delta <1$. In the case
$\delta \geq 1$ the proof follows the same lines. It is obvious that
it is enough to prove (\ref{In1}) in the case $r=1<s=p$ and $b=1$.
In order to simplify the formulas we introduce the following
notations $\phi\theta$ for inner product,
$F_{1,2}(\phi\theta)=\phi\theta\pm\sqrt{\phi\theta^2+\delta^2-1}$,
$H_{1,2}(\phi\theta)=F_{1,2}^{n-2}(\phi\theta)$
$\cdot\sqrt{F_{1,2}^2(\phi\theta)+F_{1,2}^{'2}(\phi\theta)\left(1-\phi\theta^2\right)}$
and $I(\phi\theta)$ for the condition
$\phi\theta\geq\sqrt{1-\delta^2}$. Using above transformations and
triangle inequality we obtain
\begin{align}\label{MMIn1}
&\lefteqn{\left[\frac{1}{\left|B\right|_{\alpha_2}}\int_{B}\left(\frac{1}{\left|S(\x;\delta
|\x|)\right|_{\alpha_1}}\int_{S(\x;\delta |\x|)}f(\y
)|\y|^{\alpha_1}ds(\y )\right)^p|\x|^{\alpha_2}d\x\right]^{1/p}}
\nonumber \\
&  \leq \frac{1}{\left|B\right|_{\alpha_2}^{1/p}\left|S(\delta
)\right|_{\alpha_1}}\nonumber\\
&\quad \cdot
\left[\left(\int_{t=0}^{1}\int_{\theta}\left(\int_{I(\phi\theta)}
f\left(tF_1\left(\phi\theta\right)\phi\right)F_{1}^{\alpha_1}\left(\phi\theta\right)H_1(\phi\theta
)d\phi \right)^pt^{\alpha_2+n-1}dtd\theta\right)^{1/p}\right.
\nonumber
\\&\quad  +\left. \left(\int_{t=0}^{1}\int_{\theta}\left(\int_{I\left(\phi\theta\right)}
f\left(tF_2\left(\phi\theta\right)\phi\right)F_{2}^{\alpha_1}(\phi\theta)H_2(\phi\theta
)d\phi \right)^pt^{\alpha_2+n-1}dtd\theta\right)^{1/p} \right].
\end{align}
Using integral equality (\ref{Haar}) and rotational invariance of
the induced Lebesgue measure on $S^{n-1}$, we transform the first
term in square brackets in (\ref{MMIn1}) as follows
\begin{align}\label{MMIn2}
&\lefteqn{\left|S\right|^{-\frac{1}{p}}
\left(\int_{t=0}^{1}\int_{\theta}\left(\int_{I(\phi\theta)}
f\left(tF_1\left(\phi\theta\right)\phi\right)F_{1}^{\alpha_1}(\phi\theta)H_1(\phi\theta
)d\phi \right)^pt^{\alpha_2+n-1}dtd\theta\right)^{1/p}} \nonumber \\
&  = \left(\int_{t=0}^{1}\int_{\sigma}\left(\int_{I(\phi \sigma
\textbf{e})} f\left(tF_1\left(\phi\sigma \textbf{
e}\right)\phi\right)F_{1}^{\alpha_1}(\phi\sigma \textbf{
e})H_1(\phi\sigma \textbf{e} )d\phi
\right)^pt^{\alpha_2+n-1}dtd\sigma\right)^{1/p} \nonumber
\\ & =\Big(\int_{t=0}^{1}\int_{\sigma}\left(\int_{I\left(\sigma^{-1}\phi \textbf{
e}\right)} f\left(tF_1\left(\sigma^{-1}\phi \textbf{
e}\right)\phi\right)F_{1}^{\alpha_1}(\sigma^{-1}\phi \textbf{
e})H_1(\sigma^{-1}\phi\textbf{e} )d\phi
\right)^p \nonumber\\
& \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad
\qquad\qquad \qquad \qquad \qquad
t^{\alpha_2+n-1}dtd\sigma\Big)^{1/p} \nonumber\\
&=\left(\int_{t=0}^{1}\int_{\sigma}\left(\int_{I\left(\phi \textbf{
e}\right)} f\left(tF_1\left(\phi \textbf{
e}\right)\sigma\phi\right)F_{1}^{\alpha_1}(\phi\textbf{e})H_1(\phi\textbf{
e} )d\phi \right)^pt^{\alpha_2+n-1}dtd\sigma\right)^{1/p} \nonumber
\\  & \leq  \int_{I\left(\phi\cdot \textbf{
e}\right)}\left(\int_{t=0}^{1}\int_{\sigma}
f^p\left(tF_1\left(\phi\textbf{
e}\right)\sigma\phi\right)t^{\alpha_2+n-1}dtd\sigma\right)^{\frac{1}{p}}F_{1}^{\alpha_1}(\phi\textbf{
e})H_1(\phi \textbf{e} )d\phi \nonumber \\ & =\int_{I\left(\phi
\textbf{e}\right)}
\left(\frac{\left|B\right|_{\alpha_2}}{\left|B\left(F_1(\phi
\textbf{ e})\right)\right|_{\alpha_2}}\int_{0}^{F_1\left(\phi
\textbf{
e}\right)}\int_{\sigma}f^p\left(t\sigma\phi\right)t^{\alpha_2+n-1}d\sigma
dt\right)^{\frac{1}{p}}F_{1}^{\alpha_1}(\phi\textbf{e})H_1\left(\phi
\textbf{e}\right)d\phi \nonumber \\ &
=\frac{\left|B\right|_{\alpha_2}^{\frac{1}{p}}}{\left|S\right|^{\frac{1}{p}}}\int_{I\left(\phi
\textbf{e}\right)}\left(\frac{1}{\left|B\left(F_1(\phi\textbf{
e})\right)\right|_{\alpha_2}}\int_{0}^{F_1\left(\phi \textbf{
e}\right)}\int_{\theta}f^p\left(t\theta\right)t^{\alpha_2+n-1}d\theta
dt\right)^{\frac{1}{p}} \nonumber\\
&  \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad
\qquad\qquad \qquad \qquad
F_{1}^{\alpha_1}(\phi\textbf{e})H_1\left(\phi
\textbf{e}\right)d\phi \nonumber \\
&=\frac{\left|B\right|_{\alpha_2}^{\frac{1}{p}}}{\left|S\right|^{\frac{1}{p}}}\int_{S_{+}\left(\textbf{
e};\delta\right)} \left(\frac{1}{\left|B\left(|\y
|\right)\right|_{\alpha_2}}\int_{B(|\y
|)}f^p\left(\x\right)\left|\x\right|^{\alpha_2}d\x\right)^{\frac{1}{p}}
|\y|^{\alpha_1}ds(\y),
\end{align}
where integral Minkowski inequality is used, and integral equality
(\ref{Haar}) again.

Analogous arguing for the second term in (\ref{MMIn1}) gives
\begin{eqnarray}\label{MMIn3}
\lefteqn{\left(\int_{t=0}^{1}\int_{\theta}\left(\int_{I\left(\phi\theta\right)}
f\left(tF_2\left(\phi\theta\right)\phi\right)F_{2}^{\alpha_1}(\phi\theta)H_2(\phi\theta
)d\phi \right)^pt^{\alpha_2+n-1}dtd\theta\right)^{1/p}}  \\
& & \leq
\left|B\right|_{\alpha_2}^{\frac{1}{p}}\int_{S_{-}\left(\textbf{
e};\delta\right)} \left(\frac{1}{\left|B\left(|\y
|\right)\right|_{\alpha_2}}\int_{B(|\y
|)}f^p\left(\x\right)\left|\x\right|^{\alpha_2}d\x\right)^{\frac{1}{p}}
|\y|^{\alpha_1}ds(\y). \nonumber
\end{eqnarray}
Using (\ref{MMIn1}), (\ref{MMIn2}) and (\ref{MMIn3}), inequality
(\ref{In1}) follows.

Finally, it is straightforward to check that both sides of
inequality (\ref{In1}), rewritten for the function
$f(\x)=|\x|^{\lambda}$, are equal to
$$b^{\lambda}M_{r}\left(|\y|^{\lambda}; S^{n-1}(\textbf{
e};\delta);\alpha_1\right)M_{s}\left(|\x|^{\lambda};
B;\alpha_2\right),$$ which gives the sharpness of the inequality.
\end{proof}

Mixed-mean inequality for uncentered case is given in the following
theorem.

\begin{theorem}\label{Unc1}Let $r, s, b, \delta ,\alpha_1
,\alpha_2\in\mathbb{R}$ be such that $r\leq s$, $r,s\neq 0$, $b>0$,
$\delta\neq 1$, $\alpha_2>-n$ and $\alpha_1>-n+1$ in the case
$\delta=1/2$. If $f$ is a non-negative function on
$B\left((|\delta|+|1-\delta|)b )\right)$ ($f$ positive in the case
$r<0$) and $\textbf{b}=b\textbf{e}$, $|\textbf{e}|=1$, then the
inequality
\begin{align}\label{Unc2}
&\lefteqn{\left[\ff{1}{|B(b)|_{\alpha_2}}\int_{B(b)}
\left(\ff{1}{|S(\delta\textbf{x},|1-\delta|
|\textbf{x}|)|_{\alpha_1}} \int_{S(\delta\textbf{x},|1-\delta|
|\textbf{ x}|)}f^r(\textbf{y})|\y|^{\alpha_1}ds(\y)\right)^
{\ff{s}{r}}
|\x|^{\alpha_2}d\x\right]^{\ff{1}{s}}} \nonumber \\
& \leq\left[\ff{1}{|S(\delta\textbf{b},|1-\delta| b)|_{\alpha_1}}
\int_{S(\textbf{b},\delta b)} \left(\ff{1}{|B(|\textbf{
x}|)|_{\alpha_2}} \int_{B(|\textbf{x}|)}f^{s}(\textbf{
y})|\y|^{\alpha_2}d\y\right)^{\ff{r}{s}}
|\x|^{\alpha_1}ds(\x)\right]^{\ff{1}{r}}.    \nonumber \\
\end{align}
holds. Inequality (\ref{In1}) is sharp and equality holds for functions of the form
$f(\x )=C|\x|^{\lambda}$, $C>0$. In the case $r\geq s$ the sign of
inequality in (\ref{In1}) is reversed.\end{theorem}
\begin{proof}
The proof is analogous to the proof of Theorem \ref{Th1}. In
transforming inequality (\ref{Unc2}) in polar coordinates using
$\x=t\theta$, $\y=u\phi$, the relation $\y\in
S\left(\delta\x,|1-\delta||\x|\right)$ is equivalent to equation
$$u^2-2ut\delta \phi\cdot\theta+(2\delta-1)t^2=0.$$
Three cases should be considered. For $0\leq \delta\leq 1/2$ the
equation of the sphere is given by
$$u=t\delta\left(\phi\cdot\theta+\sqrt{\left(\phi\cdot\theta\right)^2+\frac{1-2\delta}{\delta^2}}\right),$$
for $\delta<0$
$$u=t|\delta|\left(\sqrt{\left(\phi\cdot\theta\right)^2+\frac{1-2\delta}{\delta^2}}-\phi\cdot\theta\right)$$
and for $\delta>1/2,\delta\neq 1$ we must decompose the sphere into
two parts given by
$$u=t\delta\left(\phi\cdot\theta\pm\sqrt{\left(\phi\cdot\theta\right)^2+\frac{1-2\delta}{\delta^2}}\right),$$
with the condition
$\phi\cdot\theta\geq\frac{\sqrt{1-2\delta}}{\delta}$.

The rest of the proof is as in the proof of Theorem \ref{Th1}.
\end{proof}



\section{Hardy and Carleman type inequalities}

The mixed means can be used in proving various integral
inequalities, such as the Hardy and the Carleman inequality (for the
classical theory see \cite{CL, HLP, LE, MPF} and for the
multidimensional case see for example \cite{DHK}). Analogously to
the procedure given in \cite{CP1, CPP, CizPer}, we apply the mixed
mean inequality (\ref{In1}) to deduce the Hardy-type inequalities
for the operators ${\mathcal S}_{\delta,\alpha}^{\mathrm{c}}$ and
${\mathcal S}_{\delta,\alpha}^{\mathrm{unc}}$ defined in the
Introduction.

\begin{theorem}\label{Hardy1} Let $p>1$, $0<b\leq \infty$,
$\alpha_1$, $\alpha_2\in\R$, $\delta >0$ be such that $\alpha_2>-n$
and $\alpha_1>-n+1$ in the case $\delta =1$. If $f$ is a nonnegative
function on $B\left((1+\delta)b\right)$ and $|\textbf{e}|=1$, then
\begin{eqnarray}\label{Hardy2}
\lefteqn{\left[\int_{B(b)} \left(\ff{1}{|S(\textbf{x},\delta
|\textbf{ x}|)|_{\alpha_1}} \int_{S(\textbf{x},\delta
|\textbf{x}|)}f(\textbf{ y})|\y|^{\alpha_1}ds(\y)\right)^ {p}
|\x|^{\alpha_2}d\x\right]^{\ff{1}{p}}} \nonumber \\
& & \leq C(n,p;\delta;\alpha_1;\alpha_2) \left(
\int_{B((1+\delta)b)}f^{p}(\textbf{
y})|\y|^{\alpha_2}d\y\right)^{\ff{1}{p}},    \nonumber \\
\end{eqnarray}
where
\begin{eqnarray*}
C(n,p;\delta ;\alpha_1;\alpha_2) =\frac{1}{\left|S(\textbf{e},\delta
)\right|_{\alpha_1}}\int_{S(\textbf{ e},\delta )}|\x
|^{-\frac{n+\alpha_2}{p}}|\x |^{\alpha_1}ds(\x),
\end{eqnarray*}
is the best possible constant.
\end{theorem}
\begin{proof}Let $0<b<\infty$. Inequality (\ref{In1}) for $r=1$ and
$s=p$ and obvious estimation $\int_{B(|\x |)}f^p(\y
)|\y|^{\alpha_2}d\y\leq \int_{B((1+\delta )b)}f^p(\y
)|\y|^{\alpha_2}d\y$, which holds for every $\x\in S({\mathbf
b},\delta b)$, implies the inequality
\begin{eqnarray*}\label{Hardy4}
\lefteqn{\left[\int_{B(b)} \left(\ff{1}{|S(\textbf{x},\delta
|\textbf{ x}|)|_{\alpha_1}} \int_{S(\textbf{x},\delta
|\textbf{x}|)}f(\textbf{ y})|\y|^{\alpha_1}ds(\y)\right)^ {p}
|\x|^{\alpha_2}d\x\right]^{\ff{1}{p}}} \\
& & \leq \frac{\left|B(b)\right|_{\alpha_2}^{1/p}}{\left|S({\mathbf
b},\delta b)\right|_{\alpha_1}}\int_{S({\mathbf b},\delta
b)}\left|B(|\x|)\right|_{\alpha_2}^{-1/p}|\x|^{\alpha_1}ds(\x)
\left( \int_{B((1+\delta)b)}f^{p}(\textbf{
y})|\y|^{\alpha_2}d\y\right)^{\ff{1}{p}}.\end{eqnarray*} Using
$|B(b)|_{\alpha_2}=b^{n+\alpha_2}|B(1)|_{\alpha_2}$, $|S(\textbf{
b},\delta b)|_{\alpha_1}=b^{n+\alpha_1-1}|S(\textbf{
e},\delta)|_{\alpha_1}$, and simple substitution $\x'=\x/b$ and
radiallity of the involved function  we obtain (\ref{Hardy2}). Since
the constant $C(n,p;\delta ;\alpha_1;\alpha_2)$  is independent of
$b$, inequality (\ref{Hardy2}) obviously holds for $b=\infty$ as
well.

In the usual manner, for the best possibility of inequality
(\ref{Hardy2}) consider the functions
$f_{\epsilon}(\x)=|\x|^{-(\alpha_2+n)/p+\epsilon}$. It is
straightforward to check that the quotient of the integral
expressions on the left side and the right side of inequality
(\ref{Hardy2}), in this particular choice of functions, tends to the
constant $C(n,p;\delta ;\alpha_1;\alpha_2)$ as $\epsilon$ tends to
0.\end{proof}


\begin{theorem}\label{Hardy5} Let $0\neq p<1$, $0<b\leq \infty$,
$\alpha_1$, $\alpha_2\in\R$, $\delta >0$ be such that $\alpha_2>-n$
and $\alpha_1>-n+1$ in the case $\delta =1$. If $f$ is a nonnegative
function on $B\left((1+\delta)b\right)$, then
\begin{eqnarray*}
\lefteqn{\int_{B(b)} \left(\ff{1}{|S(\textbf{x},\delta |\textbf{
x}|)|_{\alpha_1}} \int_{S(\textbf{x},\delta
|\textbf{x}|)}f^p(\textbf{
y})|\y|^{\alpha_1}ds(\y)\right)^{\frac{1}{p}}
|\x|^{\alpha_2}d\x} \nonumber \\
& & \leq C_1(n,p;\delta;\alpha_1;\alpha_2)
\int_{B((1+\delta)b)}f(\textbf{
y})|\y|^{\alpha_2}d\y,    \nonumber \\
\end{eqnarray*}
where
\begin{equation}\label{Hardy7}C_1(n,p;\delta ;\alpha_1;\alpha_2)
=\left(\frac{1}{\left|S(\textbf{e},\delta
)\right|_{\alpha_1}}\int_{S(\textbf{e},\delta )}|\x
|^{-p\left(n+\alpha_2\right)}|\x
|^{\alpha_1}ds(\x)\right)^{\frac{1}{p}},
\end{equation}
is the best possible constant.
\end{theorem}
\begin{proof}The proof is analogous to the proof of Theorem
\ref{Hardy1} taking in Theorem \ref{In1} $r=p<s=1$. For the best
possibility of the constant $C_1(n,p;\delta ;\alpha_1;\alpha_2)$,
arguing is the same as in Theorem \ref{Hardy1} using functions
$f_{\epsilon}(\y)=|\y|^{-p\left(n+\alpha_2\right)+\epsilon }$.
\end{proof}

Finally, we give the related Carleman type inequality for geometric
mean.

\begin{theorem}\label{Carl1}Let $0<b\leq \infty$,
$\alpha_1,\alpha_2\in\R$, $\delta>0$ be such that $\alpha_2>-n$ and
$\alpha_1>-n+1$ in the case $\delta =1$. If $f$ is a positive
function on $B\left((1+\delta)b\right)$, then
\begin{eqnarray}\label{Carl2}\lefteqn{\int_{B(b)}\mathrm{exp}\left[\frac{1}{\left|S\left(\x,\delta
|\x |\right)\right|_{\alpha_1}}\int_{S\left(\x,\delta |\x
|\right)}|\y |^{\alpha_1}\log{f(\y )}\;ds(\y )\right]|\x
|^{\alpha_2}d\x} \nonumber \\
& & \leq C_2\left(n;\delta
;\alpha_1,\alpha_2\right)\int_{B\left((1+\delta )b\right)}f(\y )|\y
|^{\alpha_2}\;d\y ,
\end{eqnarray}
where \begin{eqnarray*} C_2\left(n;\delta ;\alpha_1,\alpha_2\right)=
\mathrm{exp}\left[\frac{\alpha_2 +n}{\left|S\left(\textbf{e},\delta
\right)\right|_{\alpha_1}}\int_{S\left(\textbf{e},\delta \right)}|\x
|^{\alpha_1}\log{\frac{1}{|\x |}}\;ds(\x )\right]\end{eqnarray*} is
the best possible constant.
\end{theorem}
\begin{proof}Inequality (\ref{Carl2}) follows from (\ref{Hardy7}) by
taking the limiting procedure $\lim_{p\to 0}$.

We give here the proof that the constant $C_2\left(n;\delta
;\alpha_1,\alpha_2\right)$ is the best possible one. To do that
consider the functions $f_{\epsilon}(\y )=|\y
|^{-n-\alpha_2+\epsilon },\;\epsilon>0$. The integral on the right
hand side of inequality (\ref{Carl2}), for this choice of functions,
is  equal to $|S|(1+\delta )^{\epsilon}b^{\epsilon }/\epsilon$. The
integral on the left hand side of inequality (\ref{Hardy7}), for
this choice of functions, using substitution $\y\mapsto \y/|\x |$
and obvious transformations gives
\begin{eqnarray*}\lefteqn{\int_{B(b)}\mathrm{exp}\left[-\frac{n+\alpha_2-\epsilon}{\left|S\left(\x,\delta
|\x |\right)\right|_{\alpha_1}}\int_{S\left(\x,\delta |\x
|\right)}|\y |^{\alpha_1}\log{|\y |}\;ds(\y )\right]|\x
|^{\alpha_2}d\x}  \\
& &
=\int_{B(b)}\mathrm{exp}\left[-\frac{n+\alpha_2-\epsilon}{\left|S\left(\textbf{
e},\delta\right)\right|_{\alpha_1}}\int_{S\left(\textbf{e},\delta
\right)}|\y |^{\alpha_1}\left(\log{|\y |}+\log{|\x |}\right)\;ds(\y
)\right]|\x |^{\alpha_2}d\x \\
& &
=\mathrm{exp}\left[-\frac{n+\alpha_2-\epsilon}{\left|S\left(\textbf{
e},\delta\right)\right|_{\alpha_1}}\int_{S\left(\textbf{e},\delta
\right)}|\y |^{\alpha_1}\log{|\y |}ds(\y)\right]\int_{B(b)}|\x
|^{-n+\epsilon}d\x \\
& &
=\mathrm{exp}\left[-\frac{n+\alpha_2-\epsilon}{\left|S\left(\textbf{
e},\delta\right)\right|_{\alpha_1}}\int_{S\left(\textbf{e},\delta
\right)}|\y |^{\alpha_1}\log{|\y
|}ds(\y)\right]|S|\frac{b^{\epsilon}}{\epsilon},
\end{eqnarray*}
which gives that the quotient of the integrals on the left hand side
and on the right hand side of the inequality (\ref{Hardy7}), for
this particular choice of functions, tends to $C_2\left(n;\delta
;\alpha_1,\alpha_2\right)$ as $\epsilon$ tends to $0$.
\end{proof}

Keeping in mind Theorem \ref{Unc1}, it is obvious what are the
uncentered versions of Theorems \ref{Hardy1}, \ref{Hardy5} and
\ref{Carl1}, so we give just the forms of the constants in analogous
inequalities
\begin{eqnarray*}
\lefteqn{C^{unc}(n,p;\delta ;\alpha_1;\alpha_2)}  \\
& & =\frac{1}{\left|S({\delta\bf e},|1-\delta|
)\right|_{\alpha_1}}\int_{S({\delta\bf e},|1-\delta| )}|\x
|^{-\frac{n+\alpha_2}{p}}|\x |^{\alpha_1}ds(\x),\quad p>1,
\end{eqnarray*}
\begin{eqnarray*}
\lefteqn{C_{1}^{unc}(n,p;\delta
;\alpha_1;\alpha_2)}  \\
& & =\left(\frac{1}{\left|S(\delta\textbf{e},|1-\delta|
)\right|_{\alpha_1}}\int_{S(\delta\textbf{e},|1-\delta| )}|\x
|^{-p\left(n+\alpha_2\right)}|\x
|^{\alpha_1}ds(\x)\right)^{\frac{1}{p}},\quad 0\neq p<1,
\end{eqnarray*}

\begin{eqnarray*}
\lefteqn{C_{2}^{unc}\left(n;\delta ;\alpha_1,\alpha_2\right)} \\ &
&= \mathrm{exp}\left[\frac{\alpha_2 +n}{\left|S\left(\delta\textbf{
e},|1-\delta |\right)\right|_{\alpha_1}}\int_{S\left(\delta\textbf{
e},|1-\delta |\right)}|\x |^{\alpha_1}\log{\frac{1}{|\x |}}\;ds(\x
)\right], \quad p=0.
\end{eqnarray*}


\section{Concluding remarks}

We give several remarks on the constants $C(n,p;\delta
)=C(n,p;\delta ;0,0)$, $C_2(n;\delta )=C_2(n;\delta ;0,0)$,
$C^{unc}(n,p;\delta )=C^{unc}(n,p;\delta ;0,0)$,
$C_{2}^{unc}(n;\delta )=C_{2}^{unc}(n;\delta ;0,0)$.

Using Lemma \ref{SurfMeas} for $\x =\textbf{e}_n$, $\y=u\phi$ and
using $d\phi =\sin^{n-2}{\varphi_{n-1}}d\varphi_{n-1}d\bar{\phi}$,
$\bar{\phi}\in S^{n-2}$, $\phi\cdot
\textbf{e}_n=\cos{\varphi_{n-1}}$,
$\left|S^{n-1}\right|=\frac{2\pi^{n/2}}{\Gamma \left(n/2\right)}$,
we easily get
\begin{eqnarray*}
\lefteqn{C(n,p;\delta )} \\
& &
=\frac{1}{\delta^{n-2}}\frac{\Gamma\left(\frac{n}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{n-1}{2}\right)}
\int_{-1}^{1}\left(t+\sqrt{t^2+\delta^2-1}\right)^{\frac{n}{p'}-1}\frac{(1-t^2)^{\frac{n-3}{2}}dt}
{\sqrt{t^2+\delta^2-1}},\;\delta\geq 1,
\end{eqnarray*}

\begin{eqnarray*}
\lefteqn{C^{unc}(n,p;\delta )} \\
& &
=\frac{|\delta|^{\frac{n}{p'}-2}}{(1-\delta)^{n-2}}\frac{\Gamma\left(\frac{n}{2}\right)}
{\sqrt{\pi}\Gamma\left(\frac{n-1}{2}\right)}
\int_{-1}^{1}\left(t+\sqrt{t^2+\frac{1-2\delta}{\delta^2}}\right)^{\frac{n}{p'}-1}\frac{(1-t^2)^{\frac{n-3}{2}}dt}
{\sqrt{t^2+\frac{1-2\delta}{\delta^2}}},\;\delta\leq\frac{1}{2},
\end{eqnarray*}


\begin{eqnarray*}
\lefteqn{C_2(n;\delta )} \\
& &
=\mathrm{exp}\left[-\frac{n}{\delta^{n-2}}\frac{\Gamma\left(\frac{n}{2}\right)}{\sqrt{\pi}\Gamma
\left(\frac{n-1}{2}\right)}\right. \nonumber \\
& & \cdot \left.
\int_{-1}^{1}\log{\left(t+\sqrt{t^2+\delta^2-1}\right)}\left(t+\sqrt{t^2+\delta^2-1}\right)^{n-1}\frac{(1-t^2)^{\frac{n-3}{2}}dt}
{\sqrt{t^2+\delta^2-1}}\right],\;\delta\geq 1,
\end{eqnarray*}

\begin{eqnarray*}
\lefteqn{C_{2}^{unc}(n;\delta )} \\
& &
=\mathrm{exp}\left[-\frac{n|\delta|^{n-2}}{(1-\delta)^{n-2}}\frac{\Gamma\left(\frac{n}{2}\right)}{\sqrt{\pi}\Gamma
\left(\frac{n-1}{2}\right)}\right. \\
& & \cdot \left.
\int_{-1}^{1}\log{\left[|\delta|\left(t+\sqrt{t^2+\frac{1-2\delta}{\delta^2}}\right)\right]}\left(t+\sqrt{t^2+
\frac{1-2\delta}{\delta^2}}\right)^{n-1}\frac{(1-t^2)^{\frac{n-3}{2}}dt}
{\sqrt{t^2+\frac{1-2\delta}{\delta^2}}}\right],\\
& & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad
\qquad \qquad \qquad \qquad \qquad \qquad \;\delta\leq\frac{1}{2}.
\end{eqnarray*}

It is not necessary to have complementary formulas (in the centered
case $0<\delta\leq 1$, in uncentered case $\delta>1/2$) since it is
easy to see that the  following identities hold
\begin{equation}\label{Id1}
C^{unc}(n,p;\delta)=C^{unc}(n,p;1-\delta),\;\delta\leq
1/2,
\end{equation}
\begin{equation}\label{Id2}
C(n,p;\delta
)=\delta^{-\frac{n}{p}}C\left(n,p;\frac{1}{\delta}\right),
\end{equation}
\begin{equation}\label{Id3}C^{unc}(n,p;\delta
)=\delta^{-\frac{n}{p}}C\left(n,p;\frac{1}{\delta}-1\right),\;0<\delta<1.
\end{equation}
In some cases we can explicitly calculate the above constants as
functions of $\delta$. The easiest case is $p=\frac{n}{n-2}$. This
is the case when the function $\x\mapsto |\x|^{-\frac{n}{p}}$ is a
harmonic function. We get
$C(n,p=\frac{n}{n-2};\delta)=1,\;0<\delta\leq 1$ and
$C(n,p=\frac{n}{n-2};\delta)=\delta^{2-n},\;\delta\geq 1$. Also,
$C^{unc}(n,p=\frac{n}{n-2};\delta)=\delta^{2-n},\;\delta\geq 1/2$
and $C^{unc}(n,p=\frac{n}{n-2};\delta)=(1-\delta)^{2-n},\;\delta\leq
1/2$. Note that $\sup_{\delta>0}C(n,p=\frac{n}{n-2};\delta)=1$ and
$\sup_{\delta}C^{unc}(n,p=\frac{n}{n-2};\delta)=2^{n-2}$. It is easy
to see using (\ref{Id2}) that in harmonic case $p=n/(n-2)$ and in
super-harmonic case $p>n/(n-2)$, we get
$\sup_{\delta>0}C(n,p;\delta)=1$. Only in sub-harmonic cases
$n/(n-1)<p<n/(n-2)$ we obtain non-trivial lower bounds. For example,
$C(p=2,n=3;\delta)=\frac{\sqrt{1+\delta}-\sqrt{|1-\delta|}}{\delta}$,
so $\sup_{\delta>0}C(p=2,n=3;\delta)=\sqrt{2}$.

The identities (\ref{Id1}), (\ref{Id2}), (\ref{Id3}) and previous
examples suggest to consider $C(n,p;1)$ and $C^{unc}(n,p;1/2)$ in
order to obtain the best possible lower bounds for operator norms
for appropriate maximal functions. We easily get
\begin{eqnarray*}
C(n,p;1)=2^{\frac{n}{p'}-2}\frac{\Gamma\left(\frac{n}{2}\right)}{\sqrt{\pi}
\Gamma\left(\frac{n-1}{2}\right)}B\left(\frac{n}{2p'}-\frac{1}{2},\frac{n-1}{2}\right),\;p>n'=\frac{n}{n-1},
\end{eqnarray*}
and
\begin{eqnarray*}
C^{unc}(n,p;\frac{1}{2})=2^{n-2}\frac{\Gamma\left(\frac{n}{2}\right)}{\sqrt{\pi}
\Gamma\left(\frac{n-1}{2}\right)}B\left(\frac{n}{2p'}-\frac{1}{2},\frac{n-1}{2}\right),\;p>n'=\frac{n}{n-1}.
\end{eqnarray*}
Also,
\begin{eqnarray*}
C_2(n;1)=2^{-n}\mathrm{exp}\left[\frac{n}{2}\left(H(n-2)-H\left(\frac{n-3}{2}\right)\right)\right],
\end{eqnarray*}
and
\begin{eqnarray*}
C_{2}^{unc}(n;1)=\mathrm{exp}\left[\frac{n}{2}\left(H(n-2)-H\left(\frac{n-3}{2}\right)\right)\right],
\end{eqnarray*}
where $H=H(s),\;s>-1,$ are harmonic numbers.

Finally, using Stirling asymptotic formula $\Gamma (x)\sim
e^{-x}x^{x-\frac{1}{2}}\sqrt{2\pi}$, we can give asymptotic behavior
of the above constants for fixed $p>1$ and large $n$. For the
similar discussion in the case of balls see \cite{GM-S}.
Straightforward calculation gives that $C(n,p;\delta)$
asymptotically  behaves as
$$\left(\frac{4^{\frac{1}{p'}}\left(\frac{1}{p'}-\frac{1}{n}\right)^{\frac{1}{p'}}}
{\left(\frac{1}{p'}+1-\frac{2}{n}\right)^{1+\frac{1}{p'}}}\right)^{\frac{n}{2}},$$
which shows that $C(n,p;\delta)$ has exponential decay, since by
Bernoulli inequality $4/p'<(1+1/p')^{1+p'}$. Analogous arguing gives
that $C^{unc}(n,p;\delta)$ asymptotically behaves as
$$\left(\frac{4\left(\frac{1}{p'}-\frac{1}{n}\right)^{\frac{1}{p'}}}
{\left(\frac{1}{p'}+1-\frac{2}{n}\right)^{1+\frac{1}{p'}}}\right)^{\frac{n}{2}},$$
which shows that $C^{unc}(n,p;\delta)$ has exponential growth, since
$4^{p'}/p'>(1+1/p')^{1+p'}$. Using
$\lim_{n\to\infty}\left(H(2k)-H(k)\right)=\log 2$, we get that
$C_2(n;1)$ asymptotically behaves as $2^{-n/2}$ and
$C_{2}^{unc}(n;1/2)$ behaves as $2^{n/2}$.\\

\textbf{Acknowledgement.} This research was supported by the
\textit{Croatian Ministry of Science, Education and Sports} under
Research Grant 058-1170889-1050.

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\end{document}

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