Hölder–Besov boundedness for periodic pseudo-differential operators

Abstract

Abstract. In this work we give Hölder-Besov estimates for periodic Fourier multipliers. We present a class of bounded pseudo-differential operators on periodic Besov spaces with symbols of limited regularity. MSC 2010. Primary 43A22, 43A77; Secondary 43A15.

On some spectral properties of pseudo-differential operators on T

“…The simplest example in this direction is the case of the one dimensional torus, on which we have the concept of periodic pseudo-differential operators acting on periodic functions. These operators have been widely studied [10,5,4,25,17,19,18,29,13] and remarkable results have been obtained, being one of the most important, perhaps the most, the global definition of pseudo-differential operators on the unit circle in [1], proposed by Agranovich in 1979 crediting L.R. Volevich.…”

On the boundedness of periodic pseudo-differential operators

“…It is important to mention that there exists a connection between the L p boundedness of periodic operators and its continuity on Besov spaces. This relation has been studied by the author on general compact Lie groups in [7,Section 3]. Although some results in this paper consider the L p -boundedness of pseudo-differential operators on the torus (for 1 ≤ p < ∞), this problem has been addressed on general compact Lie groups in the references [10,14] for all 1 < p < ∞.…”

Mapping properties for operator-valued pseudodifferential operators on toroidal Besov spaces

“…Mapping properties of toroidal pseudodifferential operators in L p -spaces were studied studied by Delgado [10], Molahajloo-Shahla-Wong [15], Wong [19], Cardona [9] and others. In particular, in Cardona [9] mapping properties in Besov and Hölder spaces are shown.…”

“…Mapping properties of toroidal pseudodifferential operators in L p -spaces were studied studied by Delgado [10], Molahajloo-Shahla-Wong [15], Wong [19], Cardona [9] and others. In particular, in Cardona [9] mapping properties in Besov and Hölder spaces are shown. The global quantization approach mentioned above can be generalized to compact Lie groups, see Ruzhansky-Turunen [17], Ruzhansky-Turunen-Wirth [18], Cardona [8] and references therein.…”