Characterization of Lipschitz spaces via commutators of the Hardy–Littlewood maximal function

We give some characterizations of the boundedness of the maximal or nonlinear commutators of the Hardy–Littlewood maximal function and sharp maximal function in variable exponent Lebesgue spaces when the symbols b belong to the Lipschitz spaces, by which some new characterizations of Lipschitz spaces and nonnegative Lipschitz functions are obtained. Some equivalent relations between the Lipschitz norm and the variable exponent Lebesgue norm are also given.

Characterization of boundedness of some commutators of maximal functions in terms of Lipschitz spaces

In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator $$M_{\alpha }$$
 on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator $$M_{b,\alpha }$$
 and nonlinear commutator of fractional maximal operator $$[b,M_{\alpha }]$$
 on Orlicz spaces, when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.

Fractional maximal function and its commutators on Orlicz spaces

Let $$0\le \alpha <n$$
 , $$[b,M_{\alpha }]$$
 and $$[b,M^{\sharp }]$$
 be the nonlinear commutators of the fractional maximal function $$M_{\alpha }$$
 and the sharp maximal function $$M^{\sharp }$$
 with a locally integrable function b. In this note, we give necessary and sufficient conditions for the boundedness of $$[b,M_{\alpha }]$$
 and $$[b,M^{\sharp }]$$
 on Orlicz spaces when the symbol b belongs to Lipschitz spaces, by which some new characterizations of non-negative Lipschitz functions (in terms of the Orlicz norm) are obtained.

Commutators of Some Maximal Functions with Lipschitz Function on Orlicz Spaces

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function b is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when b belongs to Lipschitz or $\mathit{BMO}({\mathbb{R}}^{n})$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $\mathit{BMO}({\mathbb{R}}^{n})$ spaces are obtained.

/pdf/some-notes-on-commutators-of-the-fractional-maximal-function-3rxh56r5ww.pdf

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Let ${M_\beta }$ be the fractional maximal function. The commutator generated by ${M_\beta }$ and a suitable function b is defined by $[{M_\beta },b]f = {M_\beta }(bf) - b{M_\beta }(f)$. Denote by P(ℝn) the set of all measurable functions p(·): ℝn → [1,∞) such that 

$$1 < p_ - : = \mathop {es\sin fp(x)}\limits_{x \in \mathbb{R}^n } andp_ + : = \mathop {es\operatorname{s} \sup p(x) < \infty }\limits_{x \in \mathbb{R}^n } ,$$

and by B(ℝn) the set of all p(·) ∈ P(ℝn) such that the Hardy-Littlewood maximal function M is bounded on Lp(·)(ℝn). In this paper, the authors give some characterizations of b for which $[{M_\beta },b]$ is bounded from Lp(·)(ℝn) into Lq(·)(ℝn), when p(·) ∈ P(ℝn), 0 < β < n/p+ and 1/q(·) = 1/p(·) − β/n with q(·)(n − β)/n ∈ B(ℝn).

/pdf/commutators-of-the-fractional-maximal-function-on-variable-189mkys2bx.pdf

Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Let M be the Hardy-Littlewood maximal function, the commutator generated by M and a suitable function b is defined by [M,b] f = M(b f )−bM f . In this paper, the authors give some characterizations of b for which [M,b] is bounded on the Lebesgue spaces with variable exponent. The similar results are also proved for the commutator of the sharp maximal function. Mathematics subject classification (2010): 42B25, 46E30.

/pdf/commutators-for-the-maximal-functions-on-lebesgue-spaces-49our7cy62.pdf

Commutators for the maximal functions on Lebesgue spaces with variable exponent

Let $0<\alpha<n$ and $M_{\alpha}$ be the fractional maximal function. The nonlinear commutator of $M_{\alpha}$ and a locally integrable function $b$ is given by $[b,M_{\alpha}](f)=bM_{\alpha}(f)-M_{\alpha}(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_{\alpha}]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.

/pdf/some-notes-on-commutators-of-the-fractional-maximal-function-2fbdltz70f.pdf

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Papers

Commutators of the fractional maximal function on variable exponent Lebesgue spaces

Commutators of Some Maximal Functions with Lipschitz Function on Orlicz Spaces

Commutators for the maximal functions on Lebesgue spaces with variable exponent

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces

Some notes on commutators of the fractional maximal function on variable Lebesgue spaces