<p>In this paper, we give sufficient conditions on a measurable function <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2971_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="147" /> </InlineMediaObject> <EquationSource Format="TEX">$p:(0,\infty )^{n}\rightarrow [1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>:</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <mrow> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in order that harmonic analysis operators (maximal operators, Riesz transforms, Littlewood–Paley functions, and multipliers) associated with <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2971_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-Laguerre polynomial expansions are bounded on the variable Lebesgue space <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2971_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="122" /> </InlineMediaObject> <EquationSource Format="TEX">$L^{p(\cdot )} ((0,\infty )^{n}, \mu _\alpha )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> <mo>,</mo> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2971_Article_IEq4.gif" Format="GIF" Height="42" Rendition="HTML" Resolution="72" Type="Linedraw" Width="215" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathrm{{d}}\mu _\alpha (x)=2^{n}\prod _{j=1}^{n} \frac{x_{j}^{2\alpha _{j}+1} e^{-x_{j}^{2}}}{\Gamma (\alpha _{j}+1)} \mathrm{{d}}x$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">d</mi> <msub> <mi>μ</mi> <mi>α</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <msubsup> <mo>∏</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mfrac> <mrow> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> </mrow> <mrow> <mn>2</mn> <msub> <mi>α</mi> <mi>j</mi> </msub> <mo>+</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mi>e</mi> <mrow> <mo>-</mo> <msubsup> <mi>x</mi> <mrow> <mi>j</mi> </mrow> <mn>2</mn> </msubsup> </mrow> </msup> </mrow> <mrow> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mi>j</mi> </msub> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mfrac> <mi mathvariant="normal">d</mi> <mi>x</mi> </mrow> </math></EquationSource> </InlineEquation>, being <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2971_Article_IEq5.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="192" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha =(\alpha _{1}, \dots , \alpha _{n})\in [0,\infty )^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>α</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>α</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2971_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="190" /> </InlineMediaObject> <EquationSource Format="TEX">$x=(x_1,\dots ,x_n)\in (0,\infty )^{n}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>x</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>.</p>

Harmonic Analysis Operators Associated with Laguerre Polynomial Expansions on Variable Lebesgue Spaces

期刊论文
发表日期：

2025年10月17日

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Jorge J. Betancor,
Estefanía Dalmasso,
Pablo Quijano,
Roberto Scotto

摘要

In this paper, we give sufficient conditions on a measurable function $p:(0,\infty )^{n}\rightarrow [1,\infty )$ $p : {(0, \infty)}^{n} \to [1, \infty)$ in order that harmonic analysis operators (maximal operators, Riesz transforms, Littlewood–Paley functions, and multipliers) associated with $\alpha $ $α$ -Laguerre polynomial expansions are bounded on the variable Lebesgue space $L^{p(\cdot )} ((0,\infty )^{n}, \mu _\alpha )$ $L^{p (\cdot)} ({(0, \infty)}^{n}, μ_{α})$ , where $\mathrm{{d}}\mu _\alpha (x)=2^{n}\prod _{j=1}^{n} \frac{x_{j}^{2\alpha _{j}+1} e^{-x_{j}^{2}}}{\Gamma (\alpha _{j}+1)} \mathrm{{d}}x$ $d μ_{α} (x) = 2^{n} \prod_{j = 1}^{n} \frac{x_{j}^{2 α_{j} + 1} e^{- x_{j}^{2}}}{Γ (α_{j} + 1)} d x$ , being $\alpha =(\alpha _{1}, \dots , \alpha _{n})\in [0,\infty )^{n}$ $α = (α_{1}, \dots, α_{n}) \in {[0, \infty)}^{n}$ and $x=(x_1,\dots ,x_n)\in (0,\infty )^{n}$ $x = (x_{1}, \dots, x_{n}) \in {(0, \infty)}^{n}$ .