<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> be a Radon measure on <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </math></EquationSource> </InlineEquation> which satisfies the growth condition <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu (Q(x,l))\le C_{0}l^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <msub> <mi>C</mi> <mn>0</mn> </msub> <msup> <mi>l</mi> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> for any cube <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ Q(x,l)\subset \mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>Q</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>l</mi> <mo stretchy="false">)</mo> </mrow> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$x\in \mathbb {R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$l(Q)>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>l</mi> <mo stretchy="false">(</mo> <mi>Q</mi> <mo stretchy="false">)</mo> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, with some fixed constants <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$C_{0}>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>C</mi> <mn>0</mn> </msub> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$n\in (0,d]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>d</mi> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>. We introduce a new type of fractional maximal operator <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$M^{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mi>a</mi> </msup> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq10"> <EquationSource Format="TEX">$0\le a<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>0</mn> <mo>≤</mo> <mi>a</mi> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>) under the measure <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>. If <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mu (\mathbb {R}^{d})<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> <mo stretchy="false">)</mo> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, we obtain a lower oscillation estimate of <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$M^{a}f$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>M</mi> <mi>a</mi> </msup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> and the boundedness of <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$M^{a}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>M</mi> <mi>a</mi> </msup> </math></EquationSource> </InlineEquation> from <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$RBMO(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>B</mi> <mi>M</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> to <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$RBLO(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>B</mi> <mi>L</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which is a generalization of R. Gibara and J. Kline’s result (JFA, 2023) on the measure <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation>. Finally, we obtain some estimates for natural fractional maximal operators on <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$RBLO(\mu )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mi>B</mi> <mi>L</mi> <mi>O</mi> <mo stretchy="false">(</mo> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> spaces, which is a generalization of Dachun Yang et al. (LNM, 2017).</p>

Fractional Maximal Operators on BMO-Type Spaces Under a Growth Condition

期刊论文
发表日期：

2026年5月22日

查看全文

Shining Li,
Baode Li

摘要

Let $\mu $ $μ$ be a Radon measure on $\mathbb {R}^{d}$ $R^{d}$ which satisfies the growth condition $\mu (Q(x,l))\le C_{0}l^{n}$ $μ (Q (x, l)) \leq C_{0} l^{n}$ for any cube $ Q(x,l)\subset \mathbb {R}^{d}$ $Q (x, l) \subset R^{d}$ , $x\in \mathbb {R}^{d}$ $x \in R^{d}$ and $l(Q)>0$ $l (Q) > 0$ , with some fixed constants $C_{0}>0$ $C_{0} > 0$ and $n\in (0,d]$ $n \in (0, d]$ . We introduce a new type of fractional maximal operator $M^{a}$ $M^{a}$ ( $0\le a<1$ $0 \leq a < 1$ ) under the measure $\mu $ $μ$ . If $\mu (\mathbb {R}^{d})<\infty $ $μ (R^{d}) < \infty$ , we obtain a lower oscillation estimate of $M^{a}f$ $M^{a} f$ and the boundedness of $M^{a}$ $M^{a}$ from $RBMO(\mu )$ $R B M O (μ)$ to $RBLO(\mu )$ $R B L O (μ)$ , which is a generalization of R. Gibara and J. Kline’s result (JFA, 2023) on the measure $\mu $ $μ$ . Finally, we obtain some estimates for natural fractional maximal operators on $RBLO(\mu )$ $R B L O (μ)$ spaces, which is a generalization of Dachun Yang et al. (LNM, 2017).