## 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.

### MSC:

 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

### Keywords:

Morrey–Orlicz space; frational integral; Riesz potential
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### References:

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