an: Zbl 1200.47029
au: Jabbarzadeh, M.R.
ti: Conditional multipliers and essential norm of $uC_\varphi$ between $L^p$
    spaces.
la: EN
so: Banach J. Math. Anal. 4, No. 2, 158-168, electronic only (2010).
py: 2010
dt: J
cc: *47B20 47B38
ut: conditional expectation; multipliers; multiplication operators; weighted
    composition operators; essential norm
ab: Let $(X,\Sigma,\mu)$ be a $\sigma$-finite measure space. The space of
    $\Sigma$-measurable functions on $X$ is denoted by $L^0(\Sigma)$, and if
    $\le p<\infty$, then $L^p(\Sigma)$ denotes $\{[f]:\Vert [f]\Vert_p=\Vert
    p\Vert_p< \infty\}$, where $[f]$ is the equivalence class of functions which
    differ from $f$ on sets of measure $0$, and for $1\le p\le\infty$,
    $\Vert\cdot\Vert_p$ is the $L^p$-norm.
 
 In the paper, ${\Cal
    A}\subseteq\Sigma$ denotes a complete $\sigma$-finite sub-algebra of
    $\Sigma$, and the operator $E^{\Cal A}=E$ is said to be a conditional
    expectation operator if for $f\ge 0$, $f\in L^p(\Sigma)$, $E(f)$ is ${\Cal
    A}$-measurable and $\int_{\Cal A}E(f)\,d\mu= \int_{\Cal A}f\,d\mu$, whenever
    the integrals are finite. If $1\le p,q<\infty$, then the set of conditional
    multipliers $K_{p,q}= K_{p,q}({\Cal A},\Sigma)$ is defined to be $\{u\in
    L^0(\Sigma): uL^p({\Cal A})\subseteq L^q(\Sigma)\}$, $M_u: L^p({\Cal A})\to
    L^q(\Sigma)$ denotes the multiplication operator $M_u(f)= uf$, and if
    $\varphi: X\to X$ is a non-singular measurable transformation with measure
    $u\circ\varphi^{-1}$ which is absolutely continuous with respect to $\mu$,
    the derivative $d(p\circ\varphi^{-1})/d\mu$ is denoted by $h$. If
    $C_\varphi$ denotes the composition operator, $C_\varphi(f)(x)=
    f\circ\varphi(x)= f(\varphi(x))$, then $K^\varphi_{p,q}$ is defined to be
    $\{u\in L^0(\Sigma): u\text{\,Range}(C_\varphi)\subseteq L^q(\Sigma)\}$, and
    
 $$\Vert uC_\varphi f\Vert^q_q= \int_X S|f|^q\,d\mu,$$
  $1\le q<\infty$,
    where $S= hE^{\varphi^{-1}}(\Sigma)/(E^{\Cal A}(|u|^q))\circ\varphi^{-1}$.
    The results of this paper include:
 {\parindent=7mm
 \item{(1)} $(K_{p,q},
    \Vert\cdot\Vert_{p, q})$ is a Banach space, where $\Vert u\Vert_{p,q}=\Vert
    E(|u|^q)^{1/q}\Vert_r$, if $1\le q\le p$ and $1/r= (1/q)-(1/p)$, and $\Vert
    u\Vert_{p,q}= \{\sup_n E(|u|^q)(A_n)/\mu(A_n)^{1/r}\}$, if $1\le p\le q$,
    $1/r= (1/p)- (1/q)$, and $\{A_n\}$ is a countable set of pairwise disjoint
    ${\Cal A}$-atoms such that $X=B\cup(\bigcup A_n)$;
 \item{(2)}
    characterizations indicating that $u\in K^\varphi_{p,q}$ if and only if
    $S\in L^{1/(1-(q(p))}(\Sigma)$, where $1\le q\le p$; $u\in K^\varphi_{p,q}$
    if and only $S= 0$ a.e on $B$ and $\sup_n\{S(A_n)/\mu(A_n)^{(q(p))-1}\}<
    \infty$, where $1\le p\le q$. \par}
rv: George O. Okikiolu (London)