Optimal norm estimate of operators related to the harmonic Bergman projection on the ball

Double integral characterizations of harmonic Bergman spaces

“…for all x, y ∈ B; see [11,17,4]. Also, one can see from the various representations of R α (x, y) mentioned above that one of the variables x and y can be allowed to be a boundary point and that the resulting function is still harmonic in the other variable.…”

“…Fix ζ ∈ ∂B. By [4,Lemma 4.2] there exist numbers = (β) > 0, r 0 = r 0 (β) ∈ (0, 1) and C = C (β) > 0 such that…”

“…For various representations of R α (x, y) in terms of series expansion, or fractional derivatives, or integrals based on extended Poisson kernel, we refer to [11,17,4]. In particular, for the unweighted case, we have (modulo a constant factor)…”

Some Estimates for the Norm of the Bergman Projection on Besov Spaces

“…Here the the relation X ≈ Y means that the coefficient X/Y is bounded by some positive constants from above and belove. Also, the sharp estimations of the function I α,s were done in [4] in case when 0 < α < ν, 0 < s < ν for the given ν.…”

Norm of an Integral Operator Related to the Harmonic Bergman Projection

“…This is in the spirit of a result of Zhu [9] dealing with holomorphic Bergman projections. Actually, the true harmonic analog of Zhu's result was also given in [1]: Π α L p α →L p α ≈ p 2 /(p − 1). Note that (1.2) also provides some information on the behavior of T α L p α →L p α as α → −1, that is, T α L p α →L p α grows at most like (α + 1) −1 as α → −1.…”