Boundedness of the Riesz potential in central Morrey-Orlicz spaces. (English) Zbl 1490.42017

The Orlicz space \(L^\Phi\) on \(\mathbb{R}^n\) consists of those \(f\) whose Luxembourg norm \[ \|f\|_{L^\Phi}=\inf\left\{\epsilon>0: \int_\Omega \Phi\left(\frac{|f(x)|}{\epsilon}\right)d\mu(x)\leq 1\right\} \] is finite. Here, \(\Phi\) is an Orlicz function (it is increasing, continuous and convex on \([0,\infty)\) and \(\Phi(0)=0\)).
For such \(\Phi\), \(\lambda\in\mathbb{R}\) and \(A\subset\mathbb{R}^n\) set \[ \|f\|_{\Phi,\lambda,A}=\inf\left\{\epsilon>0:\frac{1}{|A|^\lambda}\int_A \Phi\left(\frac{|f(x)|}{\epsilon}\right)\, dx\leq 1 \right\} \] and set \[ \|f\|_{\Phi,\lambda,A,\infty}=\inf\left\{\epsilon>0:\sup_{u>0}\Phi\left(\frac{u}{\epsilon}\right)\frac{|\{x\in A: |f(x)|>u\}|}{|A|^\lambda} \leq 1\right\}\, . \] The central Morrey-Orlicz space is \(M^{\Phi,\lambda}(0)=\{f\in L^1_{\mathrm{loc}}(\mathbb{R}^n): \|f\|_{M^{\Phi,\lambda}(0)} =\sup_{r>0} \|f\|_{\Phi,\lambda,B(r)}<\infty\}\) where \(B(r)\) is the ball centered at zero of radius \(r>0\). The weak central Morrey-Orlicz space is \(WM^{\Phi,\lambda}(0)=\{f\in L^1_{\mathrm{loc}}(\mathbb{R}^n): \|f\|_{WM^{\Phi,\lambda}(0)} =\sup_{r>0} \|f\|_{\Phi,\lambda,B(r),\infty}<\infty\}\). The Riesz potential \(I_\alpha\) on \(\mathbb{R}^n\) is convolution with \(|x|^{\alpha-n}\). The main results establish necessary and sufficient conditions for boundedness of the Riesz potential between suitable central Morrey-Orlicz spaces on \(\mathbb{R}^n\).
Theorem 2 provides a necessary condition for boundedness. It states that if \(0<\alpha<n\), \(\Phi\), \(\Psi\) are Orlicz functions and \(0\leq \lambda\), \(\mu<1\), and if \(I_\alpha\) is bounded from \(M^{\Phi,\lambda}(0)\) to \(M^{\Psi,\mu}(0)\) then there are constants \(C_1\), \(C_2\) such that \(u^{\alpha/n}\Phi^{-1}(u^{\lambda-1})\leq C_1 \Psi^{-1}(u^{\mu-1})\) (\(u>0\)) and \(s_{\Psi^{-1}}(u^{\mu-1}) \leq C_2 u^{\alpha/n} s_{\Phi^{-1}}(u^{\lambda-1})\). Here, \(s_{\Phi^{-1}}(t)=\sup_{s>0}\Phi^{-1}(st)/\Phi^{-1}(s)\). In addition, if \(\liminf_{t\to\infty} \Phi^{-1}(ct^\lambda)/\Psi^{-1}(ct^\mu)=\infty\) (the constant \(c>0\) depends on the \(M^{\Phi,\lambda}(0)\)-norm of \(\chi_B\) for a certain ball \(B\)) then \(I_\alpha\) is not bounded from \(M^{\Phi,\lambda}(0)\) to \(M^{\Psi,\mu}(0)\).
Theorem 3 provides a sufficient condition for boundedness. It states that if \(0<\lambda\), \(\mu<1\), \(\lambda\neq \mu\) or if \(\lambda=\mu=0\), and if there are constants \(C_4\), \(C_5\) such that \(\int_u^\infty t^{\alpha/n}\Phi^{-1}(t^{\lambda-1})\frac{dt}{t}\leq C_4 \Psi^{-1}(u^{\mu-1})\) (\(u>0\)) and \(\int_u^\infty t^{\alpha/n}\Phi^{-1}(r^\lambda/t)\frac{dt}{t}\leq C_5 \Psi^{-1}(r^\mu/u)\) (\(u>0,r>0\)), and if the complementary function \(\Phi^\ast(v)=\sup_{u>0}[uv-\Phi(u)]\) (\(v>0\)) satisfies \(\Phi^\ast(2u)\leq D\Phi^\ast(u)\) for a fixed \(D>0\), then \(I_\alpha\) is bounded from \(M^{\Phi,\lambda}(0)\) to \(M^{\Psi,\mu}(0)\). Moreover, \(I_\alpha\) is bounded from \(M^{\Phi,\lambda}(0)\) to \(WM^{\Psi,\mu}(0)\) regardless of whether \(\Phi^\ast(2u)\leq D\Phi^\ast(u)\).
The boundedness results for \(I_\alpha\) extend prior work of the authors regarding boundedness and weak-type boundedness of the maximal function and Calderón-Zygmund singular integral operators on central Morley-Orlicz spaces. Various remarks in this work place the results in their context of previously known boundedness results in Morley-Orlicz and central Morley-Orlicz spaces.


42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B35 Function spaces arising in harmonic analysis
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