REGULARITY OF THE WEIGHTED BERGMAN PROJECTION ON THE FOCK–BARGMANN–HARTOGS DOMAIN
/ Authors
/ Abstract
<jats:p>The Fock–Bargmann–Hartogs domain <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline2.png" /><jats:tex-math>$D_{n,m}(\,\unicode[STIX]{x1D707}):=\{(z,w)\in \mathbb{C}^{n}\times \mathbb{C}^{m}:\Vert w\Vert ^{2}<e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}\}$</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline3.png" /><jats:tex-math>$\unicode[STIX]{x1D707}>0$</jats:tex-math></jats:alternatives></jats:inline-formula>, is an unbounded strongly pseudoconvex domain with smooth real-analytic boundary. We compute the weighted Bergman kernel of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline4.png" /><jats:tex-math>$D_{n,m}(\,\unicode[STIX]{x1D707})$</jats:tex-math></jats:alternatives></jats:inline-formula> with respect to the weight <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline5.png" /><jats:tex-math>$(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}$</jats:tex-math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline6.png" /><jats:tex-math>$\unicode[STIX]{x1D70C}(z,w):=\Vert w\Vert ^{2}-e^{-\unicode[STIX]{x1D707}\Vert z\Vert ^{2}}$</jats:tex-math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline7.png" /><jats:tex-math>$\unicode[STIX]{x1D6FC}>-1$</jats:tex-math></jats:alternatives></jats:inline-formula>. Then, for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline8.png" /><jats:tex-math>$p\in [1,\infty ),$</jats:tex-math></jats:alternatives></jats:inline-formula> we show that the corresponding weighted Bergman projection <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline9.png" /><jats:tex-math>$P_{D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}}}$</jats:tex-math></jats:alternatives></jats:inline-formula> is unbounded on <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline10.png" /><jats:tex-math>$L^{p}(D_{n,m}(\,\unicode[STIX]{x1D707}),(-\unicode[STIX]{x1D70C})^{\unicode[STIX]{x1D6FC}})$</jats:tex-math></jats:alternatives></jats:inline-formula>, except for the trivial case <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline11.png" /><jats:tex-math>$p=2$</jats:tex-math></jats:alternatives></jats:inline-formula>. This gives an example of an unbounded strongly pseudoconvex domain whose ordinary Bergman projection is <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline12.png" /><jats:tex-math>$L^{p}$</jats:tex-math></jats:alternatives></jats:inline-formula> irregular when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline13.png" /><jats:tex-math>$p\in [1,\infty )\setminus \{2\}$</jats:tex-math></jats:alternatives></jats:inline-formula>, in contrast to the well-known positive <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0004972719001424_inline14.png" /><jats:tex-math>$L^{p}$</jats:tex-math></jats:alternatives></jats:inline-formula> regularity result on a bounded strongly pseudoconvex domain.</jats:p>
Journal: Bulletin of the Australian Mathematical Society