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On non-effective weights in Orlicz spaces. (English) Zbl 1167.46020

Let \(\Phi:\mathbb{R}\to [0,\infty)\) be a Young function such that \(\Phi\) is even, convex on \(\mathbb{R}\) and \(\Phi(0)=0\). Let \(\Omega\subset\mathbb{R}^n\) have finite Lebesgue measure and \(w\) be a weight on \(\Omega\), i.e., \(w\) is a positive locally integrable real function defined on \(\Omega\). The Orlicz (resp., weighted Orlicz) space \(L_\Phi(w)\) is induced by the modular
\[ \rho(f) = \int_\Omega \Phi(f(x))\, dx \quad (\text{resp.,} \quad \rho(f,w) = \int_\Omega \Phi(f(x)) w(x)\, dx), \]
and equipped with either the Luxemburg or Orlicz norm, which are equivalent. The definition can be formally generalized by replacing the Lebesgue measure by a general \(\sigma\)-finite measure space \((\Omega, \nu)\), where \(\Omega\) is an abstract set. Below there are samples of the main results which also hold true for general non-atomic \(\sigma\)-finite measure spaces.
Theorem 1. Let \(\Phi\) be a Young function and for \(K>0\) put \(L_K(t) = \Phi(K\Phi^{-1}(t))\). Let \(S_K\) be the complementary function to \(L_K\) and assume that \(w\) is a weight function, bounded away from zero. Then \(L_\Phi(\chi_\Omega) = L_\Phi (w)\) if and only if there exists \(K>1\) such that \[ \int_\Omega S_K(w(x))\,dx < \infty. \]
Theorem 2. Let \(w_j\), \(j=1,2\), be weight functions on \(\Omega\), \(w_j \geq 1\), and \(S_K\) be the function from Theorem 1 for some \(K>1\). Then \(L_\Phi(w_1) \hookrightarrow L_\Phi(w_2)\) if and only if \[ \int_\Omega S_k(w_2(x)/w_1(x)) w_1(x)\, dx < \infty. \]
Theorem 3. For any Young function \(\Phi\), there exists an essentially unbounded weight \(w\) such that \(L_\Phi (\chi_\Omega) = L_\Phi(w)\) if and only if
\[ \liminf_{t\to\infty} \Phi(Kt)/\Phi(t) =\infty \]
for some \(K>1\).
The applications of the above theorems allow to provide criteria for a composition operator to be continuous on \(L_\Phi(\Omega)\). This is an improvement and simplification of results in the literature. The main theorems are proved by using the techniques developed in the theory of Musiełak–Orlicz spaces.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B25 Classical Banach spaces in the general theory
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
47B33 Linear composition operators

Citations:

Zbl 1073.47032
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References:

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