A computation of Poisson kernels for some standard weighted biharmonic operators in the unit disc

A Computation of Poisson Kernels for Some Standard Weighted Biharmonic Operators in the Unit Disc

We compute Poisson kernels for integer weight parameter standard weighted biharmonic operators in the unit disc with Dirichlet boundary conditions. The computations performed extend the supply of explicit examples of such kernels and suggest similar formulas for these Poisson kernels to hold true in more generality. Computations have been carried out using the open source computer algebra packa...

A Representation Formula for Radially Weighted Biharmonic Functions in the Unit Disc

Weighted Inequalities for Integral Operators with Some Homogeneous Kernels

In this paper we study integral operators of the form Tf(x) = |x − a1y| −α1 . . . |x − amy| mf(y) dy, α1 + . . . + αm = n. We obtain the L (w) boundedness for them, and a weighted (1, 1) inequality for weights w in Ap satisfying that there exists c > 1 such that w(aix) 6 cw(x) for a.e. x ∈ n , 1 6 i 6 m. Moreover, we prove ‖Tf‖BMO 6 c‖f‖∞ for a wide family of functions f ∈ L( ).

Some new properties of biharmonic heat kernels

Contrary to the second order case, biharmonic heat kernels are sign-changing. A deep knowledge of their behaviour may however allow to prove positivity results for solutions of the Cauchy problem. We establish further properties of these kernels, we prove some Lorch-Szegö-type monotonicity results and we give some hints on how to obtain similar results for higher polyharmonic parabolic problems.

Composition Operators from the Weighted Bergman Space to the nth Weighted Spaces on the Unit Disc