Wiener’s lemma for singular integral operators of Bessel potential type

Spectra of Bochner-Riesz Means onLp

“…Spectral invariance for different function spaces is closely related to algebra of singular integral operators [4,6,13,23] and Wiener's lemma for infinite matrices [10,22,24]. It has been established for singular integral operators with kernels being Hölder continuous and having certain off-diagonal decay [1,6,7,19,23], but it is not well studied yet for Calderon-Zygmund operators, oscillatory integrals, and many other linear operators in Fourier analysis.…”

Differential subalgebras and norm-controlled inversion

“…The concept of differential subalgebras of order θ was introduced in [7,26,32] for θ = 1 and [12,20,36] for θ ∈ (0, 1). We also refer the reader to [3,15,19,20,21,25,33,34,41,42,43,45] for various differential subalgebras of infinite matrices, convolution operators, and integral operators with certain off-diagonal decay.…”

“…For m = 2, the estimate (4.3) to the inverse A −1 ∈ A is essentially established in [19,20] for θ = 1 and [36,40] for θ ∈ (0, 1), and a similar estimate is given in [34]. The reader may refer to [15,21,42,43,45] for norm estimation of the inverse of elements in Banach algebras of infinite matrices and integral operators with certain off-diagonal decay. Remark 4.2.…”

“…For the applications to some mathematical and engineering fields, the widely-used algebras B of infinite matrices and integral operators are the operator algebras B(ℓ p ) and B(L p ), 1 ≤ p ≤ ∞, which are symmetric only when p = 2. In [1,15,36,38,42,48], inverse-closedness of localized matrices and integral operators in B(ℓ p ) and B(L p ), 1 ≤ p ≤ ∞, are discussed, and in [14], Beurling algebras B p,α with 1 ≤ p < ∞ and α > d(1 − 1/p) are shown to admit polynomial norm-controlled inversion in nonsymmetric algebras B(ℓ p ), 1 ≤ p < ∞. It is still widely open to discuss Wiener's lemma and norm-controlled inversion when B and A are not * -algebras and B is not a symmetric algebra.…”

Stability of integral operators on a space of homogeneous type

“…The ‐stability of linear operators arises in the fields of approximation , wavelet and affine frames , Gabor frames and non‐uniform sampling , pseudo‐differential operators and many more. It is also closely related with Wiener's lemma for infinite matrices and localized integral operators .…”