<p>We study some necessary and sufficient conditions for the boundedness of the Riesz potential operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$I_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> and its commutator on the total Morrey spaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L_{p,\lambda ,\mu }(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We characterize the strong and weak Spanne type and Adams type boundedness of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$I_{\alpha }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mi>α</mi> </msub> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_{p,\lambda ,\mu }(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, respectively. We also give necessary and sufficient conditions for the boundedness of the commutator of the Riesz potential operator <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$[b,I_{\alpha }]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mi>b</mi> <mo>,</mo> <msub> <mi>I</mi> <mi>α</mi> </msub> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$L_{p,\lambda ,\mu }(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>L</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when <i>b</i> belongs to the spaces <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$BMO(\mathbb {R}^n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>B</mi> <mi>M</mi> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. As applications, we obtain some estimates for the Marcinkiewicz operator and fractional powers of some analytic semigroups on the total Morrey spaces.</p>

Hardy-Littlewood-Sobolev inequality in total Morrey spaces

期刊论文
发表日期：

2026年1月5日

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V. S. Guliyev

摘要

We study some necessary and sufficient conditions for the boundedness of the Riesz potential operator $I_{\alpha }$ $I_{α}$ and its commutator on the total Morrey spaces $L_{p,\lambda ,\mu }(\mathbb {R}^n)$ $L_{p, λ, μ} (R^{n})$ . We characterize the strong and weak Spanne type and Adams type boundedness of $I_{\alpha }$ $I_{α}$ on $L_{p,\lambda ,\mu }(\mathbb {R}^n)$ $L_{p, λ, μ} (R^{n})$ , respectively. We also give necessary and sufficient conditions for the boundedness of the commutator of the Riesz potential operator $[b,I_{\alpha }]$ $[b, I_{α}]$ on $L_{p,\lambda ,\mu }(\mathbb {R}^n)$ $L_{p, λ, μ} (R^{n})$ when b belongs to the spaces $BMO(\mathbb {R}^n)$ $B M O (R^{n})$ . As applications, we obtain some estimates for the Marcinkiewicz operator and fractional powers of some analytic semigroups on the total Morrey spaces.