<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nu = (\nu _1, \ldots , \nu _n) \in (-1/2, \infty )^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>ν</mi> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>ν</mi> <mn>1</mn> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>ν</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>2</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, with <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\Delta _\nu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>ν</mi> </msub> </math></EquationSource> </InlineEquation> be the multivariate Bessel operator defined by <Equation ID="Equ51"> <EquationSource Format="TEX">$ \Delta _{\nu } = -\sum _{j=1}^n\left( \frac{\partial ^2}{\partial x_j^2} - \frac{\nu _j^2 - 1/4}{x_j^2} \right) . $</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi mathvariant="normal">Δ</mi> <mi>ν</mi> </msub> <mo>=</mo> <mo>-</mo> <munderover> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </munderover> <mfenced close=")" open="("> <mfrac> <msup> <mi>∂</mi> <mn>2</mn> </msup> <mrow> <mi>∂</mi> <msubsup> <mi>x</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mrow> </mfrac> <mo>-</mo> <mfrac> <mrow> <msubsup> <mi>ν</mi> <mi>j</mi> <mn>2</mn> </msubsup> <mo>-</mo> <mn>1</mn> <mo stretchy="false">/</mo> <mn>4</mn> </mrow> <msubsup> <mi>x</mi> <mi>j</mi> <mn>2</mn> </msubsup> </mfrac> </mfenced> <mo>.</mo> </mrow> </math></EquationSource> </Equation>In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Delta _\nu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>ν</mi> </msub> </math></EquationSource> </InlineEquation>. We then study the higher-order Riesz transforms associated with <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Delta _\nu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>ν</mi> </msub> </math></EquationSource> </InlineEquation>. First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Delta _\nu $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Δ</mi> <mi>ν</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

Hardy spaces, Campanato spaces and higher order Riesz transforms associated with Bessel operators

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2026年4月2日

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The Anh Bui

摘要

Let $\nu = (\nu _1, \ldots , \nu _n) \in (-1/2, \infty )^n$ $ν = (ν_{1}, \dots, ν_{n}) \in {(- 1 / 2, \infty)}^{n}$ , with $n \ge 1$ $n \geq 1$ , and let $\Delta _\nu $ $Δ_{ν}$ be the multivariate Bessel operator defined by $ \Delta _{\nu } = -\sum _{j=1}^n\left( \frac{\partial ^2}{\partial x_j^2} - \frac{\nu _j^2 - 1/4}{x_j^2} \right) . $ $Δ_{ν} = - \sum_{j = 1}^{n} (\frac{\partial^{2}}{\partial x_{j}^{2}} - \frac{ν_{j}^{2} - 1 / 4}{x_{j}^{2}}) .$ In this paper, we develop the theory of Hardy spaces and BMO-type spaces associated with the Bessel operator $\Delta _\nu $ $Δ_{ν}$ . We then study the higher-order Riesz transforms associated with $\Delta _\nu $ $Δ_{ν}$ . First, we show that these transforms are Calderón-Zygmund operators. We further prove that they are bounded on the Hardy spaces and BMO-type spaces associated with $\Delta _\nu $ $Δ_{ν}$ .