Table of Contents
ISRN Mathematical Analysis
Volumeย 2012, Article IDย 379491, 22 pages
http://dx.doi.org/10.5402/2012/379491
Research Article

Sobolev Regularity in Neutron Transport Theory

Laboratoire de Mathรฉmatiques et Applications, Facultรฉ des Sciences et Techniques, Universitรฉ Sultan Moulay Slimane, BP 523, Beni-Mellal 23000, Morocco

Received 20 September 2011; Accepted 9 November 2011

Academic Editors: M.ย Escobedo, K. A.ย Lurie, and D. X.ย Zhou

Copyright ยฉ 2012 Ahmed Zeghal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The main purpose of this paper is to extend the ๐‘Š1,๐‘ regularity results in neutron transport theory, with respect to the Lebesgue measure due to Mokhtar-Kharroubi, (1991), and to abstract measures covering, in particular, the continuous models or multigroup models. The results are obtained for vacuum boundary conditions as well as periodic boundary conditions. ๐ป2 regularity results are derived when the velocity space is endowed with an appropriate class of measures (signed in the multidimensional case).

1. Introduction

Let ๐ท be an open bounded convex subset of โ„๐‘. Let ๐‘‘๐œ‡ be a positive bounded measure on โ„๐‘ supported by ๐‘‰={๐‘ฃโˆˆโ„๐‘;|๐‘ฃ|<๐‘…},0<๐‘…<+โˆž. We denote ๐‘‹๐‘=๎‚ป๐œ“โˆˆ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ));๐‘ฃโ‹…๐œ•๐œ“๐œ•๐‘ฅโˆˆ๐ฟ๐‘๎‚ผ(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)),(1.1)

where ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))=๐ฟ๐‘(๐ทร—๐‘‰;๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)), 1โ‰ค๐‘<+โˆž, and ๐‘‘๐‘ฅ is the Lebesgue measure on ๐ท.

The streaming operator ๐‘‡ is defined ๐‘‡๐œ“=โˆ’๐‘ฃโ‹…๐œ•๐œ“=๎€ฝ๐œ•๐‘ฅโˆ’๐œŽ(๐‘ฃ)๐œ“,๐ท(๐‘‡)๐œ“โˆˆ๐‘‹๐‘;๐œ“|ฮ“โˆ’๎€พ,=0(1.2) where๐œŽ(โ‹…)โˆˆ๐ฟโˆžฮ“(๐‘‰;๐‘‘๐œ‡(๐‘ฃ)),โˆ’={(๐‘ฅ,๐‘ฃ)โˆˆ๐œ•๐ทร—๐‘‰;๐‘ฃโ‹…๐‘›(๐‘ฅ)<0}.(1.3) Here, ๐‘›(๐‘ฅ) is the unit outward normal at ๐‘ฅโˆˆ๐œ•๐ท. It is well known that under the hypothesis ๐‘‘๐œ‡({0})=0, any function ๐œ“โˆˆ๐‘‹๐‘ possesses traces on ฮ“โˆ’ (see [1โ€“4]).

Let ๐‘˜๐‘–(โ‹…,โ‹…)(๐‘–=1,2) be two measurable functions on ๐‘‰ร—๐‘‰ such that๐พ๐‘–โˆˆโ„’(๐ฟ๐‘(๐‘‰;๐‘‘๐œ‡(๐‘ฃ))),whereKiistheoperatorwithkernel๐‘˜๐‘–,i.e.๐พ๐‘–โˆถ๐œ‘โˆˆ๐ฟ๐‘๎€œ(๐‘‰;๐‘‘๐œ‡(๐‘ฃ))โŸถ๐‘‰๐‘˜๐‘–๎€ทโ‹…,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ.(1.4)

Note that, for all ๐œ†>โˆ’๐œ†โˆ—=โˆ’๐‘‘๐œ‡ess-inf๐œŽ(โ‹…), the operators ๐’ฉ๐œ†=๐พ1(๐œ†โˆ’๐‘‡)โˆ’1๐พ2 and โ„ณ๐œ†=๐พ1(๐œ†โˆ’๐‘‡)โˆ’1 map continuously ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) into itself. In [5], several ๐‘Š1,๐‘ regularity results of the operators ๐’ฉ๐œ† and โ„ณ๐œ† were established when the velocity space ๐‘‰ is endowed with the Lebesque measure. These results play a cornerstone role in the proof of the neutron transport approximations [6โ€“8].

The main purpose of this paper is to extend the results of [5] in two directions. First, we deal with a general class of abstract measures covering, in particular, the continuous models or multigroup models. Secondly, we extend the results to periodic boundary conditions on the torus.

We also give a proof of the ๐ป1/2 Sobolev regularity result, due to Agoshkov, for a general class of measures considered in [9]. We use analogous arguments as in [9] but the result is not a consequence of [9, Theorem 4], (see the commentary on the beginning of Section 4).

Finally, we prove how suitable assumptions on the abstract velocity measure (signed for ๐‘>1) can be useful to derive the ๐ป2 Sobolev regularity for the operator โ„ณ๐œ†.

Before closing the introduction, let us recall that the smoothing effect of velocity averages (โ„ณ0, with ๐‘˜1=1), in terms of the ๐ป1/2 Sobolev regularity, is given in [10] when the velocity space is endowed with the Lebesque measure, while a systematic analysis of the fractional Sobolev regularity, for general velocity measures in ๐ฟ๐‘ space (๐‘>1), is given in [9] (see also [11โ€“16]). These results play a cornerstone role in the analysis of kinetic models (see, e.g., [17, 18]). Finally, velocity averages turn out to play an important role in the context of inverse problems (see [19, 20]).

2. ๐‘Š๐‘ฅ1,๐‘ Regularity of the Operator ๐’ฉ๐œ†=๐พ1(๐œ†โˆ’๐‘‡)โˆ’1๐พ2

Let ๐‘‘๐œ‡(๐‘ฃ)=๐‘‘๐›ผ(๐œŒ)โŠ—๐‘‘๐‘†(๐‘ค), where ๐‘‘๐‘† is the Lebesgue measure on ๐•Š๐‘โˆ’1 (the unit sphere of โ„๐‘) and ๐‘‘๐›ผ is a positive bounded measure on [0,1) such that (for simplicity we assume that ๐‘…=1)๎€œ10๐‘‘๐›ผ(๐œŒ)๐œŒ<โˆž.(2.1) For (๐‘ง,๐‘ฃ,๐‘ฃ๎…ž)โˆˆ๐‘‰, we set๐บ๎€ท๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆถ=๐‘˜1(๐‘ฃ,๐‘ง)๐‘˜2๎€ท๐‘ง,๐‘ฃ๎…ž๎€ธ,๐œ†โˆ—โˆถ=๐‘‘๐œ‡ess-inf๐œŽ(โ‹…).(2.2) Let ๐œ“โˆˆ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)). It is easy to show that for any ๐œ†>โˆ’๐œ†โˆ—๐’ฉ๐œ†๐œ“(๐‘ฅ,๐‘ฃ)=๐พ1(๐œ†โˆ’๐‘‡)โˆ’1๐พ2=๎€œ๐œ“(๐‘ฅ,๐‘ฃ)๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€œ๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎€œ๐‘ก(๐‘ฅ,๐‘ฃโ€ฒโ€ฒ)0๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘’โˆ’๐‘ก(๐œ†+๐œŽ(๐‘ฃโ€ฒโ€ฒ))๐œ“๎€ท๐‘ฅโˆ’๐‘ก๐‘ฃ๎…ž๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ก,(2.3) where ๐‘ก(๐‘ฅ,๐‘ฃ)=inf{๐‘ก>0,๐‘ฅโˆ’๐‘ก๐‘ฃโˆ‰๐ท}.(2.4)

To prove that ๐’ฉ๐œ† is an integral operator, we set๎€œ๐ผ=๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎€œ๐‘ก(๐‘ฅ,๐‘ฃโ€ฒโ€ฒ)0๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘’โˆ’๐‘ก(๐œ†+๐œŽ(๐‘ฃโ€ฒโ€ฒ))๐œ“๎€ท๐‘ฅโˆ’๐‘ก๐‘ฃ๎…ž๎…ž,๐‘ฃ๎…ž๎€ธ=๎€œ๐‘‘๐‘ก10๎€œ๐‘‘๐›ผ(๐œŒ)๐‘†๐‘โˆ’1๎€œ๐‘‘๐‘†(๐‘ค)0๐‘ก(๐‘ฅ,๐‘ค)/๐œŒ๐บ๎€ท๐œŒ๐‘ค,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘’โˆ’๐‘ก(๐œ†+๐œŽ(๐œŒ๐‘ค))๐œ“๎€ท๐‘ฅโˆ’๐‘ก๐œŒ๐‘ค,๐‘ฃ๎…ž๎€ธ=๎€œ๐‘‘๐‘ก10๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œ๐‘†๐‘โˆ’1๎€œ๐‘‘๐‘†(๐‘ค)0๐‘ก(๐‘ฅ,๐‘ค)๐บ๎€ท๐œŒ๐‘ค,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘’โˆ’(๐‘ก/๐œŒ)(๐œ†+๐œŽ(๐œŒ๐‘ค))๐œ“๎€ท๐‘ฅโˆ’๐‘ก๐‘ค,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ก.(2.5) Since ๐ท is convex, the change of variables ๐‘ฅโ€ฒ=๐‘ฅโˆ’๐‘ก๐‘ค (๐‘‘๐‘ฅโ€ฒ=๐‘ก๐‘โˆ’1๐‘‘๐‘ก๐‘‘๐‘†(๐‘ค)) leads to๎€œ๐ผ=10๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œ๐ท๐บ๎€ท๐œŒ๎€ท๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ/||๐‘ฅโˆ’๐‘ฅ๎…ž||๎€ธ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘’โˆ’(|๐‘ฅโˆ’๐‘ฅโ€ฒ|/๐œŒ)(๐œ†+๐œŽ(๐œŒ((๐‘ฅโˆ’๐‘ฅโ€ฒ)/|๐‘ฅโˆ’๐‘ฅ๎…ž|)))||๐‘ฅโˆ’๐‘ฅ๎…ž||๐‘โˆ’1๐œ“๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ฅโ€ฒ.(2.6) Thus, ๐’ฉ๐œ† is an integral operator with kernel๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ=๎€œ10๐บ๎€ท๐œŒ๎€ท๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž๎€ธ/||๐‘ฅโˆ’๐‘ฅ๎…ž||๎€ธ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘’โˆ’(|๐‘ฅโˆ’๐‘ฅโ€ฒ|/๐œŒ)(๐œ†+๐œŽ(๐œŒ((๐‘ฅโˆ’๐‘ฅโ€ฒ)/|๐‘ฅโˆ’๐‘ฅโ€ฒ|)))|๐‘ฅโˆ’๐‘ฅโ€ฒ|๐‘โˆ’1๐‘‘๐›ผ(๐œŒ)๐œŒ.(2.7) Let us now introduce the following hypotheses: (๐’œ1)๐บ๎€ท๐œŽ(๐‘ง)=๐œŽ(โˆ’๐‘ง),๐‘‘๐œ‡(๐‘ฃ)a.e.,๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎€ท=๐บโˆ’๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ,๐‘‘๐œ‡(๐‘ฃ)a.e.,(2.8)(๐’œ2)๐’ขโˆžโˆˆโ„’(๐ฟ๐‘(๐’ข๐‘‰;๐‘‘๐œ‡(๐‘ฃ))),whereโˆžistheintegraloperatorwithkernel๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆถ=sup๐‘งโˆˆ๐‘‰||๐บ๎€ท๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||,(2.9)(๐’œ3)๐’ขโˆˆโ„’(๐ฟ๐‘(๐‘‰;๐‘‘๐œ‡(๐‘ฃ))),where๐’ขistheintegraloperatorwithkernel๐บ๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆถ=sup๐‘งโˆˆ๐‘‰||๐บ๎€ท๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||,|๐‘ง|(2.10)(๐’œ4)๐’ข๐‘–โˆˆโ„’(๐ฟ๐‘(๐’ข๐‘‰;๐‘‘๐œ‡(๐‘ฃ))),where๐‘–istheintegraloperatorwithkernel๐บ๐‘–๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆถ=sup๐‘งโˆˆ๐‘‰||||๐œ•๐บ๐œ•๐‘ง๐‘–๎€ท๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||||,1โ‰ค๐‘–โ‰ค๐‘,(2.11)(๐’œ5)๐œŽ(โ‹…)โˆˆ๐‘Š1,โˆž(๐‘‰;๐‘‘๐œ‡(๐‘ฃ)).(2.12)

Remark 2.1. (1) (๐’œ1) is the key assumption.
(2) Assumption (๐’œ2) is a consequence of (๐’œ3).
(3) Assumption (๐’œ3) is not necessary if ๐‘‘๐œ‡ is the Lebesgue measure (see [5]).
Throughout this paper we set ๐œŽ(โ‹…)=๐œ†+๐œŽ(โ‹…).(2.13)

Now, we are ready to state our first main result.

Theorem 2.2. Let assumptions (๐’œ1),(๐’œ3)โ€“(๐’œ5) be satisfied. Then, for any ๐œ†>โˆ’๐œ†โˆ— and ๐‘>1, the operator ๐’ฉ๐œ† maps continuously ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) into ๐‘Š๐‘ฅ1,๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)), where ๐‘Š๐‘ฅ1,๐‘๎‚ป(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))โˆถ=๐œ“โˆˆ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))suchthat๐œ•๐œ“๐œ•๐‘ฅ๐‘–โˆˆ๐ฟ๐‘๎‚ผ(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)).(2.14) Moreover, โ€–๐’ฉ๐œ†โ€–โ„’(๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ));๐‘Š๐‘ฅ1,๐‘) is locally bounded with respect to ๐œ†>โˆ’๐œ†โˆ—.

Proof . Let us first compute ๐œ•๐’ฉ๐œ†๐œ‘/๐œ•๐‘ฅ๐‘– in the distributional sense for ๐œ‘โˆˆ๐’Ÿ(๐ทร—๐‘‰). To this end, let ๐œ“โˆˆ๐’Ÿ(๐ทร—๐‘‰), and extend ๐‘˜๐‘–(โ‹…,โ‹…), ๐‘–=1,2, by zero outside of ๐‘‰; then ๎ƒก๐œ•๐’ฉ๐œ†๐œ‘๐œ•๐‘ฅ๐‘–๎ƒข,๐œ“๐’Ÿ๎…ž,๐’Ÿ๎€œ=โˆ’๐ทร—๐‘‰๐œ•๐œ“๐œ•๐‘ฅ๐‘–๎€œ(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅ๐‘‘๐œ‡(๐‘ฃ)๐ทร—๐‘‰๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ฅ๎…ž๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ=โˆ’lim๐œ€โ†’0๎€œ๐‘‰ร—๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡(๐‘ฃ)๐‘‘๐œ‡๎…ž๎€ธ๎€œโ„๐‘๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ฅ๎…žร—๎€œ||๐‘ฅโˆ’๐‘ฅโ€ฒ||โ‰ฅ๐œ–๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ•๐œ“๐œ•๐‘ฅ๐‘–(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅ.(2.15) The use of the Green formula leads to lim๐œ€โ†’0๎‚ธ๎€œ๐‘‰ร—๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡(๐‘ฃ)๐‘‘๐œ‡๎…ž๎€ธ๎€œโ„๐‘๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๎€œ๐‘‘๐‘ฅโ€ฒ|๐‘ฅโˆ’xโ€ฒ|โ‰ฅ๐œ–๐œ•๐‘๐œ†๐œ•๐‘ฅ๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆ’๎€œ๐œ“(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅ๐‘‰ร—๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡(๐‘ฃ)๐‘‘๐œ‡๎…ž๎€ธ๎€œโ„๐‘๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ฅ๎…ž๎€œ||๐‘ฅโˆ’๐‘ฅโ€ฒ||=๐œ–๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ“(๐‘ฅ,๐‘ฃ)๐œˆ๐‘–๎ƒญ.๐‘‘๐œŽ(๐‘ฅ)(2.16)
We claim that the second part of (2.16) goes to zero as ๐œ€โ†’0.
Indeed, ๎€œ|๐‘ฅโˆ’๐‘ฅโ€ฒ|=๐œ–๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ“(๐‘ฅ,๐‘ฃ)๐œˆ๐‘–๎€ท๐‘ฅ๐‘‘๐œŽ(๐‘ฅ)=๐œ“๎…ž๎€ธ๎€œ,๐‘ฃ|๐‘ฅโˆ’๐‘ฅโ€ฒ|=๐œ–๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œˆ๐‘–+๎€œ๐‘‘๐œŽ(๐‘ฅ)|๐‘ฅโˆ’๐‘ฅโ€ฒ|=๐œ–๐‘๐œ†๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||๐‘ฅโˆ’๐‘ฅ๎…ž||๐‘‚๎€ท||๐‘ฅโˆ’๐‘ฅ๎…ž||๎€ธ๐œˆ๐‘–๐‘‘๐œŽ(๐‘ฅ)=๐ผ1+๐ผ2.(2.17) Clearly ๐ผ1=0 (because ๐‘๐œ†(๐‘ฆ,๐‘ฃ,๐‘ฃโ€ฒ)=๐‘๐œ†(โˆ’๐‘ฆ,๐‘ฃ,๐‘ฃโ€ฒ)). Thus, using the dominated convergence theorem of Lebesgue, it is enough to prove that ๐ผ2โ†’0 as ๐œ€ goes to zero. This follows from the estimate ||๐ผ2||โ‰ค๎‚ต๎€œ10๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎€œ|๐‘ฅโˆ’๐‘ฅโ€ฒ|=๐œ€๐‘‘๐œŽ(๐‘ฅ)๐œ€๐‘โˆ’2=๎‚ต๎€œ10๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ||๐•Š๐‘โˆ’1||๐œ€,(2.18) which proves the claim.
Next, we compute the derivative of the kernel ๐‘๐œ†.
Indeed, by straightforward calculations ๐œ•๐‘๐œ†๐œ•๐‘ฅ๐‘–๎€ท๐‘ฅ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎“=โˆ’๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+2๐บ๐‘—๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))+๎€œ๐‘‘๐›ผ(๐œŒ)10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+2๐บ๐‘–๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))+๎“๐‘‘๐›ผ(๐œŒ)๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+1๐œŽ๐‘—๎‚ต๐œŒ๐‘ฅ|๎‚ถ๐บ๎‚ต๐œŒ๐‘ฅ๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+1๐œŽ๐‘–๎‚ต๐œŒ๐‘ฅ๎‚ถ๐บ๎‚ต๐œŒ๐‘ฅ|๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10๐‘ฅ๐‘–|๐‘ฅ|๐‘๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ๐บ๎€ท|๐‘ฅ|๐œŒ(๐‘ฅ/|๐‘ฅ|),๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŒ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œโˆ’(๐‘โˆ’1)10๐‘ฅ๐‘–|๐‘ฅ|๐‘+1๐บ๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒ,(2.19) where ๐œŽ๐‘—(โ‹…)โˆถ=๐œ•๐œŽ๐œ•๐‘ง๐‘—(โ‹…),๐บ๐‘—๎€ทโ‹…,๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆถ=๐œ•๐บ๐œ•๐‘ง๐‘—๎€ทโ‹…,๐‘ฃ,๐‘ฃ๎…ž๎€ธ.(2.20)
It is easily seen that ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))=1โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐ต(๐‘ฅ,๐œŒ), where ๐ต(โ‹…,โ‹…) is a bounded function.
Therefore, ๐œ•๐‘๐œ†/๐œ•๐‘ฅ๐‘– may be decomposed as ๐œ•๐‘๐œ†๐œ•๐‘ฅ๐‘–๎€ท๐‘ฅ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎“=โˆ’๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+2๐บ๐‘—๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๎€œ๐‘‘๐›ผ(๐œŒ)+10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+2๐บ๐‘–๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ+๎“๐‘‘๐›ผ(๐œŒ)๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+1๐บ๐‘—๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ|๐‘ฅ|๐ต(๐‘ฅ,๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+1๐บ๐‘–๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ๐ต|๐‘ฅ|(๐‘ฅ,๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒ+๎“๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+1๐œŽ๐‘—๎‚ต๐œŒ๐‘ฅ|๎‚ถ๐บ๎‚ต๐œŒ๐‘ฅ๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+1๐œŽ๐‘–๎‚ต๐œŒ๐‘ฅ|๎‚ถ๐บ๎‚ต๐œŒ๐‘ฅ๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10๐‘ฅ๐‘–|๐‘ฅ|๐‘๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ๐บ๎€ท|๐‘ฅ|๐œŒ(๐‘ฅ/|๐‘ฅ|),๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŒ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œโˆ’(๐‘โˆ’1)10๐‘ฅ๐‘–|๐‘ฅ|๐‘+1๐บ๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œ+(๐‘โˆ’1)10๐‘ฅ๐‘–|๐‘ฅ|๐‘๐บ๎€ท๐œŒ(๐‘ฅ/|๐‘ฅ|),๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŒ๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ|๐‘ฅ|๐ต(๐‘ฅ,๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒ.(2.21) Set ๐‘†๐‘–๎€ท๐‘ฅ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎“โˆถ=โˆ’๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|2๐บ๐‘—๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๎€œ๐‘‘๐›ผ(๐œŒ)+10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|2๐บ๐‘–๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๎€œ๐‘‘๐›ผ(๐œŒ)โˆ’(๐‘โˆ’1)10๐‘ฅ๐‘–๐บ๎‚ต๐œŒ๐‘ฅ|๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘‘๐›ผ(๐œŒ)๐œŒ,๐‘…๐‘–๎€ท๐‘ฅ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎“โˆถ=๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+1๐บ๐‘—๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ|๐‘ฅ|๐ต(๐‘ฅ,๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+1๐บ๐‘–๎‚ต๐œŒ๐‘ฅ|๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ|๐‘ฅ|๐ต(๐‘ฅ,๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒ+๎“๐‘—โ‰ ๐‘–๎€œ10๐‘ฅ๐‘–๐‘ฅ๐‘—|๐‘ฅ|๐‘+1๐œŽ๐‘—๎‚ต๐œŒ๐‘ฅ๎‚ถ๐บ๎‚ต๐œŒ๐‘ฅ|๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|)๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10|๐‘ฅ|2โˆ’๐‘ฅ2๐‘–|๐‘ฅ|๐‘+1๐œŽ๐‘–๎‚ต๐œŒ๐‘ฅ๎‚ถ๐บ๎‚ต๐œŒ๐‘ฅ|๐‘ฅ||๐‘ฅ|,๐‘ฃ,๐‘ฃ๎…ž๎‚ถ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|)๐‘‘๐›ผ(๐œŒ)๐œŒโˆ’๎€œ10๐‘ฅ๐‘–|๐‘ฅ|๐‘๐œŽ๎‚ต๐œŒ๐‘ฅ๎‚ถ|๐‘ฅ|๐บ(๐œŒ(๐‘ฅ/|๐‘ฅ|),๐‘ฃ,๐‘ฃโ€ฒ)๐œŒ๐‘’โˆ’(|๐‘ฅ|/๐œŒ)๐œŽ(๐œŒ(๐‘ฅ/|๐‘ฅ|))๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œ+(๐‘โˆ’1)10๐‘ฅ๐‘–|๐‘ฅ|๐‘๐บ(๐œŒ(๐‘ฅ/|๐‘ฅ|),๐‘ฃ,๐‘ฃโ€ฒ)๐œŒ๐œŽ๎‚ต๐œŒ๐‘ฅ|๎‚ถ๐‘ฅ|๐ต(๐‘ฅ,๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒ.(2.22)
Now, (2.16) becomes ๎ƒก๐œ•๐‘๐œ†๐œ‘๐œ•๐‘ฅ๐‘–๎ƒข,๐œ“๐’Ÿ๎…ž,๐’Ÿ=lim๐œ€โ†’0๎‚ธ๎€œ๐‘‰ร—๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡(๐‘ฃ)๐‘‘๐œ‡๎…ž๎€ธ๎€œโ„๐‘๎€œ๐œ“(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅ|๐‘ฅโˆ’๐‘ฅโ€ฒ|โ‰ฅ๐œ–๐‘…๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ+๎€œ๐‘‘๐‘ฅโ€ฒ๐‘‰ร—๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡(๐‘ฃ)๐‘‘๐œ‡๎…ž๎€ธ๎€œโ„๐‘๎€œ๐œ“(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅ|๐‘ฅโˆ’๐‘ฅโ€ฒ|โ‰ฅ๐œ–๐‘†๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ||๐‘ฅโˆ’๐‘ฅ๎…ž||๐‘๎ƒญ๐‘‘๐‘ฅโ€ฒ=๐ฝ1+๐ฝ2.(2.23) Clearly, ||๐‘…๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||โ‰ค๐ถ๎‚€๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ+๐บ๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ+โˆ‘๐‘๐‘—=1๐บ๐‘—๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎‚||๐‘ฅโˆ’๐‘ฅ๎…ž||๐‘โˆ’1,(2.24) where ๎‚ต๎€œ๐ถ=10๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๎ƒฉmax๐‘๎“๐‘—=1โ€–โ€–๐œŽ๐‘—โ€–โ€–โˆž,๎€ท๐‘โ€–๐ตโ€–โˆž+1๎€ธ๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธ๎ƒช.(2.25) Hence, according to (๐’œ2)โ€“(๐’œ4) and the dominated convergence theorem of Lebesgue ๐ฝ2=๎€œ๐ทร—๐‘‰๎€œ๐œ“(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅ๐‘‘๐œ‡(๐‘ฃ)๐ทร—๐‘‰๐‘…๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ฅ๎…ž๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ.(2.26) Therefore, using the Hรถlder inequality and the boundedness of ๐ท, the mapping ๎€œ๐œ“โŸถ๐ทร—๐‘‰๐‘…๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ“๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ๐‘‘๐‘ฅ๎…ž๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ(2.27)defines a bounded operator โ„›๐‘–โˆˆโ„’(๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡)).
Note, parenthetically, that from (2.25) we deduce the second part of the theorem.
Next, to deal with ๐ฝ1, we first consider the truncated operator ๐’ฎ๐‘–,๐œ–๎€œโˆถ๐œ‘โŸถ|๐‘ฅโˆ’๐‘ฅ๎…ž|โ‰ฅ๐œ–๐‘†๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ|๐‘ฅโˆ’๐‘ฅโ€ฒ|๐‘๎€ท๐‘‘๐‘ฅโ€ฒfor๏ฌxed๐‘ฃ,๐‘ฃ๎…ž๎€ธ.(2.28) It is easily seen that ๐‘†๐‘–(โ‹…,๐‘ฃ,๐‘ฃโ€ฒ) is positively homogeneous of degree 0 and even and satisfies ||๐‘†๐‘–๎€ท๐‘ฅ,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||โ‰ค๎‚ต๎€œ10๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๎ƒฉ(๐‘โˆ’1)๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ+๐‘๎“๐‘—=1๐บ๐‘—๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎ƒช.(2.29) So, ๐’ฎ๐‘–,๐œ– is a Caldรฉron-Zygmund operator which converges in ๐ฟ๐‘, as ๐œ€โ†’0 to ๐’ฎ๐‘– (see [3, 21]), where ๐’ฎ๐‘–๎€œ๐œ‘โˆถ=๐‘โ‹…๐‘ฃโ‹…โ„๐‘๐‘†๐‘–๎€ท๐‘ฅโˆ’๐‘ฅ๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฅ๎…ž,๐‘ฃ๎…ž๎€ธ|๐‘ฅโˆ’๐‘ฅโ€ฒ|๐‘๐‘‘๐‘ฅโ€ฒ.(2.30) Furthermore, the truncated maximal operator ๐’ฎโˆ—๐‘– defined by (see [3, 21]) ๐’ฎโˆ—๐‘–โˆถ=sup๐œ€>0||๐’ฎโˆ—๐‘–๐œ‘||(2.31) satisfies โ€–๐’ฎโˆ—๐‘–โ€–๐ฟ๐‘๐‘ฅ(โ„๐‘)โ‰ค๐ถโ€ฒโ€–๐‘†๐‘–๎€ทโ‹…,๐‘ฃ,๐‘ฃ๎…ž๎€ธโ€–๐ฟโˆž(๐‘†๐‘โˆ’1)๎€ทโ€–๐œ‘โ‹…,๐‘ฃ๎…ž๎€ธโ€–๐ฟ๐‘๐‘ฅ(โ„๐‘).(2.32) By the Hรถlder inequality and (2.29) ||||๎€œ๐œ“(๐‘ฅ,๐‘ฃ)๐’ฎ๐‘–,๐œ€๐œ‘||||(๐‘ฅ,๐‘ฃ)๐‘‘๐‘ฅโ‰ค๐ถ๎…ž๎…ž๎ƒฌ(๐‘โˆ’1)๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ+๐‘๎“๐‘—=1๐บ๐‘—๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎ƒญ๎‚ต๎€œ|๐œ“(๐‘ฅ,๐‘ฃ)|๐‘ž๎‚ถ1/๐‘ž๎€ทโ€–๐œ‘โ‹…,๐‘ฃ๎…ž๎€ธโ€–๐ฟ๐‘๐‘ฅ(โ„๐‘),(2.33) where ๐‘ž is the conjugate of ๐‘, that is, ๐‘ž=๐‘/(๐‘โˆ’1). Finally, by the dominated convergence theorem of Lebesque ๐ฝ2=(๐’ฎ๐‘–๐œ‘,๐œ“)๐ฟ๐‘,๐ฟ๐‘ž,๐’ฎ๐‘–๎€ท๐ฟโˆˆโ„’๐‘๎€ทโ„๐‘,ร—๐‘‰;๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)๎€ธ๎€ธ(2.34) which amounts to ๎ƒก๐œ•๐’ฉ๐œ†๐œ‘๐œ•๐‘ฅ๐‘–๎ƒข,๐œ“๐’Ÿ๎…ž,๐’Ÿ=๎€ทโ„›๐‘–๎€ธ๐œ‘,๐œ“๐ฟ๐‘,๐ฟ๐‘ž+๎€ท๐’ฎ๐‘–๎€ธ๐œ‘,๐œ“๐ฟ๐‘,๐ฟ๐‘ž.(2.35) The density of ๐’Ÿ in ๐ฟ๐‘(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡) concludes the proof.

Remark 2.3. We note that Theorem 2.2 has already been obtained by [5, Theorem 1] of Mokhtar-Kharroubi when the velocity space is endowed with the Lebesque measure. The choice of an abstract measure is motivated by our desire to give a unified treatment which covers either continuous models (the Lebesque measures on open subsets of โ„๐‘) or multigroup models (the Lebesque measure on spheres).

3. ๐ป1๐‘ฅ Regularity of the Operator ๐’ฉ๐œ†=๐พ1(๐œ†โˆ’๐‘‡๐‘)โˆ’1๐พ2 in Periodic Transport

Let us first precise the functional setting of our problem. Let ๐œ‡ be a positive bounded measure on ๐‘‰ satisfying the following: ๐‘‘๐œ‡isinvariantbysymmetrywithrespecttotheorigin(3.1)ess-sup๐‘คโˆˆ๐•Š๐‘โˆ’1๐œ‡{๐‘ฃโˆˆ๐‘‰;|๐‘ฃโ‹…๐‘ค|โ‰ค๐œ–}โ‰ค๐ถ๐œ–,(3.2) where ๐ถ is a positive constant.

We define the streaming operator ๐‘‡๐‘ on ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) by๐‘‡๐‘๐œ“=โˆ’๐‘ฃโ‹…๐œ•๐œ“๐ท๎€ท๐‘‡๐œ•๐‘ฅโˆ’๐œŽ(๐‘ฃ)๐œ“,๐‘๎€ธ=๎€ฝ๐œ“โˆˆ๐‘‹2;๐œ“|๐‘ฅ๐‘–=0=๐œ“|๐‘ฅ๐‘–=2๐œ‹๎€พ,(1โ‰ค๐‘–โ‰ค๐‘)(3.3) where ๐ท=(0,2๐œ‹)๐‘. We expand ๐œ“โˆˆ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) into the Fourier series with respect to ๐‘ฅ:๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๐‘“๐‘˜(๐‘ฃ)๐‘’๐‘–(๐‘ฅโ‹…๐‘˜),(3.4) where ๐‘“๐‘˜(1๐‘ฃ)=(2๐œ‹)๐‘›/2๎€œ๐ท๐œ“(๐‘ฅ,๐‘ฃ)๐‘’โˆ’๐‘–(๐‘ฅโ‹…๐‘˜)๐‘‘๐‘ฅโˆˆ๐ฟ2(๐‘‰;๐‘‘๐œ‡(๐‘ฃ)),๐‘˜โˆˆโ„ค๐‘›,(3.5) and by the Parseval formula โ€–๐œ“โ€–2๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))=๎“๐‘˜โˆˆโ„ค๐‘๎€œ๐‘‰|๐‘“๐‘˜(๐‘ฃ)|2๐‘‘๐œ‡(๐‘ฃ)<โˆž.(3.6) By simple computations we have๐’ฉ๐œ†๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๎ƒฌ๎€œ๐‘‰๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€œ๐‘‰๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)+๐‘–(๐‘ฃ๎…ž๎…ž๎€ท๐‘ฃโ‹…๐‘˜)๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎ƒญ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(3.7) The regularity result of ๐’ฉ๐œ† is obtained under the following two weaker assumptions: (๐’œ7)๐œŽ(๐‘ฃ)=๐œŽ(โˆ’๐‘ฃ),๐‘‘๐œ‡(๐‘ฃ)a.e.,๐œŽโˆˆ๐ฟโˆž(๐‘‰;๐‘‘๐œ‡(๐‘ฃ)),(3.8)(๐’œ8)๐บ๎€ท๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎€ท=๐บโˆ’๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ,๐บ,๐‘‘๐œ‡(๐‘ฃ)a.e.โˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธโˆถ=ess-sup๐‘งโˆˆ๐‘‰||๐บ๎€ท๐‘ง,๐‘ฃ,๐‘ฃ๎…ž๎€ธ||โˆˆ๐ฟ2(๐‘‰ร—๐‘‰;๐‘‘๐œ‡โŠ—๐‘‘๐œ‡),(3.9)

where ๐บ is defined by (2.2). The following lemma will play a crucial role in the proof of the main result.

Lemma 3.1. Let ๐‘˜โˆˆโ„ค๐‘ such that ๐‘˜โ‰ (0,โ€ฆ,0). Then, ๎€œ๐‘‰๐‘‘๐œ‡(๐‘ฃ)๎€ท๐œ†+๐œ†โˆ—๎€ธ2+(๐‘ฃโ‹…๐‘˜)2โ‰ค๐ถ0||๐‘˜||,(3.10) where ๐ถ0 is a positive constant.

Proof. We write ๎€œ๐‘‰๐‘‘๐œ‡(๐‘ฃ)(๐œ†+๐œ†โˆ—)2+(๐‘ฃโ‹…๐‘˜)2=๎€œ|๐‘ฃโ‹…(๐‘˜/|๐‘˜|)|<๐›ผ๐‘‘๐œ‡(๐‘ฃ)(๐œ†+๐œ†โˆ—)2+(๐‘ฃโ‹…๐‘˜)2+๎€œ|๐‘ฃโ‹…(๐‘˜/|๐‘˜|)|โ‰ฅ๐›ผ๐‘‘๐œ‡(๐‘ฃ)(๐œ†+๐œ†โˆ—)2+(๐‘ฃโ‹…๐‘˜)2,(3.11) where ๐›ผ<1 is to be chosen later. It is clear that ๎€œ|๐‘ฃโ‹…(๐‘˜/|๐‘˜|)|<๐›ผ๐‘‘๐œ‡(๐‘ฃ)(๐œ†+๐œ†โˆ—)2+(๐‘ฃโ‹…๐‘˜)2โ‰ค1(๐œ†+๐œ†โˆ—)2๐œ‡๎ƒฏ||||๐‘˜๐‘ฃโˆˆ๐‘‰,๐‘ฃโ‹…||๐‘˜||||||๎ƒฐโ‰ค<๐›ผ๐ถ๐›ผ(๐œ†+๐œ†โˆ—)2.(3.12) Now, let ๐›ฝ be the image of ๐œ‡ under the orthogonal projection on the direction ๐‘˜/|๐‘˜|. Hence, ๎€œ|๐‘ฃโ‹…(๐‘˜/|๐‘˜|)|โ‰ฅ๐›ผ๐‘‘๐œ‡(๐‘ฃ)๎€ท๐œ†+๐œ†โˆ—๎€ธ2+(๐‘ฃโ‹…๐‘˜)2=๎€œ1๐›ผ๐‘‘๐›ฝ(๐‘ก)๎€ท๐œ†+๐œ†โˆ—๎€ธ2+||๐‘˜||2๐‘ก2=1||๐‘˜||2๎€œ1๐›ผ||๐‘˜||2๐‘ก2๎€ท๐œ†+๐œ†โˆ—๎€ธ2+||๐‘˜||2๐‘ก2๐‘‘๐›ฝ(๐‘ก)๐‘ก2โ‰ค1||๐‘˜||2๎€œ1๐›ผ๐‘‘๐›ฝ(๐‘ก)๐‘ก2.(3.13) Let โˆซ๐œ(๐‘ก)=๐‘ก๐›ผ๐‘‘๐›ฝ(๐‘ ) for ๐‘ก>๐›ผ. An integration by parts yields ๎€œ1๐›ผ๐‘‘๐›ฝ(๐‘ก)๐‘ก2=๎‚ธ๐œ(๐‘ก)๐‘ก2๎‚น1๐›ผ๎€œ+21๐›ผ๐œ(๐‘ก)๐‘ก3๐‘‘๐‘ก.(3.14) Since ๐œ(๐‘ก)=๐›ฝ([๐›ผ,๐‘ก])โ‰ค๐›ฝ([โˆ’๐‘ก,๐‘ก])โ‰ค๐ถ๐‘ก by (3.2), it follows that ๎€œ1๐›ผ๐‘‘๐›ฝ(๐‘ก)๐‘ก2โ‰ค๐ถ+2๐ถ๐›ผโˆ’2๐ถโ‰ค2๐ถ๐›ผ.(3.15) By choosing ๐›ผ=๐œ€/|๐‘˜| with ๐œ€<1, we obtain ๎€œ๐‘‰๐‘‘๐œ‡(๐‘ฃ)๎€ท๐œ†+๐œ†โˆ—๎€ธ2+(๐‘ฃโ‹…๐‘˜)2โ‰ค๐ถ0||๐‘˜||,(3.16) where ๐ถ0 is a positive constant. This achieves the proof of the lemma.

Theorem 3.2. Under assumptions (๐’œ7)โ€“(๐’œ8), the operator ๐’ฉ๐œ†(๐œ†>โˆ’๐œ†โˆ—) maps continuously ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) into ๐ป1๐‘ฅ(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)), where ๐ป1๐‘ฅ๎‚ป(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))=๐œ“โˆˆ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)),๐œ•๐œ“๐œ•๐‘ฅ๐‘–โˆˆ๐ฟ2๎‚ผ(๐‘‘xโŠ—๐‘‘๐œ‡(๐‘ฃ)),for1โ‰ค๐‘–โ‰ค๐‘.(3.17) Moreover, โ€–๐’ฉ๐œ†โ€–โ„’(๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ));๐ป1๐‘ฅ(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))) is locally bounded with respect to ๐œ†>โˆ’๐œ†โˆ—.

Proof. Let ๐œ“โˆˆ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)). We recall that ๐’ฉ๐œ†๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๎ƒฌ๎€œ๐‘‰๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€œ๐‘‰๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)+๐‘–(๐‘ฃ๎…ž๎…ž๎€ท๐‘ฃโ‹…๐‘˜)๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎ƒญ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(3.18) Since ๎€œ๐‘‰๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)+๐‘–(๐‘ฃ๎…ž๎…ž๎€ท๐‘ฃโ‹…๐‘˜)๐‘‘๐œ‡๎…ž๎…ž๎€ธ=๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎…ž๎€ธ๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)2+(๐‘ฃ๎…ž๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎€œโˆ’๐‘–๐‘‰๎€ท๐‘ฃ๎…ž๎…ž๎€ธ๐บ๎€ท๐‘ฃโ‹…๐‘˜๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)2+(๐‘ฃ๎…ž๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ,(3.19) and thanks to the evenness of ๐œŽ(โ‹…),๐บ(โ‹…,๐‘ฃ,๐‘ฃโ€ฒ) together with the fact that ๐‘‘๐œ‡ is invariant by symmetry with respect to the origin, the last integral vanishes. Therefore, ๐’ฉ๐œ†๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๎ƒฌ๎€œ๐‘‰๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎…ž๎€ธ๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)2+(๐‘ฃ๎…ž๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎ƒญ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜),(3.20) and consequently ๐œ•๐œ•๐‘ฅ๐‘—๐’ฉ๐œ†๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๐‘–๐‘˜๐‘—๎ƒฌ๎€œ๐‘‰๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎…ž๎€ธ๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)2+(๐‘ฃ๎…ž๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ๎ƒญ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(3.21) Now, (๐’œ8) and Lemma 3.1 lead to ||||๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎…ž๎€ธ๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)2+(๐‘ฃ๎…ž๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ||||โ‰ค๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธ๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎€œ๐‘‰๐‘‘๐œ‡(๐‘ฃ)๎€ท๐œ†+๐œ†โˆ—๎€ธ2+(๐‘ฃโ‹…๐‘˜)2โ‰ค๐ถ0๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธ๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ||๐‘˜||.(3.22) By the Hรถlder inequality, we obtain ||||๐‘˜๐‘—๎€œ๐‘‰๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎…ž๎€ธ๐บ๎€ท๐‘ฃ๎…ž๎…ž,๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž๎…ž)2+(๐‘ฃ๎…ž๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎…ž๎€ธ||||โ‰ค๐ถ0๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธ๎‚ต๎€œ๐‘‰||๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ||2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎‚ถ1/2๎‚ต๎€œ๐‘‰๐บโˆž๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎‚ถ1/2(3.23) and by the Parseval formula โ€–โ€–โ€–๐œ•๐œ•๐‘ฅ๐‘–๐’ฉ๐œ†๐œ“โ€–โ€–โ€–๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))โ‰ค๐ถ0๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธโ€–๐บโˆžโ€–๐ฟ2(๐‘‰ร—๐‘‰)โ€–๐œ“โ€–๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)).(3.24) This ends the proof of the theorem.

Remark 3.3. The measure ๐‘‘๐œ‡(๐‘ฃ)=๐‘‘๐›ผ(๐œŒ)โŠ—๐‘‘๐‘†(๐‘ค), with โˆซ10(๐‘‘๐›ผ(๐œŒ)/๐œŒ)<+โˆž, satisfies (3.2). Thus, it covers the continuous models as well as the multigroup models in periodic Transport.

4. ๐ป1 Regularity of the Operator โ„ณ๐œ†=๐พ(๐œ†โˆ’๐‘‡๐‘)โˆ’1

Our notations and assumptions on ๐‘‘๐œ‡ are the same as in the preceding section. Let ๐‘˜(โ‹…,โ‹…) be a measurable function on ๐‘‰ร—๐‘‰ such that ๎€œ๐พโˆถ๐œ‘โŸถ๐‘‰๐‘˜๎€ทโ‹…,๐‘ฃ๎…ž๎€ธ๐œ‘๎€ท๐‘ฃ๎…ž๎€ธ๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎€ท๐ฟโˆˆโ„’2(๎€ธ.๐‘‰;๐‘‘๐œ‡(๐‘ฃ))(4.1)

It is announced without proof, in [10], that โ„ณ๐œ† maps continuously ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) into ๐ป1/2, where ๐‘‘๐œ‡ is the Lebesgue measure and ๐‘˜(โ‹…,โ‹…)=1. We propose a proof for this result (see Theorem 4.1) for a class of measures ๐‘‘๐œ‡ satisfying (3.2). The proof is inspired from [9] but is not a consequence of the results of [9] because the traces of ๐œ“โˆˆ๐ท(๐‘‡๐‘) do not lie in ๐ฟ2(๐œ•๐ทร—๐‘‰;|๐‘ฃโ‹…๐‘›(๐‘ฅ)|๐‘‘๐œŽ(๐‘ฅ)๐‘‘๐œ‡(๐‘ฃ)), but in a certain greatest weighted ๐ฟ2 space (see [1, 2, 4] for details).

We define โ€–๐œ‘โ€–1/2=(2๐œ‹)๐‘/2๎ƒฌ๎“๐‘˜โˆˆโ„ค๐‘||๐‘“๐‘˜||2๎‚€||๐‘˜||1+2๎‚1/2๎ƒญ1/2,(4.2) where ๎“๐œ‘(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘๐‘“๐‘˜๐‘’๐‘–(๐‘ฅโ‹…๐‘˜)โˆˆ๐ฟ2(๐ท).(4.3)

Theorem 4.1. Assume that ๐‘˜(โ‹…,โ‹…)=1. Then, for any ๐œ†>โˆ’๐œ†โˆ—, the operator โ„ณ๐œ† maps continuously ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)) into ๐ป1/2(๐ท).

Proof. Let ๐œ“โˆˆ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)). We expand ๐œ“ into the Fourier series: ๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๐‘“๐‘˜(๐‘ฃ)๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(4.4) We have ๐พ๎€ท๐œ†โˆ’๐‘‡๐‘๎€ธโˆ’1๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๎‚ต๎€œ๐‘‰๐‘“๐‘˜(๐‘ฃ)๎‚ถ๐‘’๐œŽ(๐‘ฃ)+๐‘–(๐‘ฃโ‹…๐‘˜)๐‘‘๐œ‡(๐‘ฃ)๐‘–(๐‘ฅโ‹…๐‘˜).(4.5) By the Hรถlder inequality and Lemma 3.1๎‚€||๐‘˜||1+2๎‚1/2||||๎€œ๐‘‰๐‘“๐‘˜(๐‘ฃ)||||๐œŽ(๐‘ฃ)+๐‘–(๐‘ฃโ‹…๐‘˜)๐‘‘๐œ‡(๐‘ฃ)2โ‰ค๎‚€||๐‘˜||1+2๎‚1/2๎‚ต๎€œ๐‘‰||๐‘“๐‘˜||(๐‘ฃ)2๎‚ถ๎€œ๐‘‘๐œ‡(๐‘ฃ)๐‘‰๐‘‘๐œ‡(๐‘ฃ)๎€ท๐œ†+๐œ†โˆ—๎€ธ2+(๐‘ฃโ‹…๐‘˜)2โ‰ค๐ถ0๎‚€||๐‘˜||1+2๎‚1/2||๐‘˜||๎€œ๐‘‰||๐‘“๐‘˜||(๐‘ฃ)2๐‘‘๐œ‡(๐‘ฃ).(4.6) The proof is achieved by the Parseval formula and the boundedness of (1+|๐‘˜|2)1/2/|๐‘˜|.

Remark 4.2. Assumption (3.1) is unnecessary in the proof of Theorem 4.1 but it plays a key role in the following theorem.

Now, we are going to prove the ๐ป1 regularity of โ„ณ๐œ† when restricted to a subspace of ๐ฟ2โจ‚(๐‘‘๐‘ฅ๐‘‘๐œ‡) consisting of even source term.

Theorem 4.3. Let (๐’œ7) be satisfied. In addition, suppose that (๐’œ9)๐‘˜๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๎€ท=๐‘˜๐‘ฃ,โˆ’๐‘ฃ๎…ž๎€ธ,๐‘‘๐œ‡(๐‘ฃ)a.e.,ess-sup๐‘ฃโ€ฒโˆˆ๐‘‰||๐‘˜๎€ทโ‹…,๐‘ฃ๎…ž๎€ธ||=๐‘˜โˆž(โ‹…)โˆˆ๐ฟ2(๐‘‰;๐‘‘๐œ‡(๐‘ฃ)).(4.7)Then, for any ๐œ†>โˆ’๐œ†โˆ—, the operator โ„ณ๐œ† maps continuously ๐’ช into ๐ป1๐‘ฅ(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)), where ๎‚ป๐’ช=๐œ“โˆˆ๐ฟ2๎‚ต(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(v)),๐œ“(๐‘ฅ,๐‘ฃ)=๐œ“(๐‘ฅ,โˆ’๐‘ฃ),sup๐‘ฃโˆˆ๐‘‰||๐‘“๐‘˜||๎‚ถ(๐‘ฃ)๐‘˜โˆˆโ„ค๐‘โˆˆ๐‘™2๎‚ผ(4.8) and ๐‘“๐‘˜(โ‹…)=(1/(2๐œ‹)๐‘)โˆซ๐ท๐œ“(๐‘ฅ,โ‹…)๐‘’โˆ’๐‘–(๐‘ฅโ‹…๐‘˜)๐‘‘๐‘ฅ.
Moreover, โ€–โ„ณ๐œ†โ€–โ„’(๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ));๐ป1๐‘ฅ(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ))) is locally bounded with respect to ๐œ†>โˆ’๐œ†โˆ—.

Proof. We proceed as in the proof of Theorem 3.2. We expand ๐œ“โˆˆ๐’ช into the Fourier series: ๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๐‘“๐‘˜(๐‘ฃ)๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(4.9) We have ๐พ๎€ท๐œ†โˆ’๐‘‡๐‘๎€ธโˆ’1๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๎ƒฉ๎€œ๐‘‰๐‘˜๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž)+๐‘–(๐‘ฃ๎…ž๎€ท๐‘ฃโ‹…๐‘˜)๐‘‘๐œ‡๎…ž๎€ธ๎ƒช๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(4.10) According to (๐’œ7) and (๐’œ9)๐œ•๐œ•๐‘ฅ๐‘—โ„ณ๐œ†๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๐‘–๐‘˜๐‘—๎ƒฉ๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎€ธ๐‘˜๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž)2+(๐‘ฃ๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎ƒช๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(4.11) The use of Lemma 3.1 gives ||||๐‘˜๐‘—๎€œ๐‘‰๐œŽ๎€ท๐‘ฃ๎…ž๎€ธ๐‘˜๎€ท๐‘ฃ,๐‘ฃ๎…ž๎€ธ๐‘“๐‘˜๎€ท๐‘ฃ๎…ž๎€ธ๐œŽ(๐‘ฃ๎…ž)2+(๐‘ฃ๎…žโ‹…๐‘˜)2๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ||||โ‰ค||๐‘˜๐‘—||๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธ๐‘˜โˆž๎‚ต(๐‘ฃ)sup๐‘ฃโˆˆ๐‘‰||๐‘“๐‘˜||๎‚ถร—๎ƒฉ๎€œ(๐‘ฃ)๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ(๐œ†+๐œ†โˆ—)2+(๐‘ฃ๎…žโ‹…๐‘˜)2๎ƒชโ‰ค๐ถ0๎€ท๐œ†+โ€–๐œŽโ€–โˆž๎€ธ๐‘˜โˆž(๐‘ฃ)sup๐‘ฃโˆˆ๐‘‰||๐‘“๐‘˜(||.๐‘ฃ)(4.12) Since (sup๐‘ฃโˆˆ๐‘‰|๐‘“๐‘˜(๐‘ฃ)|)๐‘˜โˆˆโ„ค๐‘โˆˆ๐‘™2, we deduce from the Parseval formula that (๐œ•/๐œ•๐‘ฅ๐‘—)โ„ณ๐œ†๐œ“โˆˆ๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)), which concludes the proof.

Remark 4.4. Arguing as in the proof of Lemma 3.1, one sees that ๎€œ๐‘‰๎€ท๐‘ฃ๐‘‘๐œ‡๎…ž๎€ธ๎‚ƒ๎€ท๐œ†+๐œ†โˆ—๎€ธ2+(๐‘ฃ๎…žโ‹…๐‘˜)2๎‚„2โ‰ค๎‚Š๐ถ0||๐‘˜||,(4.13) where ๎‚Š๐ถ0 is a positive constant. So, by the Hรถlder inequality Theorem 4.3 is still true if we replace ๐’ช by ๎ƒฏ๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘—โˆˆ๐ฝ,๐ฝis๏ฌnite๐œ“๐‘—(๐‘ฅ)๐œ‘๐‘—(๐‘ฃ),๐œ“๐‘—โˆˆ๐ฟ2(๐‘‘๐‘ฅ),๐œ‘๐‘—โˆˆ๐ฟโˆž(๐‘‘๐œ‡)and๐œ‘๐‘—(๐‘ฃ)=๐œ‘๐‘—๎ƒฐ.(โˆ’๐‘ฃ)๐‘‘๐œ‡a.e.(4.14) which is of interest for neutron transport approximations.
The subspace ๐’ช can be also replaced by ๎ƒฏ๐œ“โˆˆ๐ฟ2๎“(๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡(๐‘ฃ)),๐œ“(๐‘ฅ,๐‘ฃ)=๐œ“(๐‘ฅ,โˆ’๐‘ฃ)and๐‘˜โˆˆโ„ค๐‘๎‚€||๐‘˜||1+2๎‚1/2๎€œ๐‘ฃโˆˆ๐‘‰|๐‘“๐‘˜(๐‘ฃ)|2๎ƒฐ๐‘‘๐œ‡<โˆž,(4.15) where ๐‘“๐‘˜(โ‹…)=(1/2๐œ‹๐‘)โˆซ๐ท๐œ“(๐‘ฅ,โ‹…)๐‘’โˆ’๐‘–(๐‘ฅโ‹…๐‘˜)๐‘‘๐‘ฅ.

5. ๐ป2 Regularity of the Operator โ„ณ๐œ†=๐พ(๐œ†โˆ’๐‘‡๐‘)โˆ’1

In this section we focus our attention on the smoothing effect of the velocity averages (๐ป2) under some particular assumptions on the abstract measure ๐‘‘๐œ‡. For technical reasons, we treat separately the cases ๐‘=1 and ๐‘>1.

5.1. ๐ป2๐‘ฅ of Regularity โ„ณ๐œ† in One Dimension

Let ๐‘‘๐›ผ be a bounded measure (not necessarily positive) on (โˆ’1,1) satisfying the following assumptions: (๐’œ10)d๐›ผisinvariantbysymmetrywithrespecttozero,(5.1)(๐’œ11)๎€œ10๐‘‘|๐›ผ|(๐œ‡)๐œ‡2<โˆž,(5.2)

where ๐‘‘|๐›ผ| is the absolute value of the measure ๐‘‘๐›ผ, (๐’œ12)๐œŽโˆˆ๐ฟโˆž((โˆ’1,1);๐‘‘|๐›ผ|),๐œŽiseven๐‘‘|๐›ผ|a.e.(5.3)

Let ๐‘˜(โ‹…,โ‹…) be a ๐‘‘|๐›ผ| measurable function such that (๐’œ13)๐‘˜๎€ทโ‹…,๐œ‡๎…ž๎€ธ๎€ท=๐‘˜โ‹…,โˆ’๐œ‡๎…ž๎€ธ๐‘‘|๐›ผ|a.e.,๐‘‘|๐›ผ|ess-sup๐œ‡โ€ฒโˆˆ(โˆ’1,1)||๐‘˜๎€ทโ‹…,๐œ‡๎…ž๎€ธ||=๐‘˜โˆž(โ‹…)โˆˆ๐ฟ2((โˆ’1,1);๐‘‘|๐›ผ|).(5.4)

Let us introduce the following subspace of ๐ฟ2((0,2๐œ‹)ร—(โˆ’1,1);๐‘‘๐‘ฅโŠ—(๐‘‘|๐›ผ|/๐œ‡2)): ๎๎‚ป๐’ช=๐œ“โˆˆ๐ฟ2๎‚ต(0,2๐œ‹)ร—(โˆ’1,1);๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|๐œ‡2๎‚ถ๎‚ผ,๐œ“(โ‹…,๐œ‡)=๐œ“(โ‹…,โˆ’๐œ‡)๐‘‘|๐›ผ|a.e..(5.5) Then, we have the following.

Theorem 5.1. Let assumptions (๐’œ10)โ€“(๐’œ13) be satisfied. Then, for any operator ๐œ†>โˆ’๐œ†โˆ—=โˆ’๐‘‘|๐›ผ|ess-inf๐œŽ(โ‹…) the operator โ„ณ๐œ† maps continuously ๎๐’ช into ๐ป2๐‘ฅ๎‚ป(๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|)=๐œ“โˆˆ๐ฟ2๐œ•(๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|),2๐œ“๐œ•๐‘ฅ2โˆˆ๐ฟ2๎‚ผ.(๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|)(5.6) Moreover, โ€–โ„ณ๐œ†โ€–โ„’(๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|(๐œ‡)/๐œ‡2));๐ป2๐‘ฅ(๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|)) is locally bounded with respect to ๐œ†>โˆ’๐œ†โˆ—.

Proof. For ๎๐’ช๐œ“โˆˆ we have ๐พ๎€ท๐œ†โˆ’๐‘‡๐‘๎€ธโˆ’1๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๎ƒฉ๎€œ1โˆ’1๐‘˜๎€ท๐œ‡,๐œ‡๎…ž๎€ธ๐‘“๐‘˜๎€ท๐œ‡๎…ž๎€ธ๐œ†+๐œŽ(๐œ‡๎…ž)+๐‘–๐œ‡๎…ž๐‘˜๎€ท๐œ‡๐‘‘๐›ผ๎…ž๎€ธ๎ƒช๐‘’๐‘–๐‘ฅ๐‘˜.(5.7) Thus, ๐œ•2โ„ณ๐œ†๐œ“๐œ•๐‘ฅ2๎“(๐‘ฅ,๐‘ฃ)=โˆ’๐‘˜โˆˆโ„ค๐‘˜2๎ƒฉ๎€œ1โˆ’1๎€ท๎€ท๐œ‡๐œ†+๐œŽ๎…ž๐‘˜๎€ท๎€ธ๎€ธ๐œ‡,๐œ‡๎…ž๎€ธ๐‘“๐‘˜๎€ท๐œ‡๎…ž๎€ธ๐œ†+๐œŽ(๐œ‡๎…ž)+๐‘–๐œ‡๎…ž๐‘˜๎€ท๐œ‡๐‘‘๐›ผ๎…ž๎€ธ๎ƒช๐‘’๐‘–๐‘ฅ๐‘˜๎“=โˆ’๐‘˜โˆˆโ„ค๐‘˜2๎ƒฉ๎€œ1โˆ’1๎€ท๎€ท๐œ‡๐œ†+๐œŽ๎…ž๐‘˜๎€ท๎€ธ๎€ธ๐œ‡,๐œ‡๎…ž๎€ธ๐‘“๐‘˜๎€ท๐œ‡๎…ž๎€ธ(๐œ†+๐œŽ(๐œ‡๎…ž))2+๐œ‡๎…ž2๐‘˜2๎€ท๐œ‡๐‘‘๐›ผ๎…ž๎€ธ๎€œ+๐‘–๐‘˜1โˆ’1๐œ‡๎…ž๐‘˜๎€ท๐œ‡,๐œ‡๎…ž๎€ธ๐‘“๐‘˜๎€ท๐œ‡๎…ž๎€ธ(๐œ†+๐œŽ(๐œ‡๎…ž))2+๐œ‡๎…ž2๐‘˜2๎€ท๐œ‡๐‘‘๐›ผ๎…ž๎€ธ๎ƒช๐‘’๐‘–๐‘ฅ๐‘˜.(5.8) Note that the last integral of (5.8) vanishes because its integrand is odd, in view of assumptions (๐’œ10),(๐’œ12)โ€“(๐’œ13) and the evenness of ๐‘“๐‘˜(โ‹…). Now, using the Hรถlder inequality together with assumption (๐’œ10), we obtain ๐‘˜2||||๎€œ1โˆ’1๐‘˜๎€ท๐œ‡,๐œ‡๎…ž๎€ธ๐‘“๐‘˜๎€ท๐œ‡๎…ž๎€ธ๐œ†+๐œŽ(๐œ‡๎…ž)+๐‘–๐œ‡๎…ž๐‘˜๎€ท๐œ‡๐‘‘๐›ผ๎…ž๎€ธ||||๎€ทโ‰ค2๐œ†+โ€–๐œŽโ€–๐ฟโˆž(๐‘‘|๐›ผ|)๎€ธ๐‘˜โˆž๎€œ(๐œ‡)10๐‘˜2๐œ‡2(๐œ†+๐œŽ(๐œ‡))2+๐œ‡2๐‘˜2||๐‘“๐‘˜||๐‘‘(๐œ‡)|๐›ผ|(๐œ‡)๐œ‡2๎€ทโ‰ค2๐œ†+โ€–๐œŽโ€–๐ฟโˆž(๐‘‘|๐›ผ|)๎€ธ๐‘˜โˆž๎‚ต๎€œ(๐œ‡)10๐‘‘|๐›ผ|(๐œ‡)๐œ‡2๎‚ถ1/2๎‚ต๎€œ10||๐‘“๐‘˜||(๐œ‡)2๐‘‘|๐›ผ|(๐œ‡)๐œ‡2๎‚ถ1/2.(5.9) Next, since ๐‘˜โˆž(โ‹…)โˆˆ๐ฟ2((โˆ’1,1);๐‘‘|๐›ผ|) and โˆ‘๐‘˜โˆˆโ„คโˆซ10|๐‘“๐‘˜(๐œ‡)|2(๐‘‘|๐›ผ|(๐œ‡)/๐œ‡2)<โˆž, it follows from the Parseval formula that โ€–โ€–โ€–๐œ•2โ„ณ๐œ†๐œ“๐œ•๐‘ฅ2โ€–โ€–โ€–๐ฟ2(๐‘‘๐‘ฅโŠ—๐‘‘|๐›ผ|)โ‰ค๐ถโ€–๐œ“โ€–๐ฟ2(๐‘‘๐‘ฅโŠ—(๐‘‘|๐›ผ|/๐œ‡2)),(5.10) where ๎€ท๐ถ=2๐œ†+โ€–๐œŽโ€–๐ฟโˆž(๐‘‘|๐›ผ|)๎€ธโ€–๐‘˜โˆžโ€–๐ฟ2(๐‘‘|๐›ผ|)๎‚ต๎€œ10๐‘‘|๐›ผ|(๐œ‡)๐œ‡2๎‚ถ1/2(5.11) is locally bounded in ๐œ†>โˆ’๐œ†โˆ—, and the proof is complete.

5.2. ๐ป2 Regularity of โ„ณ๐œ†=๐‘€(๐œ†โˆ’๐‘‡๐‘)โˆ’1(๐‘>1)

Let ๐‘‘๐œ‡(๐‘ฃ)=๐‘‘๐›ผ(๐œŒ)โŠ—๐‘‘๐‘†(๐‘ค), where ๐‘‘๐‘†(๐‘ค) is the Lebesgue measure on ๐•Š๐‘โˆ’1 and ๐‘‘๐›ผ is a signed measure on [0,1] satisfying the following conditions: (๐’œ14)๎€œ10๐‘‘๐›ผ(๐œŒ)๐œŒ=0,(5.12)(๐’œ15)๎€œ10๐‘‘|๐›ผ|(๐œŒ)๐œŒ2<โˆž.(5.13)

The more technical assumption (๐’œ14) plays a key role for our analysis. In the sequel, we show the optimality of this assumption. We need also the following assumption: (๐’œ16)๐œŽ(๐‘ฃ)=๐œŽ(|๐‘ฃ|),๐œŽ(โ‹…)โˆˆ๐‘Š1,โˆž([]0,1).(5.14)

Theorem 5.2. Let the hypotheses (๐’œ14)โ€“(๐’œ16) be satisfied. Then, for any ๐œ†>โˆ’๐œ†โˆ—=โˆ’๐‘‘|๐›ผ|ess-inf๐œŽ(โ‹…), the operator โ„ณ๐œ†=๐‘€(๐œ†โˆ’๐‘‡๐‘)โˆ’1 maps continuously ๐ฟ2(๐ท) into ๐ป2(๐ท), where ๐‘€โˆถ๐œ‘โˆˆ๐ฟ2(๎€œ๐‘‘๐‘ฅโŠ—๐‘‘๐œ‡)โŸถ๐‘‰๐œ‘(๐‘ฅ,๐‘ฃ)๐‘‘๐œ‡(๐‘ฃ)โˆˆ๐ฟ2(๐ท),(5.15) is the averaging operator.

Proof. Let ๐œ“โˆˆ๐ฟ2(๐ท). We use the Fourier series of โˆ‘๐œ“(๐‘ฅ)=๐‘˜โˆˆโ„ค๐‘๐‘“๐‘˜๐‘’๐‘–(๐‘ฅโ‹…๐‘˜). We recall that โ„ณ๐œ†๎“๐œ“(๐‘ฅ,๐‘ฃ)=๐‘˜โˆˆโ„ค๐‘๐‘“๐‘˜๎€œ10๎€œ๐‘‘๐›ผ(๐œŒ)๐•Š๐‘โˆ’1๐‘‘๐‘†(๐‘ค)๐‘’๐œŽ(๐œŒ)+๐‘–๐œŒ(๐‘คโ‹…๐‘˜)๐‘–(๐‘ฅโ‹…๐‘˜),(5.16) where ๐œŽ(๐œŒ)=๐œ†+๐œŽ(๐œŒ).
Let us recall [22] that for all fixed ๐‘ค0โˆˆ๐’ฎ๐‘โˆ’1๎€œ๐‘†๐‘โˆ’1๐‘“๎€ท๐‘คโ‹…๐‘ค0๎€ธ||๐‘†๐‘‘๐‘†(๐‘ค)=๐‘โˆ’2||๎€œ1โˆ’1๎€ท๐‘“(๐‘ก)1โˆ’๐‘ก2๎€ธ(๐‘โˆ’3)/2๐‘‘๐‘ก.(5.17) Accordingly, โ„ณ๐œ†๐œ“(๐‘ฅ,๐‘ฃ)=๐‘“0||๐‘†๐‘โˆ’1||๎€œ10๐‘‘๐›ผ(๐œŒ)||๐‘†๐œŽ(๐œŒ)+2๐‘โˆ’2||๎“๐‘˜โ‰ 0๐‘“๐‘˜๎ƒฉ๎€œ10๎€œ๐‘‘๐›ผ(๐œŒ)10๎€ท1โˆ’๐‘ก2๎€ธ(๐‘โˆ’3)/2๐œŽ(๐œŒ)๐œŽ(๐œŒ)2+๐œŒ2||๐‘˜||2๐‘ก2๎ƒช๐‘’๐‘‘๐‘ก๐‘–(๐‘ฅโ‹…๐‘˜).(5.18) Set ๐น0=๐‘“0||๐‘†๐‘โˆ’1||๎€œ10๐‘‘๐›ผ(๐œŒ).๐œŽ(๐œŒ)(5.19) We give the proof for ๐‘>3 (the other cases are similar).
An integration by parts with respect to ๐‘ก yields โ„ณ๐œ†๐œ“(๐‘ฅ,๐‘ฃ)=๐น0||๐‘†+2(๐‘โˆ’3)๐‘โˆ’2||ร—๎“๐‘˜โ‰ 0๐‘“๐‘˜||๐‘˜||๎‚ต๎€œ10๐‘ก๎€ท1โˆ’๐‘ก2๎€ธ(๐‘โˆ’5)/2๎€œ๐‘‘๐‘ก10๎‚ต๐œŒ||๐‘˜||๐‘กarctan๎‚ถ๐œŽ(๐œŒ)๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(5.20) Thanks to (๐’œ14), (5.20) becomes โ„ณ๐œ†๐œ“(๐‘ฅ,๐‘ฃ)=๐น0||๐‘†โˆ’2(๐‘โˆ’3)๐‘โˆ’2||ร—๎“๐‘˜โ‰ 0๐‘“๐‘˜||๐‘˜||๎‚ต๎€œ10๐‘ก(1โˆ’๐‘ก2)(๐‘โˆ’5)/2๎€œ๐‘‘๐‘ก10๎‚ตarctan๐œŽ(๐œŒ)๐œŒ||๐‘˜||๐‘ก๎‚ถ๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜)||๐‘†+2๐œ‹(๐‘โˆ’3)๐‘โˆ’2||๎€œ10๐‘‘๐›ผ(๐œŒ)๐œŒ๎€œ10๐‘ก(1โˆ’๐‘ก2)(๐‘โˆ’5)/2๎“๐‘‘๐‘ก๐‘˜โ‰ 0๐‘“๐‘˜||๐‘˜||๐‘’๐‘–(๐‘ฅโ‹…๐‘˜)=๐น0||๐‘†โˆ’2(๐‘โˆ’3)๐‘โˆ’2||ร—๎“๐‘˜โ‰ 0๐‘“๐‘˜||๐‘˜||๎‚ต๎€œ10๐‘ก(1โˆ’๐‘ก2)(๐‘โˆ’5)/2๎€œ๐‘‘๐‘ก10๎‚ตarctan๐œŽ(๐œŒ)๐œŒ||๐‘˜||๐‘ก๎‚ถ๐‘‘๐›ผ(๐œŒ)๐œŒ๎‚ถ๐‘’๐‘–(๐‘ฅโ‹…๐‘˜).(5.21) Set ๎€œ๐‘ƒ(๐œŒ)=๐œŒ0๐‘‘๐›ผ(๐‘ )๐‘ .(5.22) An integration by parts for the Stieltjes measures yields ๎€œ10๎‚ตarctan๐œŽ(๐œŒ)๐œŒ||๐‘˜||๐‘ก๎‚ถ1๐‘‘๐‘ƒ(๐œŒ)=โˆ’||๐‘˜||๐‘ก๎€œ10๎‚ต๐‘ƒ(๐œŒ)๐œŽ(๐œŒ)๐œŒ๎‚ถ๎…ž๐œŒ2|๐‘˜|2๐‘ก2๐œŽ(๐œŒ)2+๐œŒ2|๐‘˜|2๐‘ก2+๎‚ธ๎‚ต๐‘‘๐œŒ๐‘ƒ(๐œŒ)arctan๐œŽ(๐œŒ)๐œŒ||๐‘˜||๐‘ก๎‚ถ๎‚น๐œŒ=1๐œŒ=0.(5.23) Now, thanks to (๐’œ14), the last term of (5.23) vanishes. Set ๎‚ต๐‘(๐œŒ)=๐‘ƒ(๐œŒ)๐œŽ(๐œŒ)๐œŒ๎‚ถ๎…ž.(5.24) Accordingly, (5.20) becomes โ„ณ๐œ†๐œ“(๐‘ฅ,๐‘ฃ)=๐น0||๐‘†+2(๐‘โˆ’3)๐‘โˆ’2||๎“๐‘˜โ‰ 0๐‘“๐‘˜||๐‘˜||2๐‘’๐‘–(๐‘ฅโ‹…๐‘˜)ร—๎ƒฉ๎€œ10๎€ท1โˆ’๐‘ก2๎€ธ(๐‘โˆ’5)/2๎€œ๐‘‘๐‘ก10๐œŒ๐‘(๐œŒ)2||๐‘˜||2๐‘ก2๐œŽ(๐œŒ)2+๐œŒ2||๐‘˜||2๐‘ก2๎ƒช.๐‘‘๐œŒ(5.25) Consequently, ๐œ•๐œ•๐‘ฅ๐‘–๐œ•๐œ•๐‘ฅ๐‘—๎€ทโ„ณ๐œ†๎€ธ||๐‘†๐œ“(๐‘ฅ,๐‘ฃ)=โˆ’2(๐‘โˆ’3)๐‘โˆ’2||๎“๐‘˜โ‰ 0๐‘˜๐‘–๐‘˜๐‘—||๐‘˜||2๐‘“๐‘˜๐‘’๐‘–(๐‘ฅโ‹…๐‘˜)ร—๎ƒฉ๎€œ10๎€ท1โˆ’๐‘ก2๎€ธ(๐‘โˆ’5)/2๎€œ๐‘‘๐‘ก10๐œŒ๐‘(๐œŒ)2||๐‘˜||2๐‘ก2๐œŽ(๐œŒ)2+๐œŒ2||๐‘˜||2๐‘ก2๎ƒช.๐‘‘๐œŒ(5.26) Since ||๐‘†2(๐‘โˆ’3)๐‘โˆ’2|||||||๐‘˜๐‘–๐‘˜๐‘—||๐‘˜||2๎€œ10๎€ท1โˆ’๐‘ก2๎€ธ(๐‘โˆ’5)/2๎€œ๐‘‘๐‘ก10๐œŒ๐‘(๐œŒ)2||๐‘˜||2๐‘ก2๐œŽ(๐œŒ)2+๐œŒ2||๐‘˜||2๐‘ก2|||||โ‰ค||๐‘†๐‘‘๐œŒ(๐‘โˆ’2)๐‘