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Partial integral operators in Orlicz spaces with mixed norm. (English) Zbl 0922.47025

Let \((T,{\mathcal U}(T),\mu)\), \((S,{\mathcal U}(S),\nu)\) be \(\sigma\)-finite measure spaces which are atom-free, and let \(\mathbb{R}\) be the set of real numbers. If \(l:T\times S\times T\to \mathbb{R}\), \(m:T\times S\times T\to \mathbb{R}\) are measurable functions, then the ‘partial integral operators’, \(L\), \(M\), are defined by \[ L(f)(t,s)= \int_Tl(t,s,x) f(x,s)d\mu(x);\;M(f)(t,s)= \int_S m(t,s,x) f(t,x)d\nu(x), \] so that one of the variables in terms of which functions of the domain space are expressed features in transforms of the range space.
In addition the operators \(L_s\), \(M_t\), are defined on spaces of functions of single variables by \[ L_s(u)(t)= L(u)(t,s);\;M_t(u)(s)= M(u)(t,s),\;u(x,y)= u(x)\quad\text{or}\quad u(x,y)= u(y). \] If \(U\), \(V\) are ideal spaces (i.e. \(L^\infty\)-Banach lattices) with full support on the domains \(T\), \(S\), respectively, then \([U\to V]\), \([U\leftarrow V]\) are the mixed norm spaces with norms \(\|\cdot\|_{[U\to V]}= \| \|\cdot\|_U\|_V\), \(\|\cdot\|_{[U\leftarrow V]}= \| \|\cdot\|_V\|_U\), and the multiplicator space \(W_1/W_2\) consists of measurable functions of the form \(w\) on \(\Omega\) such that \(\| w\|_{W_1/W_2}= \sup\{\| ww_2\|_{W_1}:\| w_2\|_{W_2}\leq 1\}<\infty\), where \(W_1\), \(W_2\) are two ideal spaces over a domain \(\Omega\).
If \(W_1\), \(W_2\), \(W_3\), are ideal spaces over a domain \(\Omega\), and \(\theta\) is a permutation of \((\theta_1,\theta_2,\theta_3)\), then \([W_1,W_2, W_3;\theta]\) is the mixed norm space consisting of functions on \(W_1\times W_2\times W_3\), with rearrangements of the orders of norms.
The main results of this paper involve estimates for the operators \(L\), \(M\) as bounded linear operators from \(X= [U\to V]\) into \(Y= [U\leftarrow V]\) in terms of ‘mixed norm’ estimates involving \(L_s\), \(s\in S\), \(M_t\), \(t\in T\), as bounded operators from \(U_1\) into \(U_2\) and from \(V_1\) into \(V_2\). Particular consideration is given to cases in which \(U_1= L_{\phi_1}\), \(U_2= L_{\phi_2}\), where \(\phi_1\), \(\phi_2\) are Young functions which define Orlicz spaces.
In particular, the estimates derived include results of the type \[ \| L\|_{{\mathcal L}(X,Y)}\leq c_N\| \| L_{(\cdot)}\|_{{\mathcal L}(U_1, U_2)}\|_{L_{\phi_1:\phi_2}};\;\| M\|_{{\mathcal L}(X,Y)}\leq c_M\| \| M_{(\cdot)}\|_{{\mathcal L}(X,Y)}\|_{L_{\theta_1:\theta_2}}, \] where \(\|\cdot\|_{{\mathcal L}(X,Y)}\) is the norm of a bounded linear operator from \(X\) into \(Y\), \[ (\phi_1:\phi_2)(t)= \sup\{\phi_1(tv)- \phi_2(v): 0<v<\infty\}, \] and \(L_{\phi_1}/L_{\phi_2}= L_{\phi_1:\phi_2}\) if \(\phi_2(t)/\phi_1(kt)\to 0\) as \(t\to\infty\) for \(k>0\).

MSC:

47B38 Linear operators on function spaces (general)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47G99 Integral, integro-differential, and pseudodifferential operators
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