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<p>We show that the Bergman projection on different generalizations of the Hartogs triangle fails to be well-defined as an integral operator on the entire <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> space, when <i>p</i> is the lower endpoint of the open interval for which the Bergman projection is <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> bounded. In particular, there are infinitely many such <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> functions <i>f</i>, such that the Bergman projection <i>B</i>(<i>f</i>) is not defined at every point in the Hartogs domain. Further, we show that on another type of generalized rational Hartogs triangles, the Bergman projection satisfies a weak-type (<i>p</i>,&#xa0;<i>p</i>) estimate at the upper endpoint of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(L^p\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>L</mi> <mi>p</mi> </msup> </math></EquationSource> </InlineEquation> boundedness.</p>

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Lower endpoint blowup of the Bergman projection on non-smooth Hartogs domains in \(\mathbb {C}^n\)

  • 期刊论文

  • 发表日期：

    2025年11月15日

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  • Liwei Chen,
  • Adam B. Christopherson

摘要

We show that the Bergman projection on different generalizations of the Hartogs triangle fails to be well-defined as an integral operator on the entire \(L^p\) L p space, when p is the lower endpoint of the open interval for which the Bergman projection is \(L^p\) L p bounded. In particular, there are infinitely many such \(L^p\) L p functions f, such that the Bergman projection B(f) is not defined at every point in the Hartogs domain. Further, we show that on another type of generalized rational Hartogs triangles, the Bergman projection satisfies a weak-type (p, p) estimate at the upper endpoint of \(L^p\) L p boundedness.