Microlocal properties of bisingular operators

Abstract

Abstract. We study the microlocal properties of bisingular operators, a class of operators on the product of two compact manifolds. We define a wave front set for such operators, and analyse its properties. We compare our wave front set with the SG wave front set, a global wave front set which shares with it formal similarities.

On a class of pseudodifferential operators on the product of compact Lie groups

“…• If 𝑛 𝑖 + 𝑚 𝑖 = 0 for 𝑖 = 1, 2, then there exists 𝐶 > 0 and 𝑎, 𝑏 ∈ ℕ (independent of 𝜎) such that for all 𝑦 ∉ 𝑆 • If 𝑛 𝑖 + 𝑚 𝑖 < 0 and 𝑛 𝑗 + 𝑚 𝑗 = 0 for 𝑖, 𝑗 ∈ {1, 2}, 𝑖 ≠ 𝑗, then 𝐶 > 0 and 𝛾 𝑗 ∈ ℕ 2…”

“…Afterward, a priori estimates and Fredholm properties for bisingular operators were studied by Rabinovič in [11], while in 1975 Rodino in [12] introduced the so-called calculus of bisingular pseudodifferential operators. Other related questions, such as residues and index problems, have been recently considered by Nicola and Rodino in [8], while microlocal properties have been studied by Borsero and Schulz in [2].…”

On a class of pseudodifferential operators on the product of compact Lie groups

“…This finally yields the convergence 2) and (3) the proof is the same (by reversing the roles of the parameters) and it is similar to the one in the case (1). For completeness we show the steps in case (2), that is, when λ ξ 1 > ε 1 ℓ 1 and λ ξ 2 ≤ ε 1 ℓ 2 . Under these hypotheses we have…”

“…Afterwards, a priori estimates and Fredholm properties for bisingular operators were studied by Rabinovic in [11], while in 1975 Rodino in [12] introduced the so-called calculus of bisingular pseudodifferential operators. Other related questions, such as residues and index problems, have been recently considered by Nicola and Rodino in [8], while microlocal properties have been studied by Borsero and Schulz in [2].…”

Ellipticity in the SG calculus

“…Other types of wave-front sets, capturing different notions of regularity, and somehow related to the versions of SG-wave-front set described above exist. To mention a few, we recall the works by M. Borsero and R. Schulz [9], L. Robbiano and C. Zuily [71], L. Rodino and P. Wahlberg [72].…”