Let p , 1 p ∞, be the space of all p-summable sequences and C a be the convolution operator associated with a summable sequence a. It is known that the p -stability of the convolution operator C a for different 1 p ∞ are equivalent to each other, i.e., if C a has p -stability for some 1 p ∞ then C a has q -stability for all 1 q ∞. In the study of spline approximation, wavelet analysis, time-frequency analysis, and sampling, there are many localized operators of non-convolution type whose stability is one of the basic assumptions. In this paper, we consider the stability of those localized operators including infinite matrices in the Sjöstrand class, synthesis operators with generating functions enveloped by shifts of a function in the Wiener amalgam space, and integral operators with kernels having certain regularity and decay at infinity. We show that the p -stability (or L p -stability) of those three classes of localized operators are equivalent to each other, and we also prove that the left inverse of those localized operators are well localized.

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Abstract. In this paper, we consider localized integral operators whose kernels have mild singularity near the diagonal and certain Hölder regularity and decay off the diagonal. Our model example is the Bessel potential operator J γ , γ > 0. We show that if such a localized integral operator has stability on a weighted function space L p w for some p ∈ [1, ∞) and Muckenhoupt A p -weight w, then it has stability on weighted function spaces L p ′ w ′ for all 1 ≤ p ′ < ∞ and Muckenhoupt A p ′ -weights w ′ .

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