×

Maximal operator and its commutators on generalized weighted Orlicz-Morrey spaces. (English) Zbl 1418.42028

In this paper, one investigates the boundedness of the maximal operator \(M\) and its commutators \(M^{b}\) on generalized weighted Orlicz-Morrey spaces. The results obtained provide conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.
To state the main theorems of the paper some definitions are required.
A function \(\Phi: [0,\infty)\to [0,\infty]\) is called a Young function, if \(\Phi\) is convex, left-continuous, \(\lim\limits_{r\to 0^{+}}\Phi(r)=\Phi(0)=0\) and \(\lim\limits_{r\to\infty}\Phi(r)=\infty\).
The space \(A_{p}\) of Muckenhoupt’s weights appears for \(1\le p\le\infty\) and specially for the index \(p=i_{\phi}\) where \[ i_{\Phi}=\lim_{t\to 0^{+}}\frac{\log h_{\Phi}(t)}{\log t}\quad \text{and}\quad h_{\Phi}(t)=\sup_{s>0}\frac{\Phi(st)}{\Phi(s)}. \] A Young function \(\Phi\) is said to satisfy the \(\Delta_{2}\)-condition, denoted by \(\Phi\in \Delta_{2}\), if \[ \Phi (2r)\le k\Phi (r),\quad r>0 \] for some \(k>1\). A Young function \(\Phi\) is said to satisfy the \(\nabla_{2}\)-condition, denoted also by \(\Phi\in\nabla_{2}\), if \[ \Phi(r)\le \frac{1}{2k}\Phi (kr),\quad r\ge 0 \] for some \(k>1\).
For a Young function \(\Phi\) and a weight \(w\in A_{\infty}\) set \[ L_{w}^{\Phi}(\mathbb{R}^{n})\equiv \left\{ f\text{-measurable}:\int_{\mathbb{R}_{n}}\Phi(k|f(x)|)w(x)\,dx<\infty\text{ for some }k>0\right\}. \] With these preliminaries one can define the generalized weighted Orlicz-Morrey spaces:
Let \(\varphi\) be a positive measurable function on \(\mathbb{R}^{n}\times (0,\infty)\), let \(w\) be a non-negative measurable function on \(\mathbb{R}^{n}\) and \(\Phi\) any Young function. Denote by \(M_{w}^{\Phi,\varphi}(\mathbb{R}^{n})\) the generalized weighted Orlicz-Morrey space, the space of all functions \(f\in L_{w}^{\Phi,\text{loc}}(\mathbb{R}^{n})\) such that \[ \begin{split} \|f\|_{M_{w}^{\Phi,\varphi}(\mathbb{R}^{n})}\equiv \|f\|_{M_{w}^{\Phi,\varphi}}&= \sup_{x\in\mathbb{R}^{n},\,r>0} \varphi(x,r)^{-1}\Phi^{-1}(w(B(x,r))^{-1})\|f\|_{L_{w}^{\Phi}(B(x,r))}\\ &= \sup_{B\in\mathcal{B}} \varphi(B)^{-1}\Phi^{-1}(w(B)^{-1})\|f\|_{L_{w}^{\Phi}(B)}<\infty. \end{split} \] Also, for a Young function \(\Phi\) and a non-negative measurable function \(w\), we denote by \(\mathcal{G}_{\Phi}^{w}\) the set of all almost decreasing functions \(\varphi : \mathbb{R}^{n}\times (0,\infty)\to (0,\infty)\) such that \[ \inf_{B\in\mathcal{B};\, r_{B}\le r_{B_{0}}}\varphi(B) \gtrsim \varphi (B_{0})\quad \text{for all }B_{0}\in\mathcal{B} \] and \[ \inf_{B\in\mathcal{B};\, r_{B}\ge r_{B_{0}}}\frac{\varphi(B)}{\Phi^{-1}(w(B)^{-1})}\gtrsim \frac{\varphi(B_{0})}{(w(B_{0})^{-1})}\quad \text{for all }B_{0}\in\mathcal{B}, \] where \(r_{B}\) and \(r_{B_{0}}\) denote the radius of the balls \(B\) and \(B_{0}\), respectively.
The Hardy-Littlewood maximal operator \(M\) is defined by \[ Mf(x)=\sup_{r>0}\frac{1}{|B(x,r)|} \int_{B(x,r)}|f(y)|\,dy,\quad x\in\mathbb{R}^{n} \] for a locally integrable function \(f\) on \(\mathbb{R}^{n}\).
The following theorem for the boundedness of \(M\) is proved:
\(\Phi\) be a Young function and \(\varphi_{1}\), \(\varphi_{2}\) positive measurable functions on \(\mathbb{R}^{n}\times (0,\infty)\).
If \(\Phi\in\nabla_{2}\) and \(w\in A_{i_{\Phi}}\), then the condition \[ \sup_{r<t<\infty}\left(\text{ess inf}_{t<s<\infty} \frac{\varphi_{1}(x,s)}{\Phi^{-1(w(B(x,s))^{-1})}}\right)\Phi^{-1} (w(B(x,t))^{-1})\le C\varphi_{2}(x,r), \] where \(C\) does not depend on \(x\) and \(r\), is sufficient for the boundedness of \(M\) from \(M_{w} ^{\Phi,\varphi_{1}}(\mathbb{R}^{n})\) to \(M^{\Phi,\varphi_{2}}_{w}(\mathbb{R}^{n})\).
If \(\varphi_{1}\in\mathcal{G}_{w}^{\Phi}\), then the condition \[ \varphi_{1}(x,r)\le C\varphi_{2}(x,r), \tag{1} \] where \(C\) does not depend on \(x\) and \(r\), is necessary for the boundedness of \(M\) from \(M_{w}^{\Phi,\varphi_{1}}(\mathbb{R}^{n})\) to \(M^{\Phi,\varphi_{2}}_{w}(\mathbb{R}^{n})\).
Let \(\Phi\in\nabla_{2}\) and \(w\in A_{i_{\Phi}}\). If \(\varphi_{1}\in\mathcal{G}_{w}^{\Phi}\), then the condition (1) is necessary and sufficient for the boundedness of \(M\) from \(M_{w}^{\Phi,\varphi_{1}}(\mathbb{R}^{n})\) to \(M^{\Phi,\varphi_{2}}_{w}(\mathbb{R}^{n})\).

Given a measurable function \(b\) the maximal commutator is defined by \[ M^{b} f(x)=\sup_{t>0}\frac{1}{|B(x,t)|} \int_{B(x,t)}|b(x)-b(y)| |f(y)|\,dy,\quad x\in\mathbb{R}^{n} \] for a locally integrable function \(f\) on \(\mathbb{R}^{n}\).
Necessary and sufficient conditions for the boundedness of \(M^{b}\) in generalized weighted Orlicz-Morrey spaces are given in the following result.
Let \(b\in BMO(\mathbb{R}^{n})\), \(\Phi\) be a Young function and \(\varphi_{1}\), \(\varphi_{2}\) positive measurable functions on \(\mathbb{R}^{n}\times (0,\infty)\).
Let \(\Phi\in \Delta_{2}\cap\nabla_{2}\) and \(w\in A_{1}\), then the condition \[ \sup_{r<t<\infty}\left(1+\ln \frac{t}{r}\right) \Phi^{-1} (w(B(x,t))^{-1})\text{ess inf}_{t<s<\infty} \frac{\varphi_{1}(x,s)}{\Phi^{-1(w(B(x,s))^{-1})}}\le C\varphi_{2}(x,r), \] where \(C\) does not depend on \(x\) and \(r\), is sufficient for the boundedness of \(M^{b}\) from \(M_{w}^{\Phi,\varphi_{1}}(\mathbb{R}^{n})\) to \(M^{\Phi,\varphi_{2}}_{w}(\mathbb{R}^{n})\).
If \(\Phi\in \Delta_{2}\), \(\varphi_{1}\in\mathcal{G}_{w}^{\Phi}\) and \(w\in A_{1}\), then the condition (1) is necessary for the boundedness of \(M^{b}\) from \(M_{w}^{\Phi,\varphi_{1}}(\mathbb{R}^{n})\) to \(M^{\Phi,\varphi_{2}}_{w}(\mathbb{R}^{n})\).
Let \(\Phi\in \Delta_{2}\cap \nabla_{2}\) and \(w\in A_{1}\). If \(\varphi_{1}\in\mathcal{G}_{w}^{\Phi}\) satisfies the condition \[ \sup_{r<t<\infty}\left(1+\ln\frac{t}{r}\right)\varphi_{1}(x,t)\le C\varphi_{1}(x,r), \] where \(C\) does not depend on \(x\) and \(r\), then the condition (1) is necessary and sufficient for the boundedness of \(M^{b}\) from \(M_{w}^{\Phi,\varphi_{1}}(\mathbb{R}^{n})\) to \(M^{\Phi,\varphi_{2}}_{w}(\mathbb{R}^{n})\).

In the paper, ones gives also necessary and sufficient conditions for the weak-type boundedness of the operator \(M\) on generalized weighted Orlicz-Morrey spaces. One generalizes also for this class of spaces the classical vector-valued maximal inequalities due to Fefferman and Stein.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
PDFBibTeX XMLCite
Full Text: DOI