<p>The main objective of this paper is to investigate the distributional estimates for (i) commutators with Calderón–Zygmund integral operators; (ii) Marcinkiewicz multipliers; (iii) Littlewood–Paley square function, via semigroup <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\{\mathscr {C}^{\alpha }\}_{\alpha >0}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">{</mo> <msup> <mi mathvariant="script">C</mi> <mi>α</mi> </msup> <mo stretchy="false">}</mo> </mrow> <mrow> <mi>α</mi> <mo>></mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </InlineEquation> generated by Cesàro operator. In each of the cases (i)–(iii) we obtain new estimates of the distribution of elements in the range of the underlying operators in terms of the distribution function of the input function. Our method allows us to obtain optimal estimates shedding additional light at the results due to Pérez (J Funct Anal 128(1):163–185, 1995), Tao and Wright (Rev Mat Iberoam 17(3):521–558, 2001), Bakas et al. (Rev Mat Iberoam 2024), Bourgain (On the Behavior of the Constant in the Littlewood–Paley Inequality, Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math., vol. 1376, pp. 202–208. Springer, Berlin, 1989). The main feature of the distributional form inequalities lies in its broad applicability across diverse problems in analysis, e.g. they allow obtaining estimates in wide range of symmetric quasi-Banach interpolation spaces between <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$L_q$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$1<p<q<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>q</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>), not just for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-spaces (<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$1<p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo><</mo> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>).</p>

Optimal distribution estimates for commutators and Marcinkiewicz multipliers

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发表日期：

2025年10月13日

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Fedor Sukochev,
Fulin Yang,
Dmitriy Zanin,
Dejian Zhou

摘要

The main objective of this paper is to investigate the distributional estimates for (i) commutators with Calderón–Zygmund integral operators; (ii) Marcinkiewicz multipliers; (iii) Littlewood–Paley square function, via semigroup $\{\mathscr {C}^{\alpha }\}_{\alpha >0}$ ${C^{α}}_{α > 0}$ generated by Cesàro operator. In each of the cases (i)–(iii) we obtain new estimates of the distribution of elements in the range of the underlying operators in terms of the distribution function of the input function. Our method allows us to obtain optimal estimates shedding additional light at the results due to Pérez (J Funct Anal 128(1):163–185, 1995), Tao and Wright (Rev Mat Iberoam 17(3):521–558, 2001), Bakas et al. (Rev Mat Iberoam 2024), Bourgain (On the Behavior of the Constant in the Littlewood–Paley Inequality, Geometric Aspects of Functional Analysis (1987–88), Lecture Notes in Math., vol. 1376, pp. 202–208. Springer, Berlin, 1989). The main feature of the distributional form inequalities lies in its broad applicability across diverse problems in analysis, e.g. they allow obtaining estimates in wide range of symmetric quasi-Banach interpolation spaces between $L_p$ $L_{p}$ and $L_q$ $L_{q}$ ( $1<p<q<\infty $ $1 < p < q < \infty$ ), not just for $L_p$ $L_{p}$ -spaces ( $1<p<\infty $ $1 < p < \infty$ ).