Regularity for Quasi-Elliptic Pseudo-Differential Operators with Symbols in Hölder Classes

Lp-microlocal regularity for pseudodifferential operators of quasi-homogeneous type†

“…We here only deal with the case ρ = 1 for the main reason that symbols in the classes S m M,δ := S m M,1,δ plainly satisfy the Lizorkin-Marcinkiewicz Theorem for L p -Fourier multipliers [6, Ch. IV, §6], which allows to develop the L p -theory of the pseudodiffererential operators for 1 < p < ∞, [3]. The estimates in Proposition 2.1.i yield the inclusion We write σ ∼ j σ j and we call {σ j } j≥0 asymptotic expansion of σ.…”

“…For the adjoint and the product of pseudodifferential operators in Op S m M,δ , a suitable symbolic calculus is developed in [3] with some restrictions on δ; we quote here the result Proposition 2.5: symbolic calculus.…”

“…Segàla [5] for the local solvability, Garello [1] for symbols with decay of type (1, 1), Yamazaki [10] where non-smooth symbols in the L p −framework are introduced and studied under suitable restrictive conditions on the Fourier transform of the symbols themselves. In [2], [3] the authors prove the L p −boundedness of a class of pseudodifferential operators with non-smooth symbols of quasi-homogeneous type, taking their values in Zygmund-Hölder spaces with respect to the first variable. Precisely, in [3] continuity for pseudodifferential operators of (1, δ) quasi-homogeneous type, for 0 ≤ δ ≤ 1, is considered and applied to obtain local regularity results.…”

“…In [2], [3] the authors prove the L p −boundedness of a class of pseudodifferential operators with non-smooth symbols of quasi-homogeneous type, taking their values in Zygmund-Hölder spaces with respect to the first variable. Precisely, in [3] continuity for pseudodifferential operators of (1, δ) quasi-homogeneous type, for 0 ≤ δ ≤ 1, is considered and applied to obtain local regularity results. In the present paper, the arguments in [2], [3] are suitably adapted to studying microlocal Sobolev and Zygmund-Hölder regularity of pseudodifferential operators with both smooth and non-smooth symbols of quasi-homogeneous type.…”

“…Precisely, in [3] continuity for pseudodifferential operators of (1, δ) quasi-homogeneous type, for 0 ≤ δ ≤ 1, is considered and applied to obtain local regularity results. In the present paper, the arguments in [2], [3] are suitably adapted to studying microlocal Sobolev and Zygmund-Hölder regularity of pseudodifferential operators with both smooth and non-smooth symbols of quasi-homogeneous type. In §2, quasi-homogeneous weight functions are introduced together with their main properties.…”