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\begin{document}

\title[Optimal recovery and generalized Carlson inequality]{Optimal recovery and generalized Carlson inequality for weights with symmetry properties}
\author{K.~Yu.~Osipenko}

\address{Moscow State University, Moscow and Institute for Information Transmission Problems,
Russian Academy of Sciences, Moscow}
\email{kosipenko@yahoo.com}
\subjclass[2010]{41A65, 41A46, 49N30}
\keywords{Optimal recovery, Carlson type inequalities, Fourier transform, sharp constants}
\begin{abstract}
The paper concerns problems of the recovery of operators from noisy information in weighted $L_q$-spaces with homogeneous weights. A number of general theorems are proved and applied to finding exact constants in multidimensional Carlson type inequalities with several weights and problems of the recovery of differential operators from a noisy Fourier transform. In particular, optimal methods are obtained for the recovery of powers of generalized Laplace operators from a noisy Fourier transform in the $L_p$-metric.
\end{abstract}

\maketitle

\section{Introduction}

Let $T$ be a nonempty set, $\Sigma$ be the $\sigma$-algebra of subsets of $T$, and $\mu$ be a nonnegative $\sigma$-additive measure on $\Sigma$. We denote by $L_p(T,\Sigma,\mu)$ (or simply $\lp$) the set of all $\Sigma$-measurable functions with values in $\mathbb R$ or in $\mathbb C$ for which
\begin{align*}
\|x\cd\|_{\lp}&=\biggl(\int_T|x(t)|^p\,d\mu(t)\biggr)^{1/p}<\infty,\quad1\le p<\infty,\\
\|x\cd\|_{L_\infty(T,\mu)}&=\vraisup_{t\in T}|x(t)|<\infty,\quad p=\infty.
\end{align*}
If $T\subset\mathbb R^d$ and $d\mu=dt$, $t\in\mathbb R^d$, we put $L_p(T)=\lp$.

The Carlson inequality \cite{Ca}
\begin{equation*}
\|x(t)\|_{L_1(\mathbb R_+)}\le\sqrt\pi\|x(t)\|_{L_2(\mathbb R_+)}^{1/2}
\|tx(t)\|_{L_2(\mathbb R_+)}^{1/2},\quad\mathbb R_+=[0,+\infty),
\end{equation*}
was generalized by many authors (see \cite{Le}, \cite{An}, \cite{B}, \cite{Os}, \cite{Os21}). In \cite{Os} we found sharp constants for inequalities of the form
\begin{equation*}
\|w\cd x\cd\|_{\lqq}\le K\|w_0\cd x\cd\|_{\lp)}^\gamma\|w_1\cd x\cd\|_{\lr}^{1-\gamma},
\end{equation*}
where $T$ is a cone in a linear space, $w\cd$, $w_0\cd$, and $w_1\cd$ are homogenous functions and $1\le q<p,r<\infty$ (for $T=\mathbb R^d$ the sharp inequality was obtained in \cite{B}). This problem is closely related with the following extremal problem
$$\|w\cd x\cd\|_{\lqq}\to\max,\quad\|w_0\cd x\cd\|_{\lp)}\le\delta,\quad
\|w_1\cd x\cd\|_{\lr}\le1,$$
where $\delta>0$. In this paper we study the extremal problem
\begin{multline}\label{ex1}
\|w\cd x\cd\|_{\lqq}\to\max,\quad\|w_0\cd x\cd\|_{\lp)}\le\delta,\\\|w_j\cd x\cd\|_{\lr}\le1,\ j=1,\ldots,n,
\end{multline}
where $w\cd$, $w_0\cd$, and $w_j\cd$, $j=1,\ldots,n$, are homogenous functions with some symmetry properties. Using the solution of this problem we obtain the sharp constant $K$ for the inequality
\begin{equation*}
\|w\cd x\cd\|_{\lqq}\le K\|w_0\cd x\cd\|_{\lp)}^\gamma\left(\max_{1\le j\le n}\|\omega_j\cd x\cd\|_{\lr}\right)^{1-\gamma}.
\end{equation*}
In particular, we find the sharp constant for the inequality
\begin{equation*}
\|w\cd x\cd\|_{L_q(\mathbb R_+^d)}\le C\|w_0\cd x\cd\|_{L_p(\mathbb R_+^d)}^{p\alpha}\left(\max_{1\le j\le d}\|\omega_j\cd x\cd\|_{L_r(\mathbb R_+^d)}\right)^{r\beta},
\end{equation*}
where $w(t)=(t_1^2+\ldots+t_d^2)^{\theta/2}$, $w_0(t)=(t_1^2+\ldots+t_d^2)^{\theta_0/2}$,
$w_j(t)=t_j^{\theta_1}$, $j=1,\ldots,d$, $\theta=d(1-1/q)$, $\theta_0=d-(\lambda+d)/p$, $\theta_1=d+(\mu-d)/r$,
\begin{equation*}
\alpha=\frac\mu{p\mu+r\lambda},\quad\beta=\frac\lambda{p\mu+r\lambda},\quad\lambda,\mu>0,
\end{equation*}
and $(p,q,r)\in P\cup P_1\cup P_2$, where
\begin{multline*}
P=\{\,(p,q,r):1\le q<p,r\,\},\quad P_1=\{\,(p,q,r):1\le q=r<p\,\},\\ P_2=\{\,(p,q,r):1\le q=p<r\,\}.
\end{multline*}
For $d=1$, $q=1$, and $(p,1,r)\in P$ this result was proved in \cite{Le} (see also \cite{B}).

It is appeared that the value of \eqref{ex1} is the error of optimal recovery of the operator $\Lambda x\cd=w\cd x\cd$ on the class of functions $x\cd$ such that $\|w_j\cd x\cd\|_{\lr}\le1$, $j=1,\ldots,n$, by the information about the function $w_0\cd x\cd$ given with the error $\delta$ in $L_p$-norm. Therefore, in section \ref{main} we begin with the setting of optimal recovery problem  and then in section \ref{res} we prove some general theorems. In section \ref{hom} we consider the case when weights are homogeneous in a cone of linear space and section \ref{rd} is devoted to the case of $\mathbb R^d$. In section \ref{dif} the results obtained are applied to optimal recovery and sharp inequalities of differential operators defined by Fourier transforms.

\section{General setting}\label{main}

Let $T_0$ is not empty $\mu$-measurable subset of $T$. Put
\begin{gather*}
\mathcal W=\{\,x\cd:x\cd\in\lpn,\ \|\varphi_j\cd x\cd\|_{\lr}<\infty,\ j=1,\ldots,n\,\},\\
W=\{\,x\cd\in\mathcal W:\|\varphi_j\cd x\cd\|_{\lr}\le1,\ j=1,\ldots,n\,\},
\end{gather*}
where $1\le p,r\le\infty$, and $\varphi_j\cd$ is a measurable function on $T$. Consider the problem of recovery of operator $\Lambda\colon\mathcal W\to L_q(T,\mu)$, $1\le q\le\infty$, defined by equality $\Lambda x\cd=\psi\cd x\cd$, where $\psi\cd$ is a measurable function on $T$, on the class $W$ by the information about functions $x\cd\in W$ given inaccurately (we assume that $\psi\cd$ and $\varphi_j\cd$, $j=1,\ldots,n$, such that $\Lambda$ maps $\mathcal W$ to $\lqq$). More precisely, we assume that for any function $x\cd\in W$ we know $y\cd\in\lpn$ such that $\|x\cd-y\cd\|_{\lpn}\le\delta$, $\delta>0$. We want to approximate the value $\Lambda x\cd$ knowing $y\cd$. As recovery methods we consider all possible mappings $m\colon\lpn\to\lqq$.
The error of a~method~$m$ is defined as
\begin{equation*}
e(p,q,r,m)=\sup_{\substack{x\cd\in W,\ y\cd\in\lpn\\\|x\cd-y\cd\|_{\lpn}\le\delta}}\|\Lambda x\cd-m(y)\cd\|_{\lqq}.
\end{equation*}

The quantity
\begin{equation}\label{Ep}
E(p,q,r)=\inf_{m\colon\lpn\to\lqq}e(p,q,r,m)
\end{equation}
is known as the optimal recovery error, and a method on which this infimum is attained is called optimal. Various settings of optimal recovery theory and examples of such problems may be found in \cite{MR}, \cite{TW}, \cite{Pl}, \cite{Osb}, \cite{Osb22}.

For the lower bound of $E(p,q,r)$ we use the following result which was proved (in more or less general forms) in many papers (see, for example, \cite{Os14}).

\begin{lemma}\label{L1}
\begin{equation}\label{lb}
E(p,q,r)\ge\sup_{\substack{x\cd\in W\\\|x\cd\|_{\lpn}\le\delta}}\|\Lambda x\cd\|_{\lqq}.
\end{equation}
\end{lemma}

\section{Main results}\label{res}

Set
\begin{equation*}
\chi_0(t)=\begin{cases}
1,&t\in T_0,
\\
0,&t\notin T_0,
\end{cases}\quad\sigma_r(t)=\sum_{j=1}^n\lambda_j|\varphi_j(t)|^r.
\end{equation*}

\begin{theorem}\label{T1}
Let $1\le q<p,r$, $\lambda_j\ge0$, $j=0,1,\ldots,n$, $\lambda_0+\sigma_r(t)\ne0$ for almost all $t\in T_0$, $\sigma_r(t)\ne0$ for almost all $t\in T\setminus T_0$, $\wx(t)\ge0$ be a solution of equation
\begin{equation}\label{ko1}
-q|\psi(t)|^q+p\lambda_0x^{p-q}(t)\chi_0(t)+r\sigma_r(t)x^{r-q}(t)=0,
\end{equation}
$\ov\lambda$ such that
\begin{multline}\label{nez}
\iTn\wx^p(t)\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)|^r\wx^r(t)\,d\mu(t)\le1,\ j=1,\ldots,n,\\
\lambda_0\biggl(\iTn\wx^p(t)\,d\mu(t)-\delta^p\biggr)=0,\quad
\lambda_j\biggr(\iT|\varphi_j(t)|^r\wx^r(t)\,d\mu(t)-1\biggr)=0,\ j=1,\ldots,n.
\end{multline}
Then
\begin{equation}\label{Epqr}
E(p,q,r)=\biggl(q^{-1}p\lambda_0\delta^p+q^{-1}r\sum_{j=1}^n\lambda_j\biggr)^{1/q},
\end{equation}
and the method
\begin{equation}\label{met}
\wm(y)(t)=\begin{cases}q^{-1}p\lambda_0\wx^{p-q}(t)|\psi(t)|^{-q}
\psi(t)y(t),&t\in T_0,\ \psi(t)\ne0,\\
0,&\mbox{otherwise},\end{cases}
\end{equation}
is optimal recovery method.
\end{theorem}

To prove this theorem we need some preliminary results. The first one is actually a sufficient condition in the Kuhn-Tucker theorem (the only difference is that we do not require convexity of functions).

Let $f_j\colon A\to\mathbb R$, $j=0,1,\ldots,k$, be functions defined on some set $A$. Consider the extremal problem
\begin{equation}\label{ex0}
f_0(x)\to\max,\quad f_j(x)\le0,\quad j=1,\ldots,k,\quad x\in A,
\end{equation}
and write down its Lagrange function
\begin{equation*}
\LL(x,\lambda)=-f_0(x)+\sum_{j=1}^k\lambda_jf_j(x),\quad
\lambda=(\lambda_1,\ldots,\lambda_k).
\end{equation*}

\begin{lemma}\label{L2}
Assume that there exist $\wl_j\ge0$, $j=1,\ldots,k$, and an element $\wx\in A$,
admissible for problem \eqref{ex0}, such that
\begin{align*}
(a)&\quad\min_{x\in A}\LL(x,\wl)=\LL(\wx,\wl),\quad\wl=(\wl_1,\ldots,\wl_k),\\
(b)&\quad\wl_jf_j(\wx)=0,\ j=1,\ldots,k.
\end{align*}
Then $\wx$ is an extremal element for problem \eqref{ex0}.
\end{lemma}

\begin{proof}
For any $x$ admissible for problem \eqref{ex0} we have
\begin{equation*}
-f_0(x)\ge\LL(x,\wl)\ge\LL(\wx,\wl)=-f_0(\wx).
\end{equation*}
\end{proof}

Put
$$F(u,v,\alpha)=-((1-\alpha)u+\alpha v)^q+av^p+bu^r,\quad u,v\ge0,\quad\alpha\in[0,1],$$
where $a,b\ge0$, and $1\le p,q,r<\infty$.

\begin{lemma}[\cite{Os}]\label{L3}
For all $a,b\ge0$, $a+b>0$, and all $1\le q<p,r<\infty$, there exists the unique solution $\wu>0$ of the equation
\begin{equation*}
-q+pau^{p-q}+rbu^{r-q}=0.
\end{equation*}
Moreover, for all $u,v\ge0$ and $\alpha=q^{-1}pa\wu^{p-q}=1-q^{-1}rb\wu^{r-q}$
\begin{equation*}
F(\wu,\wu,\alpha)\le F(u,v,\alpha).
\end{equation*}
In particular, for all $u\ge0$
\begin{equation*}
-\wu^q+a\wu^p+b\wu^r\le-u^q+au^p+bu^r.
\end{equation*}
\end{lemma}

\begin{proof}[Proof of Theorem~\ref{T1}]

1. Lower estimate. The extremal problem on the right-hand side of \eqref{lb} (for convenience, we raise the quantity to be maximized to the $q$-th power) is as follows:
\begin{multline}\label{ex}
\iT|\psi(t)x(t)|^q\,d\mu(t)\to\max,\quad\iTn|x(t)|^p\,d\mu(t)\le\delta^p,\\
\iT|\varphi_j(t)x(t)|^r\,d\mu(t)\le1,\ j=1,\ldots,n.
\end{multline}
If $t\in T$ such that $\psi(t)=0$, then evidently $\wx(t)=0$. If $\psi(t)\ne0$ we obtain by Lemma~\ref{L3} that that there is the unique solution $\wx(t)$ of \eqref{ko1}. It follows by \eqref{nez} that $\wx\cd$ is admissible function for problem \eqref{ex}. Therefore, by \eqref{lb} we obtain
\begin{equation*}
E(p,q,r)\ge\biggl(\iT|\psi(t)|^q\wx^q(t)\,d\mu(t)\biggr)^{1/q}.
\end{equation*}
From \eqref{ko1} we have
\begin{equation*}
|\psi(t)|^q\wx^q(t)=q^{-1}p\lambda_0\wx^p(t)\chi_0(t)+q^{-1}r\sigma_r(t)\wx^r(t).
\end{equation*}
Integrating this equality over the set $T$, we obtain
\begin{equation*}
\iT|\psi(t)|^q\wx^q(t)\,d\mu(t)=q^{-1}p\lambda_0\delta^p+q^{-1}r\sum_{j=1}^n\lambda_j.
\end{equation*}
Thus,
\begin{equation*}
E(p,q,r)\ge\biggl(q^{-1}p\lambda_0\delta^p+q^{-1}r\sum_{j=1}^n\lambda_j\biggr)^{1/q}.
\end{equation*}

2. Upper estimate. To estimate the error of method \eqref{met} we need to find the value of the extremal problem:
\begin{multline}\label{La}
\iTn|\psi(t)x(t)-\psi(t)\alpha(t)y(t)|^q\,d\mu(t)+\int_{T\setminus T_0}|\psi(t)x(t)|^q\,d\mu(t)\to\max,\\\iTn|x(t)-y(t)|^p\,d\mu(t)\le
\delta^p,\quad\iT|\varphi_j(t)x(t)|^r\,d\mu(t)\le1,\ j=1,\ldots,n,
\end{multline}
where
\begin{equation*}
\alpha(t)=\begin{cases}q^{-1}p\lambda_0\wx^{p-q}(t)|\psi(t)|^{-q},&t\in T_0,\ \psi(t)\ne0,\\
0,&\mbox{otherwise}.\end{cases}
\end{equation*}
Put
\begin{equation*}
z(t)=\begin{cases}x(t)-y(t),&t\in T_0,\\
0,&t\in T\setminus T_0.\end{cases}
\end{equation*}
Then \eqref{La} may be rewritten as follows:
\begin{multline*}
\iT|\psi(t)|^q|(1-\alpha(t))x(t)+\alpha(t)z(t)|^q\,d\mu(t)\to\max,\\
\iTn|z(t)|^p\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)x(t)|^r\,d\mu(t)\le1,\ j=1,\ldots,n.
\end{multline*}
The value of this problem does not exceed the value of the problem
\begin{multline}\label{uv}
\iT|\psi(t)|^q((1-\alpha(t))u(t)+\alpha(t)v(t))^q\,d\mu(t)\to\max,\\\iTn v^p(t)\,d\mu(t)\le\delta^p,\quad
\iT|\varphi_j(t)|^ru^r(t)\,d\mu(t)\le1,\ j=1,\ldots,n,\\ u(t)\ge0,\ v(t)\ge0\quad\mbox{for almost all }\ t\in T.
\end{multline}
The Lagrange function for this problem is
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)=\iT L(t,u(t),v(t),\ov\lambda)\,d\mu(t),
\end{equation*}
where
\begin{equation*}
L(t,u,v,\ov\lambda)=-|\psi(t)|^q((1-\alpha(t))u+\alpha(t)v)^q
+\lambda_0v^p\chi_0(t)+\sigma_r(t)u^r.
\end{equation*}
By Lemma~\ref{L3} we have
\begin{equation*}
L(t,\wx(t),\wx(t),\ov\lambda)\le L(t,u(t),v(t),\ov\lambda).
\end{equation*}
Thus,
\begin{equation*}
\LL(\wx\cd,\wx\cd,\ov\lambda)\le\LL(u\cd,v\cd,\ov\lambda).
\end{equation*}
It follows by Lemma~\ref{L2} that functions $u\cd=v\cd=\wx\cd$ are extremal in \eqref{uv}. Consequently,
$$e(p,q,r,\wm)\le\biggl(\iT|\psi(t)|^q\wx^q(t)\,d\mu(t)\biggr)^{1/q}
=\biggl(q^{-1}p\lambda_0\delta^p+q^{-1}r\sum_{j=1}^n\lambda_j\biggr)^{1/q}\le E(p,q,r).$$
It means that method \eqref{met} is optimal and equality \eqref{Epqr} holds.
\end{proof}

Denote $a_+=\max\{a,0\}$.

\begin{theorem}\label{T2}
Let $1\le q=r<p$, $\lambda_0>0$, $\lambda_j\ge0$, $j=1,\ldots,n$,
\begin{equation}\label{xT2}
\wx(t)=\begin{cases}\displaystyle\left(\dfrac q{p\lambda_0}\left(|\psi(t)|^q-\sigma_q(t)
\right)_+\right)^{\frac1{p-q}},&t\in T_0,\\
0,&t\notin T_0,\end{cases}
\end{equation}
$\ov\lambda$ satisfies conditions \eqref{nez}, and $|\psi(t)|^q-\sigma_q(t)\le0$ for almost all $t\notin T_0$. Then
\begin{equation}\label{Eq=r}
E(p,q,q)=\biggl(q^{-1}p\lambda_0\delta^p+\sum_{j=1}^n\lambda_j\biggr)^{1/q},
\end{equation}
and the method
\begin{equation}\label{met1}
\wm(y)(t)=\begin{cases}\left(1-|\psi(t)|^{-q}\sigma_q(t)\right)_+\psi(t)y(t),&t\in T_0,\ \psi(t)\ne0,\\
0,&\mbox{othervise},\end{cases}
\end{equation}
is optimal.
\end{theorem}

\begin{proof}

1. Lower estimate. It follows by \eqref{nez} that $\wx\cd$ is admissible function for extremal problem in the right-hand side of \eqref{lb}. Therefore,
\begin{equation*}
E(p,q,q)\ge\biggl(\iT|\psi(t)|^q\wx^q(t)\,d\mu(t)\biggr)^{1/q}.
\end{equation*}
From the definition of $\wx\cd$ we have
\begin{equation*}
|\psi(t)|^q\wx^q(t)=q^{-1}p\lambda_0\wx^p(t)\chi_0(t)+\sigma_q(t)\wx^q(t).
\end{equation*}
Integrating this equality, we obtain
\begin{equation*}
\iT|\psi(t)|^q\wx^q(t)\,d\mu(t)=q^{-1}p\lambda_0\delta^p+\sum_{j=1}^n\lambda_j.
\end{equation*}
Thus,
\begin{equation*}
E(p,q,q)\ge\biggl(q^{-1}p\lambda_0\delta^p+\sum_{j=1}^n\lambda_j\biggr)^{1/q}.
\end{equation*}

2. Upper estimate. Put
\begin{equation*}
\alpha(t)=\begin{cases}\left(1-|\psi(t)|^{-q}\sigma_q(t)\right)_+ ,&t\in T_0,\ \psi(t)\ne0,\\
0,&\mbox{othervise},\end{cases}
\end{equation*}
To estimate the error of method \eqref{met1} we need to find the value of the extremal problem:
\begin{multline*}
\iTn|\psi(t)|^q|x(t)-\alpha(t)y(t)|^q\,d\mu(t)+\int_{T\setminus T_0}|\psi(t)x(t)|^q\,d\mu(t)\to\max,\\
\iTn|x(t)-y(t)|^p\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)x(t)|^q\,d\mu(t)\le1,\ j=1,\ldots,n.
\end{multline*}
Putting $z\cd=x\cd-y\cd$ this problem may be rewritten in the following form
\begin{multline*}
\iTn|\psi(t)|^q|(1-\alpha(t))x(t)+\alpha(t)z(t)|^q\,d\mu(t)
+\int_{T\setminus T_0}|\psi(t)x(t)|^q\,d\mu(t)\to\max,\\
\iTn|z(t)|^p\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)x(t)|^q\,d\mu(t)\le1,\ j=1,\ldots,n.
\end{multline*}
The value of this problem evidently coincides with the value of the problem
\begin{multline}\label{uv1}
\iT|\psi(t)|^q((1-\alpha(t))v(t)+\alpha(t)u(t))^q\,d\mu(t)\to\max,\\
\iTn u^p(t)\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)|^qv^q(t)\,d\mu(t)\le1,\ j=1,\ldots,n,\\ u(t),v(t)\ge0,\ \mbox{for almost all }t\in T.
\end{multline}

The Lagrange function of \eqref{uv1} has the form
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)=\iT L(u(t),v(t),\ov\lambda)\,d\mu(t),
\end{equation*}
where
\begin{equation*}
L(u,v,\ov\lambda)=\begin{cases}-|\psi(t)|^q((1-\alpha(t))v+\alpha(t)u)^q+\lambda_0u^p+
\sigma_q(t)v^q,&t\in T_0,\\
-|\psi(t)|^qv^q+\sigma_q(t)v^q,&t\notin T_0.\end{cases}
\end{equation*}
If $\alpha(t)>0$, then
\begin{equation*}
\frac{\partial L}{\partial v}=q(v^{q-1}-((1-\alpha(t))v+\alpha(t)u)^{q-1})\sigma_q(t).
\end{equation*}
Therefore, for $\alpha(t)>0$ and any $u>0$, the function $L(u,v,\ov\lambda)$, $v\in(0,+\infty)$, reaches a minimum at $v=u$. Set $T_0'=\{\,t\in T_0:\alpha(t)>0\,\}$. We have
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)\ge\int_{T_0'}L(u\cd,u\cd,\ov\lambda)\,d\mu(t).
\end{equation*}
It is easily checked that for $t\in T_0'$ for all $u(t)\ge0$
\begin{equation*}
L(u\cd,u\cd,\ov\lambda)\ge L(\wx\cd,\wx\cd,\ov\lambda).
\end{equation*}
Consequently,
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)\ge\int_{T_0'}L(\wx\cd,\wx\cd,\ov\lambda)\,d\mu(t)=
\LL(\wx\cd,\wx\cd,\ov\lambda).
\end{equation*}
Taking into account \eqref{nez} we obtain by Lemma \ref{L2} that $u\cd=v\cd=\wx\cd$ are extremal functions in \eqref{uv1}. Thus,
\begin{equation*}
e^q(p,q,q,\wm)=\iT|\psi(t)\wx(t)|^q\,d\mu(t)=q^{-1}p\lambda_0\delta^p+\sum_{j=1}^n\lambda_j\le E^q(p,q,q).
\end{equation*}
It means that the method $\wm$ is optimal and the optimal recovery error is as stated.
\end{proof}

\begin{theorem}\label{TT3}
Let $1\le q=p<r$, $\lambda_0>0$, $\lambda_j\ge0$, $j=1,\ldots,n$,
$\sigma_r(t)\ne0$ for almost all $t\in T$,
\begin{equation}\label{xT3}
\wx(t)=\begin{cases}\left(pr^{-1}\sigma_r^{-1}(t)(|\psi(t)|^p-\lambda_0)_+\right)^{\frac1{r-p}},
&t\in T_0,\\
\left(pr^{-1}\sigma_r^{-1}(t)|\psi(t)|^p\right)^{\frac1{r-p}},&t\in T \setminus T_0,\end{cases}
\end{equation}
and $\ov\lambda$ satisfies conditions \eqref{nez}. Then
\begin{equation}\label{Ep=q}
E(p,p,r)=\biggl(\lambda_0\delta^p+\frac rp\sum_{j=1}^n\lambda_j\biggr)^{1/p},
\end{equation}
and the method
\begin{equation}\label{met11}
\wm(y)(t)=\begin{cases}\alpha(t)\psi(t)y(t),&t\in T_0,\\
0,&t\in T\setminus T_0,\end{cases}\end{equation}
where
\begin{equation*}
\alpha(t)=\begin{cases}\min\left\{1,\lambda_0|\psi(t)|^{-p}\right\},&t\in T_0,\ \psi(t)\ne0,\\
0,&\mbox{othervise},\end{cases}
\end{equation*}
is optimal.
\end{theorem}

\begin{proof}
1. Lower estimate. By the definition of $\wx\cd$ we have
\begin{equation*}
|\psi(t)|^p\wx^p(t)=\lambda_0\wx^p(t)\chi_0(t)+\frac rp\sigma_r(t)\wx^r(t).
\end{equation*}
Using the similar arguments as in the proof of Theorem \ref{T1} we obtain
\begin{equation*}
E(p,p,r)\ge\biggl(\iT|\psi(t)|^p\wx^p(t)\,d\mu(t)\biggr)^{1/p}=\biggl(\lambda_0\delta^p+\frac rp\sum_{j=1}^n\lambda_j\biggr)^{1/p}.
\end{equation*}

2. Upper estimate. To estimate the error of method \eqref{met11} we need to find the value of the following extremal problem:
\begin{multline*}
\iTn|\psi(t)|^p|x(t)-\alpha(t)y(t)|^p\,d\mu(t)+\int_{T\setminus T_0}|\psi(t)x(t)|^p\,d\mu(t)\to\max,\\
\iTn|x(t)-y(t)|^p\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)x(t)|^r\,d\mu(t)\le1,\ j=1,\ldots,n.
\end{multline*}
Putting $z\cd=x\cd-y\cd$ this problem may be rewritten in the form
\begin{multline*}
\iTn|\psi(t)|^p|(1-\alpha(t))x(t)+\alpha(t)z(t)|^p\,d\mu(t)
+\int_{T\setminus T_0}|\psi(t)x(t)|^p\,d\mu(t)\to\max,\\
\iTn|z(t)|^p\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)x(t)|^r\,d\mu(t)\le1,\ j=1,\ldots,n.
\end{multline*}
The value of this problem evidently coincides with the value of the problem
\begin{multline}\label{uv11}
\iT|\psi(t)|^p((1-\alpha(t))v(t)+\alpha(t)u(t))^p\,d\mu(t)\to\max,\\
\iTn u^p(t)\,d\mu(t)\le\delta^p,\quad\iT|\varphi_j(t)|^rv^r(t)\,d\mu(t)\le1,\ j=1,\ldots,n,\\ u(t),v(t)\ge0,\ \mbox{for almost all }t\in T.
\end{multline}

The Lagrange function of \eqref{uv11} has the form
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)=\iT L(u(t),v(t),\ov\lambda)\,d\mu(t),
\end{equation*}
where
\begin{equation*}
L(u,v,\ov\lambda)=\begin{cases}-|\psi(t)|^p((1-\alpha(t))v+\alpha(t)u)^p+\lambda_0u^p+
\sigma_r(t)v^r,&t\in T_0,\\
-|\psi(t)|^pv^p+\sigma_r(t)v^r,&t\in T\setminus T_0.\end{cases}
\end{equation*}
For $t\in T_0$ and $|\psi(t)|^p>\lambda_0$ we have
\begin{equation*}
\frac{\partial L}{\partial u}=p\lambda_0(u^{p-1}-((1-\alpha(t))v+\alpha(t)u)^{p-1}).
\end{equation*}
Consequently, in this case for any $v>0$ the function $L(u,v,\ov\lambda)$, $v\in(0,+\infty)$, reaches a minimum at $v=u$. If $t\in T_0$, $0<|\psi(t)|^p\le\lambda_0$, then $\alpha(t)=1$ and $L(u,v,\ov\lambda)\ge0$. If $t\in T_0$ and $\psi(t)=0$, then again $L(u,v,\ov\lambda)\ge0$. Set
$T_1=\{t\in T_0:|\psi(t)|^p>\lambda_0\}$. Then for all $u(t),v(t)\ge0$ we have
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)\ge\iTnn L(v\cd,v\cd,\ov\lambda)\,d\mu(t)+\int_{T\setminus T_0}
L(v\cd,v\cd,\ov\lambda)\,d\mu(t).
\end{equation*}
It is easy to check that for all $v(t)\ge0$
\begin{equation*}
L(v\cd,v\cd,\ov\lambda)\ge L(\wx\cd,\wx\cd,\ov\lambda).
\end{equation*}
Therefore,
\begin{equation*}
\LL(u\cd,v\cd,\ov\lambda)\ge\int_{T_1\cup(T\setminus T_0)}L(\wx\cd,\wx\cd,\ov\lambda)\,d\mu(t)=
\LL(\wx\cd,\wx\cd,\ov\lambda).
\end{equation*}
Taking into account \eqref{nez} we obtain by Lemma \ref{L2} that $u\cd=v\cd=\wx\cd$ are extremal functions in \eqref{uv11}. Consequently,
\begin{equation*}
e^p(p,p,r,\wm)=\iT|\psi(t)\wx(t)|^q\,d\mu(t)=\lambda_0\delta^p+\frac rp\sum_{j=1}^n\lambda_j\le E^p(p,p,r).
\end{equation*}
It means that the method $\wm$ is optimal and the optimal recovery error is as stated.
\end{proof}

Note that if conditions of Theorems \ref{T1}, \ref{T2}, and \ref{TT3} are fulfilled, then we have
\begin{equation}\label{Edv}
E(p,q,r)=\sup_{\substack{\|x\cd\|_{\lpn}\le\delta\\\|\varphi_j\cd x\cd\|_{\lr}\le1,\ j=1,\ldots,n}}\|\psi\cd x\cd\|_{\lqq}.
\end{equation}

\section{The case of homogenous weight functions}
\label{hom}

Let $T$ be a cone in a linear space, $T_0=T$, $\mu\cd$ be a homogenous measure of degree $d$, $|\psi\cd|$ be homogenous function of degree $\eta$, $|\varphi_j\cd|$, $j=1,\ldots,n$, be homogenous functions of degrees $\nu$, $\psi(t)\ne0$ and $\sum_{j=1}^n|\varphi_j(t)|\ne0$ for almost all $t\in T$. Let assume, again, that $1\le p<q,r<\infty$. For $k\in[0,1)$ the function $k^{\frac1{p-q}}(1-k)^{-\frac1{r-q}}$ increases monotonically from $0$ to $+\infty$. Consequently, there exists $k\cd$ such that for almost all $t\in T$
\begin{equation}\label{kz}
\frac{k^{\frac1{p-q}}(t)}{(1-k(t))^{\frac1{r-q}}}=s_r^{-\frac1{r-q}}(t)|\psi(t)|
^{\frac{q(p-r)}{(p-q)(r-q)}},\quad s_r(t)=\sum_{j=1}^n|\varphi_j(t)|^r.
\end{equation}
Set
\begin{equation*}
k(t)=\begin{cases}\left(1-|\psi(t)|^{-q}s_q(t)\right)_+,&(p,q,r)\in P_1,\\\
\min\left\{1,|\psi(t)|^{-p}\right\}.&(p,q,r)\in P_2
\end{cases}
\end{equation*}

\begin{theorem}\label{S22}
Let $(p,q,r)\in P\cup P_1\cup P_2$ and $\nu+d(1/r-1/p)\ne0$.
Assume that for $(p,q,r)\in P\cup P_1$
\begin{align*}
I_1&=\iT|\psi(t)|^{\frac{qp}{p-q}}k^{\frac p{p-q}}(t)\,d\mu(t)<\infty,\\
I_{j+1}&=\iT|\psi(t)|^{\frac{qr}{p-q}}|\varphi_j(t)|^rk^{\frac r{p-q}}(z)\,d\mu(t)<\infty,\ j=1,\ldots,n,
\end{align*}
and for $(p,q,r)\in P_2$
\begin{align*}
I_1&=\iT\left(s_r^{-1}(t)(|\psi(t)|^p-1)_+\right)^{\frac p{r-p}}\,d\mu(t)<\infty,\\
I_{j+1}&=\iT|\varphi_j(t)|^r\left(s_r^{-1}(t)(|\psi(t)|^p-1)_+
\right)^{\frac r{r-p}}\,d\mu(t)<\infty,\ j=1,\ldots,n.
\end{align*}
Moreover, assume that $I_2=\ldots=I_{n+1}$. Then
\begin{equation}\label{EEpqr}
E(p,q,r)=\delta^\gamma
I_1^{-\gamma/p}I_2^{-(1-\gamma)/r}(I_1+nI_2)^{1/q},
\end{equation}
where
\begin{equation}\label{gam}
\gamma=\frac{\nu-\eta-d(1/q-1/r)}{\nu+d(1/r-1/p)}.
\end{equation}
The method
\begin{equation*}
\wm(y)(t)=k(\xi t)\psi(t)y(t),
\end{equation*}
where
\begin{equation}\label{yy}
\xi=\left(\delta I_1^{-1/p}I_2^{1/r}\right)^{\frac1{\nu+d(1/r-1/p)}},
\end{equation}
is optimal.
\end{theorem}

\begin{proof}
1. Let $(p,q,r)\in P$. Put
\begin{equation*}
\wx(t)=\left(\frac{q|\psi(t)|^q}{p\lambda_0}\right)^{\frac1{p-q}}k^{\frac1{p-q}}(\xi t),
\end{equation*}
where $\lambda_0$ will be specified later. We have
\begin{equation}\label{eq11}
p\lambda_0\wx^{p-q}(t)=q|\psi(t)|^qk(\xi t)
\end{equation}
and
\begin{equation*}
rc_r(t)\wx^{r-q}(t)=rc_r(t)\left(\frac{q|\psi(t)|^q}{p\lambda_0}\right)^{\frac{r-q}{p-q}}k^
{\frac{r-q}{p-q}}(\xi t).
\end{equation*}
Since $|\psi\cd|$ and $|\varphi_j\cd|$, $j=1,\ldots,n$, are homogenous it follows by \eqref{kz} that
\begin{equation*}
k^{\frac{r-q}{p-q}}(\xi t)=\frac{|\psi(\xi t)|^{\frac{q(p-r)}{p-q}}
}{c_r(\xi t)}(1-k(\xi t))
=\xi^{\eta\frac{q(p-r)}{p-q}-\nu r}\frac{|\psi(t)|^{\frac{q(p-r)}{p-q}}
}{c_r(t)}(1-k(\xi t)).
\end{equation*}
Thus,
\begin{equation*}
rc_r(t)\wx^{r-q}(t)=r\left(\frac q{p\lambda_0}\right)^{\frac{r-q}{p-q}}\xi^{\eta\frac{q(p-r)}{p-q}-\nu r}|\psi(t)|^q(1-k(\xi t)).
\end{equation*}
Put
\begin{equation}\label{l2}
\lambda=\frac qr\left(\frac q{p\lambda_0}\right)^{-\frac{r-q}{p-q}}\xi^{-\eta\frac{q(p-r)}{p-q}+\nu r}.
\end{equation}
Then
\begin{equation}\label{eq12}
r\lambda c_r(t)\wx^{r-q}(t)=q|\psi(t)|^q(1-k(\xi t)).
\end{equation}
Taking the sum of \eqref{eq11} and \eqref{eq12}, we obtain
\begin{equation*}
p\lambda_0\wx^{p-q}(t)+r\lambda c_r(t)\wx^{r-q}(t)=q|\psi(t)|^q.
\end{equation*}
It means that $\wx\cd$ satisfies \eqref{ko1} for $\lambda_1=\ldots=\lambda_n=\lambda$.

Now we show that for
\begin{equation}\label{l1}
\lambda_0=\frac qpI_1^{\frac{p-q}p}\xi^{-\eta q-d\frac{p-q}p}\delta^{q-p}
\end{equation}
the equalities
\begin{equation*}
\iT\wx^p(t)\,d\mu(t)=\delta^p,\quad\iT|\varphi_j(t)|^r\wx^r(t)\,d\mu(t)
=1,\ j=1,\ldots,n,
\end{equation*}
hold. In view of the definition of $\wx\cd$ we need to check that
\begin{align*}
\iT\left(\frac{q|\psi(t)|^q}{p\lambda_0}\right)^{\frac p{p-q}}k^{\frac p{p-q}}(\xi t)\,d\mu(t)&
=\delta^p,\\
\iT|\varphi_j(t)|^r\left(\frac{q|\psi(t)|^q}{p\lambda_0}\right)^{\frac r{p-q}}k^{\frac r{p-q}}(\xi t)\,d\mu(t)&
=1,\ j=1,\ldots,n.
\end{align*}
Changing $z=\xi t$ and taking into account that functions $|\psi\cd|$, $|\varphi_j\cd|$, $j=1,\ldots,n$, with the measure $\mu\cd$ are homogenous, we obtain
%\begin{multline*}
$$\left(\frac q{p\lambda_0}\right)^{\frac p{p-q}}I_1
=\delta^p\xi^{\frac{\eta qp}{p-q}+d},\quad
\left(\frac q{p\lambda_0}\right)^{\frac r{p-q}}I_{j+1}
=\xi^{\frac{\eta qr}{p-q}+\nu r+d},\ j=1,\ldots,n.$$
The validity of these equalities immediately follows from the definitions of $\lambda_0$ and $\xi$.

It follows by Theorem~\ref{T1}, \eqref{l1}, \eqref{l2}, and \eqref{yy} that
\begin{multline*}
E^q(p,q,r)=\frac{p\lambda_0\delta^p+nr\lambda}q=I_1^{\frac{p-q}p}
\xi^{-\eta q-d\frac{p-q}p}\delta^q\\+n\left(\frac{p\lambda_1}q\right)^{\frac{r-q}{p-q}}
\xi^{\nu r-\eta\frac{q(p-r)}{p-q}}
=\delta^{q\gamma}I_1^{-q\gamma/p}I_2^{-q(1-\gamma)/r}(I_1+nI_2).
\end{multline*}
Moreover, the same theorem states that the method
\begin{equation*}
\wm(y)(t)=q^{-1}p\lambda_0\wx^{p-q}(t)|\psi(t)|^{-q}\psi(t)y(t)=
k(\xi t)\psi(t)y(t)
\end{equation*}
is optimal.

2. Let $(p,q,r)\in P_1$. We use Theorem~\ref{T2}. Consider the function $\wx\cd$ defined by \eqref{xT2} with $\lambda_1=\ldots=\lambda_n=\lambda$. Let us find $\lambda_0$ and $\lambda$ from the conditions
\begin{equation*}
\iT\wx^p(t)\,d\mu(t)=\delta^p,\quad\iT|\varphi_j(t)|^q\wx^q(t)\,d\mu(t)=1,\ j=1,\ldots,n.
\end{equation*}
Then we obtain
\begin{gather*}
\left(\frac q{p\lambda_0}\right)^{\frac p{p-q}}\iT\left(|\psi(t)|^q-\lambda s_q(t)\right)_+^{\frac p{p-q}}\,d\mu(t)=\delta^p,\\
\left(\frac q{p\lambda_0}\right)^{\frac q{p-q}}\iT|\varphi_j(t)|^q\left(|\psi(t)|^q-\lambda s_q(t)\right)_+^{\frac q{p-q}}\,d\mu(t)=1,\ j=1,\ldots,n.
\end{gather*}
Put $\lambda=a^{(\eta-\nu)q}$, $a>0$. Changing $t=az$, we obtain
$$\left(\frac q{p\lambda_0}\right)^{\frac p{p-q}}a^{d+\frac{pq\eta}{p-q}}I_1=\delta^p,\quad
\left(\frac q{p\lambda_0}\right)^{\frac q{p-q}}a^{d+q\nu+\frac{q^2\eta}{p-q}}I_{j+1}=1,\ j=1,\ldots,n.$$
It is easy to check that these equalities are fulfilled for
\begin{equation*}
a=(I_1^{1/p}I_2^{-1/q}\delta^{-1})^{\frac1{\nu+d(1/q-1/p)}},\quad
\lambda_0=\frac qpI_1I_2^{-1}\delta^{-p}(I_1^{-q/p}I_2\delta^q)^{\frac{\eta-\nu}{\nu+d(1/q-1/p)}}.
\end{equation*}
Substituting these values in \eqref{Eq=r} and \eqref{met1} we obtain the statement of the theorem in the case under consideration.

3. Let $(p,q,r)\in P_2$. Here we use Theorem~\ref{TT3}. Put $\lambda_1=\ldots=\lambda_n=\lambda$
in the definition of $\wx\cd$ (see \eqref{xT3}). We find $\lambda_0$ and $\lambda$ from the conditions
\begin{equation*}
\iT\wx^p(t)\,d\mu(t)=\delta^p,\quad\iT|\varphi_j(t)|^r\wx^r(t)\,d\mu(t)=1,\ j=1,\ldots,n.
\end{equation*}
We have
\begin{gather*}
\left(\frac p{r\lambda}\right)^{\frac p{r-p}}\iT\left(s_r^{-1}(t)(|\psi(t)|^p-\lambda_0)_+\right)^{\frac p{r-p}}\,d\mu(t)=\delta^p,\\
\left(\frac p{r\lambda}\right)^{\frac r{r-p}}\iT|\varphi_j(t)|^r\left(s_r^{-1}(t)(|\psi(t)|^p-\lambda_0)_+\right)^{\frac r{r-p}}\,d\mu(t)=1,\ j=1,\ldots,n.
\end{gather*}
Put $\lambda_0=a^{\eta p}$, $a>0$. Changing $t=az$, we obtain
\begin{gather*}
\left(\frac p{r\lambda}\right)^{\frac p{r-p}}a^{d+\frac{p^2\eta}{r-p}-\frac{pr\nu}{r-p}}I_1=\delta^p,\\
\left(\frac p{r\lambda}\right)^{\frac r{r-p}}a^{d+r\nu+\frac{pr\eta}{r-p}-\frac{r^2\nu}{r-p}}I_{j+1}=1,\ j=1,\ldots,n.
\end{gather*}
These equalities are valid for
\begin{align*}
a&=(I_1^{1/p}I_2^{-1/r}\delta^{-1})^{\frac1{\nu+d(1/r-1/p)}},\\
\lambda&=\frac prI_1^{r/p-1}\delta^{p-r}(I_1^{r/p}I_2^{-1}\delta^{-r})^{\frac{p\eta /r-\nu-d(1/r-1/p)}{\nu+d(1/r-1/p)}}.
\end{align*}
It remains to substitute these values into \eqref{Ep=q} and \eqref{met11}.
\end{proof}

\begin{corollary}\label{Sdva}
Assume that conditions of Theorem~\ref{S22} hold. Then for all $x\cd\ne0$ such that $x\cd\in\lp$ and $\varphi_j\cd x\cd\in\lr$, $j=1,\ldots,n$, the sharp inequality
\begin{equation}\label{ncar}
\|\psi\cd x\cd\|_{\lqq}\le C\|x\cd\|_{\lp}^\gamma\left(\max_{1\le j\le n}\|\varphi_j\cd x\cd\|_{\lr}\right)^{1-\gamma}
\end{equation}
holds, where
\begin{equation*}
C=I_1^{-\gamma/p}I_2^{-(1-\gamma)/r}(I_1+nI_2)^{1/q}.
\end{equation*}
\end{corollary}

\begin{proof}
Let $x\cd\in\lp$, $\|\varphi_j\cd x\cd\|_{\lr}<\infty$, $j=1,\ldots,n$ and $x\cd\ne0$. Put
\begin{equation*}
A=\max_{1\le j\le n}\|\varphi_j\cd x\cd\|_{\lr}.
\end{equation*}
Consider $\wx\cd=x\cd/A$. Put $\delta=\|\wx\cd\|_{\lpn}$. Then $\|\varphi_j\cd \wx\cd\|_{\lr}\le1$, $j=1,\ldots,n$. In view of \eqref{Edv} and Theorem~\ref{S22} we have
\begin{equation*}
\|\psi\cd\wx\cd\|_{\lqq}\le C\|\wx\cd\|_{\lp}^\gamma.
\end{equation*}
This implies \eqref{ncar}.

If there exists a $\wCC<C$ for which \eqref{ncar} holds, then
\begin{equation*}
E(p,q,r)=\sup_{\substack{\|x\cd\|_{\lp}\le\delta\\\|\varphi_j\cd x\cd\|_{\lr}\le1,\ j=1,\ldots,n}}\|\psi\cd x\cd\|_{\lqq}\le\wCC\delta^\gamma<C\delta^\gamma.
\end{equation*}
This contradicts with \eqref{EEpqr}.
\end{proof}

Let $|w\cd|$, $|w_0\cd|$ be homogenous functions of degrees $\theta$, $\theta_0$, respectively and $|w_j\cd|$, $j=1,\ldots,n$, be homogenous functions of degree $\theta_1$. We assume that $w(t),w_0(t)\ne0$ and $\sum_{j=1}^n|w_j(t)|\ne0$ for almost all $t\in T$.

For $(p,q,r)\in P$ we define $\wk\cd$ by the equality
\begin{equation*}
\frac{\wk^{\frac1{p-q}}(t)}{(1-\wk(t))^{\frac1{r-q}}}=\left|\frac{w_0(t)}{w(t)}\right|
^{\frac p{p-q}}\biggl(\sum_{j=1}^n\left|\frac{w_j(t)}{w(t)}\right|^r\biggr)^{-\frac1{r-q}}.
\end{equation*}
For $(p,q,r)\in P_1$ set
\begin{equation*}
\wk(t)=\biggl(1-|w(t)|^{-q}\sum_{j=1}^n|w_j(t)|^q\biggr)_+.
\end{equation*}
Put
\begin{equation}\label{wg}
\wt=\theta+d/q,\quad\wt_0=\theta_0+d/p,\quad\wt_1=\theta_1+d/r,\quad
\wg=\frac{\wt_1-\wt}{\wt_1-\wt_0}.
\end{equation}

\begin{corollary}\label{Sl3}
Let $(p,q,r)\in P\cup P_1\cup P_2$ and $\wt_0\ne\wt_1$. Assume that for $(p,q,r)\in P\cup P_1$
\begin{align*}
\wII_1&=\iT\left|\frac{w(t)}{w_0(t)}\right|^{\frac{qp}{p-q}}\wk^{\frac p{p-q}}(t)\,d\mu(t)<\infty,\\
\wII_{j+1}&=\iT\frac{|w(t)|^{\frac{qr}{p-q}}}{|w_0(t)|^{\frac{pr}{p-q}}}|w_j(t)|^r\wk^{\frac r{p-q}}(t)\,d\mu(t)<\infty,\ j=1,\ldots,n,
\end{align*}
and for $(p,q,r)\in P_2$
\begin{align*}
\wII_1&=\iT|w_0(t)|^p\left(\frac{(|w(t)|^p-|w_0(t)|^p)_+}{\sum_{k=1}^n|w_k(t)|^r}\right)^{\frac p{r-p}}\,d\mu(t)<\infty,\\
\wII_{j+1}&=\iT|w_j(t)|^r\left(\frac{(|w(t)|^p-|w_0(t)|^p)_+}{\sum_{k=1}^n|w_k(t)|^r}\right)^{\frac r{r-p}}\,d\mu(t)<\infty,\ j=1,\ldots,n.
\end{align*}
Moreover, assume that $\wII_2=\ldots=\wII_{n+1}$. Then for all $x\cd\ne0$ such that $w_0\cd x\cd\in L_p(T,\mu)$ and $w_j\cd x\cd\in L_r(T,\mu)$, $j=1,\ldots,n$, the sharp inequality
\begin{equation}\label{CCC}
\|w\cd x\cd\|_{\lqq}\le\wCC\|w_0\cd x\cd\|_{\lp}^{\wg}\left(\max_{1\le j\le n}\|\omega_j\cd x\cd\|_{\lr}\right)^{1-\wg}
\end{equation}
holds, where
\begin{equation*}
\wCC=\wII_1^{-\wg/p}\wII_2^{-(1-\wg)/r}(\wII_1+n\wII_2)^{1/q}.
\end{equation*}
\end{corollary}

\begin{proof}
Set
\begin{equation*}
\psi(t)=\frac{w(t)}{w_0(t)},\quad\varphi_j(t)=\frac{w_j(t)}{w_0(t)},\ j=1,\ldots,n.
\end{equation*}
Then $|\psi\cd|$ is a homogenous function of degree $\eta=\theta-\theta_0$ and $|\varphi_j\cd|$, $j=1,\ldots,n$, are homogenous functions of degrees $\nu=\theta_1-\theta_0$. The quantity $\gamma$ which was defined by \eqref{gam} has the following form:
\begin{equation*}
\wg=\frac{\wt_1-\wt}{\wt_1-\wt_0}.
\end{equation*}
It follows by Corollary~\ref{Sdva} that for all $y\cd\ne0$ such that $y\cd\in L_p(T,\mu)$ and $\varphi_j\cd y\cd\in L_r(T,\mu)$, $j=1,\ldots,n$, the sharp inequality
\begin{equation*}
\|\psi\cd y\cd\|_{\lqq}\le\wCC\|y\cd\|_{\lp}^{\wg}\left(\max_{1\le j\le n}\|\varphi_j\cd y\cd\|_{\lr}\right)^{1-\wg}.
\end{equation*}
holds.
Substituting $y\cd=w_0\cd x\cd$, we obtain \eqref{CCC}.
\end{proof}

\section{Homogenous weights in $\mathbb R^d$}\label{rd}

Let $T$ be a cone in $\mathbb R^d$, $d\mu(t)=dt$, $|\psi\cd|$ be homogenous function of degree $\eta$, $|\varphi_j\cd|$, $j=1,\ldots,n$, be homogenous functions of degrees $\nu$, $\psi(t)\ne0$ and $\sum_{j=1}^n|\varphi_j(t)|\ne0$ for almost all $t\in T$. Consider the polar transformation
\begin{equation*}
\arraycolsep=0.08em
\begin{array}{rcl}
t_1&=&\rho\cos\omega_1,\\
t_2&=&\rho\sin\omega_1\cos\omega_2,\\
\hdotsfor{3}\\
t_{d-1}&=&\rho\sin\o