Discrete Riesz transforms on rearrangement-invariant Banach sequence spaces and maximally noncompact operators

TY - JOUR

T1 - Discrete Riesz transforms on rearrangement-invariant Banach sequence spaces and maximally noncompact operators

AU - Shargorodsky, Eugene

AU - Karlovych, Oleksiy

N1 - Publisher Copyright: © 2024, Yokohama Publications. All rights reserved.

PY - 2024/3/7

Y1 - 2024/3/7

N2 - We say that an operator between Banach spaces is maximally noncompact if its operator norm coincides with its Hausdorff measure of noncompactness. We prove that a translation-invariant operator acting from a translationinvariant Banach sequence space X(ℤ d) to a translation-invariant Banach sequence space Y(ℤ d) is maximally noncompact whenever the target space Y(ℤ d) satisfies mild additional conditions. As a consequence, we show that the discrete Riesz transforms R j, j = 1, …, d on rearrangement-invariant Banach sequence spaces with non-trivial Boyd indices are maximally noncompact. We also observe that the same results are valid for translation-invariant operators between translation-invariant Banach function spaces A(ℝ d) and Y(ℝ d).

AB - We say that an operator between Banach spaces is maximally noncompact if its operator norm coincides with its Hausdorff measure of noncompactness. We prove that a translation-invariant operator acting from a translationinvariant Banach sequence space X(ℤ d) to a translation-invariant Banach sequence space Y(ℤ d) is maximally noncompact whenever the target space Y(ℤ d) satisfies mild additional conditions. As a consequence, we show that the discrete Riesz transforms R j, j = 1, …, d on rearrangement-invariant Banach sequence spaces with non-trivial Boyd indices are maximally noncompact. We also observe that the same results are valid for translation-invariant operators between translation-invariant Banach function spaces A(ℝ d) and Y(ℝ d).

KW - Discrete Riesz transform

KW - translation-invariant operator

KW - Hausdorff measure of noncompactness

KW - essential norm

KW - translation-invariant space, rearrangement-invariant space

KW - Boyd indices

UR - http://www.scopus.com/inward/record.url?scp=85191337863&partnerID=8YFLogxK

M3 - Article

SN - 2189-3756

VL - 9

SP - 195

EP - 210

JO - Pure and Applied Functional Analysis

JF - Pure and Applied Functional Analysis

IS - 1

ER -