International Scholarly Research Notices

International Scholarly Research Notices / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 379491 | 22 pages | https://doi.org/10.5402/2012/379491

Sobolev Regularity in Neutron Transport Theory

Academic Editor: D. X. Zhou
Received20 Sep 2011
Accepted09 Nov 2011
Published26 Jan 2012

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