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Corrigendum to “Global heat kernel bounds via desingularizing weights” [J. Funct. Anal. 212 (2004) 373–398]
Journal of Functional Analysis 229 (2005) 238 – 239 www.elsevier.com/locate/jfa
Corrigendum
Corrigendum to “Global heat kernel bounds via desingularizing weights” [J. Funct. Anal. 212 (2004) 373–398]夡 Pierre D. Milman∗ , Yu. A. Semenov Department of Mathematics, University of Toronto, Toronto, Ont., Canada M5S 3G3 Received 1 March 2005; received in revised form 4 March 2005; accepted 9 March 2005
The authors regret reference on p. 389 to the inequality (following Corollary 3) in Section 2.1.6 of “Sobolev Spaces” by V.G. Maz’ya (Springer, New York, Berlin, 1985) in the case that n = 0, when it fails. The authors are grateful to Daniel Levin, who called their attention to this fact. The precise (and only) role of this inequality of Maz’ya in the paper is in the proof of Theorem 1 for = 1, i.e., 7 lines on p. 391, namely: This inequality was applied to improve the conclusion of Theorem 1 by claiming the independence of the constant C = C(d, ) on , 0 < < 1, and then to derive Theorem 1 for = 1 (proof on p. 391). This inequality of Maz’ya is valid for n > 0, which suffices to achieve the same conclusion following nearly the same structure of the proof. In the paper the sentence with the reference to Maz’ya (p. 389) should be replaced as follows: Due to the inequality from [Ma, Section 2.1.6, formula following Corollary 3] hypothesis (1 ) holds with constant cS independent of , 0 < < 1, for a new operator acting on d + n variables A() := H − (V0 ) + (−n ), where H − (V0 ) acts on the first d variables and is as defined on p. 389 and n is the Laplacian in the last n variables, n > 0 (in notations of Maz’ya d = m). The proof of Theorem 1 for 0 < < 1 applies to the operator A() directly; i.e., we verify the assumptions of Theorem A with operator A = A() in place of H − (V0 ) and without any change in the definition of function (depending on the first d variables only). It follows that the heat kernel bound for the semigroup generated by the operator A() admits constant C = C(d, ) 夡
DOI of original article: 10.1016/j.jfa.2003.12.008.
∗ Corresponding author. Fax: +1 416 978 4107.
E-mail addresses: [email protected] (P.D. Milman), [email protected] (Yu. A. Semenov). 0022-1236/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2005.03.002
P.D. Milman, Yu. A. Semenov / Journal of Functional Analysis 229 (2005) 238 – 239
239
independent of , since the constant C in the conclusion of Theorem A depends only on constants cS , j = (d + n)/(d + n − 2), c0 = 1/2 and c1 (the latter coincides for operators A() and H − (V0 ) and is also independent of by the choice of weight and the proof of Theorem 1, 0 < < 1). The heat kernel for the semigroup generated by A() is automatically a product of the heat kernels for the semigroups generated by H − (V0 ) and by −n . The heat kernel of the latter is easy to ‘factor out’ due to the well-known precise formula. In the resulting upper heat kernel bound for the semigroup generated by H − (V0 ), 0 < < 1, constant C = C(d, ) of Theorem 1 does not depend on . Therefore the arguments of the proof of Theorem 1 for = 1 on p. 391 apply, which completes the proof.
Corrigendum
Corrigendum to “Global heat kernel bounds via desingularizing weights” [J. Funct. Anal. 212 (2004) 373–398]夡 Pierre D. Milman∗ , Yu. A. Semenov Department of Mathematics, University of Toronto, Toronto, Ont., Canada M5S 3G3 Received 1 March 2005; received in revised form 4 March 2005; accepted 9 March 2005
The authors regret reference on p. 389 to the inequality (following Corollary 3) in Section 2.1.6 of “Sobolev Spaces” by V.G. Maz’ya (Springer, New York, Berlin, 1985) in the case that n = 0, when it fails. The authors are grateful to Daniel Levin, who called their attention to this fact. The precise (and only) role of this inequality of Maz’ya in the paper is in the proof of Theorem 1 for = 1, i.e., 7 lines on p. 391, namely: This inequality was applied to improve the conclusion of Theorem 1 by claiming the independence of the constant C = C(d, ) on , 0 < < 1, and then to derive Theorem 1 for = 1 (proof on p. 391). This inequality of Maz’ya is valid for n > 0, which suffices to achieve the same conclusion following nearly the same structure of the proof. In the paper the sentence with the reference to Maz’ya (p. 389) should be replaced as follows: Due to the inequality from [Ma, Section 2.1.6, formula following Corollary 3] hypothesis (1 ) holds with constant cS independent of , 0 < < 1, for a new operator acting on d + n variables A() := H − (V0 ) + (−n ), where H − (V0 ) acts on the first d variables and is as defined on p. 389 and n is the Laplacian in the last n variables, n > 0 (in notations of Maz’ya d = m). The proof of Theorem 1 for 0 < < 1 applies to the operator A() directly; i.e., we verify the assumptions of Theorem A with operator A = A() in place of H − (V0 ) and without any change in the definition of function (depending on the first d variables only). It follows that the heat kernel bound for the semigroup generated by the operator A() admits constant C = C(d, ) 夡
DOI of original article: 10.1016/j.jfa.2003.12.008.
∗ Corresponding author. Fax: +1 416 978 4107.
E-mail addresses: [email protected] (P.D. Milman), [email protected] (Yu. A. Semenov). 0022-1236/$ - see front matter © 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.jfa.2005.03.002
P.D. Milman, Yu. A. Semenov / Journal of Functional Analysis 229 (2005) 238 – 239
239
independent of , since the constant C in the conclusion of Theorem A depends only on constants cS , j = (d + n)/(d + n − 2), c0 = 1/2 and c1 (the latter coincides for operators A() and H − (V0 ) and is also independent of by the choice of weight and the proof of Theorem 1, 0 < < 1). The heat kernel for the semigroup generated by A() is automatically a product of the heat kernels for the semigroups generated by H − (V0 ) and by −n . The heat kernel of the latter is easy to ‘factor out’ due to the well-known precise formula. In the resulting upper heat kernel bound for the semigroup generated by H − (V0 ), 0 < < 1, constant C = C(d, ) of Theorem 1 does not depend on . Therefore the arguments of the proof of Theorem 1 for = 1 on p. 391 apply, which completes the proof.