TY - JOUR
T1 - Correction
T2 - Maximal Operator in Variable Stummel Spaces (Journal of Fourier Analysis and Applications, (2022), 28, 3, (50), 10.1007/s00041-022-09940-8)
AU - Almeida, Alexandre
AU - Rafeiro, Humberto
N1 - Funding Information:
The research of A. Almeida was partially supported by the Center for Research and Development in Mathematics and Applications (CIDMA) through the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia), references UIDB/04106/2020 and UIDP/04106/2020.
Publisher Copyright:
© 2023, Springer Science+Business Media, LLC, part of Springer Nature.
PY - 2023/10
Y1 - 2023/10
N2 - We use all the definitions and notations from [1]. In particular, we say that a function (Formula presented.) is locally (Formula presented.) -Hölder continuous if there exists (Formula presented.) such that (Formula presented.) The function g is said to satisfy the (Formula presented.) -Hölder continuity condition at infinity, also known as the decay condition, if there exist (Formula presented.) and (Formula presented.) such that (Formula presented.) In what follows, we always take (Formula presented.) and (Formula presented.) as the smallest constant satisfying (1.1) and (1.2), respectively. We say that g is (Formula presented.) -Hölder continuous when it satisfies conditions (1.1) and (1.2) simultaneously. The class (Formula presented.) collects all (measurable) bounded exponents (Formula presented.) which are (Formula presented.) -Hölder continuous. Given a set (Formula presented.) , let (Formula presented.) be the Stummel space consisting of all measurable functions f on (Formula presented.) such that (Formula presented.) When (Formula presented.) we write simply (Formula presented.) for the norm and (Formula presented.) for the space (as considered in [1]). The corrected version of [1, Theorem 5.2] runs as follows: Let (Formula presented.) with (Formula presented.) , (Formula presented.) satisfies (1.1), (1.2) with (Formula presented.) , and (Formula presented.). Then, for a bounded set (Formula presented.) , there holds (Formula presented.) If p and (Formula presented.) are constant, then the result holds also with (Formula presented.). In [1, Theorem 5.2] we just considered (Formula presented.). That result holds, for instance, for constant exponents (Formula presented.) and (Formula presented.). However, for variable (Formula presented.) -Hölder exponents in general we need to accommodate the influence of translations on the decay logarithmic constants. We give details below. The small changes needed in the proof essentially rely on additional quantitative information on the dependency of the main constants involved in [1, Lemmas 4.5, 4.7, 4.9 and Propositions 4.1, 4.8] with respect to the logarithmic constants of the exponents. We briefly summarize the quantitative estimates for those constants, which may have independent interest. If (Formula presented.) , (Formula presented.) , and (Formula presented.) , then we have (Formula presented.) for some constant (Formula presented.) independent of (Formula presented.) (cf. [4, Theorem 1.3]). In [1, Remark 3.2] we highlighted the dependency of the constant (Formula presented.) appearing in the inequality above with respect to the weight w by writing (Formula presented.) for some (Formula presented.) depending on n and p only. A further inspection to the proof given in [4] shows that the factor c(n, p) in (1.4) grows continuously and exponentially with the constants (Formula presented.) and (Formula presented.) of the exponent p. From the proof of [1, Lemma 4.5] it is not hard to check that the constant (Formula presented.) given there can be written more precisely as (Formula presented.) with (Formula presented.) independent of the logarithmic constants of p. The quantity (Formula presented.) can be slightly improved taking into account the exact value of the measure of the n-dimensional unit ball, but this is unimportant for our goals. In [1, Lemma 4.7] the constant (Formula presented.) indicated there should be written as (Formula presented.) also for some constant (Formula presented.) not depending on the logarithmic constants of p. Taking into account the explicit form of the constants in (1.5) and (1.6), the estimate given in [1, Proposition 4.8] should be written as (Formula presented.) where (Formula presented.) depends only on n and the infimum and supremum of p. Finally we point out a quantitative result on (Formula presented.) weights which generalizes [1, Lemma 4.9]: ([2]) Let (Formula presented.) be a function satisfying (1.1) and (1.2), and (Formula presented.). Then, for each (Formula presented.) , we have (Formula presented.) with (Formula presented.) where (Formula presented.) depends only on the dimension (Formula presented.). By the translation invariance of (Formula presented.) weights, under the same conditions of Lemma 1.2 we also have (Formula presented.) with the same (Formula presented.) constant. Following the proof given in [1, pp. 19–20], for each (Formula presented.) , we have (Formula presented.) with (Formula presented.) where (Formula presented.) (and s is taken independent of x). Since, for (Formula presented.) , (Formula presented.) the (Formula presented.) decay constant of (Formula presented.) may depend on x (note that the local (Formula presented.) -Hölder constant is translation invariant, i.e. (Formula presented.)). As observed before, the factor (Formula presented.) grows continuously and exponentially with respect to (Formula presented.) , so it may grow with respect to x. Taking into account the bounds in (1.7), (1.8) and (1.10), we see that the remaining factors on the right-hand side in (1.9) may depend also on x. Overall we can find uniform bounds with respect to x if it runs on bounded sets (Formula presented.). In the simpler case of constant exponents (Formula presented.) the translation as no influence on the bounds (with (Formula presented.) in that case), so that one can admit (Formula presented.). (Formula presented.) The formulation of [1, Theorem 5.3] remains almost the same. We only have, with the additional assumption (Formula presented.) , to add (Formula presented.) in the notation of the space and write (Formula presented.) instead. Regarding the proof given in [1, p. 21], the constant C that appears in the first three occurrences for the estimation of (Formula presented.) may depend on x via the bound of (Formula presented.) (cf. (1.8)). Consequently, we should take the supremum with respect to x on the set (Formula presented.) instead on (Formula presented.) everywhere in the proof. Full details of such proof, for the case of Riesz potentials, can be found in [3]. We take the opportunity to correct a misprint in [1, Lemma 2.1]: instead of (Formula presented.) it should be (Formula presented.).
AB - We use all the definitions and notations from [1]. In particular, we say that a function (Formula presented.) is locally (Formula presented.) -Hölder continuous if there exists (Formula presented.) such that (Formula presented.) The function g is said to satisfy the (Formula presented.) -Hölder continuity condition at infinity, also known as the decay condition, if there exist (Formula presented.) and (Formula presented.) such that (Formula presented.) In what follows, we always take (Formula presented.) and (Formula presented.) as the smallest constant satisfying (1.1) and (1.2), respectively. We say that g is (Formula presented.) -Hölder continuous when it satisfies conditions (1.1) and (1.2) simultaneously. The class (Formula presented.) collects all (measurable) bounded exponents (Formula presented.) which are (Formula presented.) -Hölder continuous. Given a set (Formula presented.) , let (Formula presented.) be the Stummel space consisting of all measurable functions f on (Formula presented.) such that (Formula presented.) When (Formula presented.) we write simply (Formula presented.) for the norm and (Formula presented.) for the space (as considered in [1]). The corrected version of [1, Theorem 5.2] runs as follows: Let (Formula presented.) with (Formula presented.) , (Formula presented.) satisfies (1.1), (1.2) with (Formula presented.) , and (Formula presented.). Then, for a bounded set (Formula presented.) , there holds (Formula presented.) If p and (Formula presented.) are constant, then the result holds also with (Formula presented.). In [1, Theorem 5.2] we just considered (Formula presented.). That result holds, for instance, for constant exponents (Formula presented.) and (Formula presented.). However, for variable (Formula presented.) -Hölder exponents in general we need to accommodate the influence of translations on the decay logarithmic constants. We give details below. The small changes needed in the proof essentially rely on additional quantitative information on the dependency of the main constants involved in [1, Lemmas 4.5, 4.7, 4.9 and Propositions 4.1, 4.8] with respect to the logarithmic constants of the exponents. We briefly summarize the quantitative estimates for those constants, which may have independent interest. If (Formula presented.) , (Formula presented.) , and (Formula presented.) , then we have (Formula presented.) for some constant (Formula presented.) independent of (Formula presented.) (cf. [4, Theorem 1.3]). In [1, Remark 3.2] we highlighted the dependency of the constant (Formula presented.) appearing in the inequality above with respect to the weight w by writing (Formula presented.) for some (Formula presented.) depending on n and p only. A further inspection to the proof given in [4] shows that the factor c(n, p) in (1.4) grows continuously and exponentially with the constants (Formula presented.) and (Formula presented.) of the exponent p. From the proof of [1, Lemma 4.5] it is not hard to check that the constant (Formula presented.) given there can be written more precisely as (Formula presented.) with (Formula presented.) independent of the logarithmic constants of p. The quantity (Formula presented.) can be slightly improved taking into account the exact value of the measure of the n-dimensional unit ball, but this is unimportant for our goals. In [1, Lemma 4.7] the constant (Formula presented.) indicated there should be written as (Formula presented.) also for some constant (Formula presented.) not depending on the logarithmic constants of p. Taking into account the explicit form of the constants in (1.5) and (1.6), the estimate given in [1, Proposition 4.8] should be written as (Formula presented.) where (Formula presented.) depends only on n and the infimum and supremum of p. Finally we point out a quantitative result on (Formula presented.) weights which generalizes [1, Lemma 4.9]: ([2]) Let (Formula presented.) be a function satisfying (1.1) and (1.2), and (Formula presented.). Then, for each (Formula presented.) , we have (Formula presented.) with (Formula presented.) where (Formula presented.) depends only on the dimension (Formula presented.). By the translation invariance of (Formula presented.) weights, under the same conditions of Lemma 1.2 we also have (Formula presented.) with the same (Formula presented.) constant. Following the proof given in [1, pp. 19–20], for each (Formula presented.) , we have (Formula presented.) with (Formula presented.) where (Formula presented.) (and s is taken independent of x). Since, for (Formula presented.) , (Formula presented.) the (Formula presented.) decay constant of (Formula presented.) may depend on x (note that the local (Formula presented.) -Hölder constant is translation invariant, i.e. (Formula presented.)). As observed before, the factor (Formula presented.) grows continuously and exponentially with respect to (Formula presented.) , so it may grow with respect to x. Taking into account the bounds in (1.7), (1.8) and (1.10), we see that the remaining factors on the right-hand side in (1.9) may depend also on x. Overall we can find uniform bounds with respect to x if it runs on bounded sets (Formula presented.). In the simpler case of constant exponents (Formula presented.) the translation as no influence on the bounds (with (Formula presented.) in that case), so that one can admit (Formula presented.). (Formula presented.) The formulation of [1, Theorem 5.3] remains almost the same. We only have, with the additional assumption (Formula presented.) , to add (Formula presented.) in the notation of the space and write (Formula presented.) instead. Regarding the proof given in [1, p. 21], the constant C that appears in the first three occurrences for the estimation of (Formula presented.) may depend on x via the bound of (Formula presented.) (cf. (1.8)). Consequently, we should take the supremum with respect to x on the set (Formula presented.) instead on (Formula presented.) everywhere in the proof. Full details of such proof, for the case of Riesz potentials, can be found in [3]. We take the opportunity to correct a misprint in [1, Lemma 2.1]: instead of (Formula presented.) it should be (Formula presented.).
KW - Maximal Operator
KW - Muckenhoupt Weights
KW - Variable Morrey Spaces
KW - Variable Stummel Spaces
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U2 - 10.1007/s00041-023-10033-3
DO - 10.1007/s00041-023-10033-3
M3 - Comment/debate
AN - SCOPUS:85168520273
SN - 1069-5869
VL - 29
JO - Journal of Fourier Analysis and Applications
JF - Journal of Fourier Analysis and Applications
IS - 5
M1 - 54
ER -