Stability of Localized Integral Operators on Normal Spaces of Homogeneous Type

Stability for localized integral operators on weighted spaces of homogeneous type

“…For a maximal 𝛾-disjoint set  𝛾 , it is observed in [15,32] that {𝐵(𝑥 𝛾,𝑖 , 𝛾 ′ ) ∶ 𝑥 𝛾,𝑖 ∈  𝛾 } forms a covering of  for any 𝛾 ′ ≥ 2𝐿 0 𝛾, and it follows from the Ahlfors 𝑑-regular property (2.5) and the arguments in the Proposition 2.2 in [15] that…”

“…We can construct a mutually disjoint covering {𝐸(𝑥 𝛾,𝑖 , 𝛾) ∶ 𝑥 𝛾,𝑖 ∈  𝛾 } of the space (, 𝜌, 𝜇) of homogeneous type by applying the similar argument in [15], that is,…”

“…for short, which can be regarded as a discretization of the space (, 𝜌, 𝜇) of homogeneous type at scale 2 −𝑛 [15]. For  = ℝ 𝑑 , the Euclidean space,  𝑛 can be considered as the lattice 2 −𝑛+1 ℤ 𝑑 .…”

“…The above equivalence of stability was investigated by Q. Fang and C.E. Shin [14,15] for the space (, 𝜌, 𝜇) of homogeneous type. In this paper, we will consider the weighted 𝐿 𝑝 𝑤 -stabilities for localized integral operators 𝜆𝐼 + 𝑇, 𝜆 ∈ ℂ, and by the bootstrap argument, the asymptotic estimate technique and the commutator trick, we establish their 𝐿 𝑝 𝑤 -stabilities equivalence for different exponents 1 ≤ 𝑝 < ∞ and 𝑤 ∈  𝑝 .…”

Polynomial control on weighted stability bounds and inversion norms of localized matrices on simple graphs

“…Remark 3.2. The equivalence of unweighted stabilities for different exponents is discussed for matrices in Baskakov-Gohberg-Sjöstrand algebras, Jaffard algebras and Beurling algebras [2,39,41,47], for convolution operators [4], and for localized integral operators of non-convolution type [16,17,34,39]. For a matrix A in the Beurling algebra B r,α with 1 ≤ r ≤ ∞ and α > d G (1−1/r), Shin and Sun use the boot-strap argument in [37] to prove that reciprocal of its optimal lower unweighted stability bound for one exponent is dominated by a polynomial of reciprocal of its optimal lower unweighted stability bound for another exponent,…”