Some Remarks on the Berndtsson’s Conjecture

This paper studies the Berndtsson’s conjecture on the sup-norm estimate for solution of \(\overline \partial\) $$\overline{\mathrm{\partial }}$$ -equation \({\overline \partial}u=f\) $$\overline{\mathrm{\partial }}u=f$$ through the Hörmander’s weighted L²-estimate where the weight function is bounded plurisubharmonic in the unit ball B_n ⊂ ℂⁿ. First we discover that if weighted function is bounded and rotation symmetric, we obtain a better pointwise estimate for the canonical solution of the equation \({\overline \partial}u=f\) $$\overline{\mathrm{\partial }}u=f$$ than the one obtained by Berndtsson. We explain more on the phenomenon that Berndtsson’s conjecture implies the solution of the Corona problem in several complex variables. Finally, we give two examples. One shows that Berndtsson’s conjecture fails for the canonical solution; the other one shows that the boundedness condition on the weight function is necessary for Berndtsson’s conjecture.