<p>In the present paper, we consider the regularity of the operator semigroups corresponding to the exponentially tempered asymmetric <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-stable processes. First, using the Fourier analysis technique, we prove that the semigroup for the tempered asymmetric stable process without drift is analytic in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$L^p(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for any <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\in [1,\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Also, it is shown that the semigroup for the process with drift is an analytic semigroup if <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha \in (1,2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and a Gevrey semigroup of order <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi>γ</mi> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\gamma >1/\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>1</mn> <mo stretchy="false">/</mo> <mi>α</mi> </mrow> </math></EquationSource> </InlineEquation> if <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Regularity of tempered asymmetric stable semigroups

期刊论文
发表日期：

2026年2月22日

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Chung-Sik Sin

摘要

In the present paper, we consider the regularity of the operator semigroups corresponding to the exponentially tempered asymmetric $\alpha $ $α$ -stable processes. First, using the Fourier analysis technique, we prove that the semigroup for the tempered asymmetric stable process without drift is analytic in $L^p(\mathbb {R})$ $L^{p} (R)$ for any $p\in [1,\infty )$ $p \in [1, \infty)$ . Also, it is shown that the semigroup for the process with drift is an analytic semigroup if $\alpha \in (1,2)$ $α \in (1, 2)$ , and a Gevrey semigroup of order $\gamma $ $γ$ with $\gamma >1/\alpha $ $γ > 1 / α$ if $\alpha \in (0,1)$ $α \in (0, 1)$ .