Bulletin des Sciences Mathematiques最新文献

{"title":"Maximal functions of the multilinear pseudo-differentials operators","authors":"Liang Huang","doi":"10.1016/j.bulsci.2025.103618","DOIUrl":"10.1016/j.bulsci.2025.103618","url":null,"abstract":"<div><div>In this paper, we consider the maximal multilinear pseudo-differential operator with symbols <span><math><mi>σ</mi><mo>∈</mo><mi>B</mi><msubsup><mrow><mi>S</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup><mo>)</mo></math></span>, and establish the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> estimate with a sharp bound <span><math><mi>O</mi><mo>(</mo><msqrt><mrow><mi>log</mi><mo>⁡</mo><mo>(</mo><mi>N</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></msqrt><mo>)</mo></math></span>. Our work improves the work of Chen, Dai and Lu <span><span>[5]</span></span> by extending the symbol <em>σ</em> from the Hörmander class <span><math><mi>B</mi><msubsup><mrow><mi>S</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>0</mn></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> to <span><math><mi>B</mi><msubsup><mrow><mi>S</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msubsup></math></span> with <span><math><mn>0</mn><mo>≤</mo><mi>δ</mi><mo>&lt;</mo><mn>1</mn></math></span>. The main tools such as localizing the maximal pseudo-differential operators and the time-frequency analysis in <span><span>[5]</span></span> may not accommodate symbols <span><math><mi>σ</mi><mo>∈</mo><mi>B</mi><msubsup><mrow><mi>S</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>δ</mi></mrow><mrow><mn>0</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup><mo>)</mo></math></span> with <span><math><mn>0</mn><mo>&lt;</mo><mi>δ</mi><mo>&lt;</mo><mn>1</mn></math></span>. In this work, we handle this difficulty by applying the inhomogeneous Littlewood-Paley-Stein decomposition to the space variable and using Taylor's expansion to track the size of those decomposed pieces. Then together with the ideas of using martingales, some related pointwise estimates and the good-<em>λ</em> inequality as in <span><span>[16]</span></span>, <span><span>[19]</span></span>, we will be able to obtain the boundedness with the optimal bound.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"202 ","pages":"Article 103618"},"PeriodicalIF":1.3,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Decomposable abelian G-curves and special subvarieties","authors":"Irene Spelta ,&nbsp;Carolina Tamborini","doi":"10.1016/j.bulsci.2025.103616","DOIUrl":"10.1016/j.bulsci.2025.103616","url":null,"abstract":"<div><div>We consider families of abelian Galois coverings of the line. When the Jacobian of the general element is totally decomposable, i.e., is isogenous to a product of elliptic curves, we prove that they yield special subvarieties of <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>g</mi></mrow></msub></math></span> if and only if a numerical condition holds, which in the general case is only known to be sufficient.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103616"},"PeriodicalIF":1.3,"publicationDate":"2025-03-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143682035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Compactness criterions for certain commutators of oscillatory singular integrals","authors":"Chenyan Wang ,&nbsp;Huoxiong Wu ,&nbsp;Weijin Yan","doi":"10.1016/j.bulsci.2025.103613","DOIUrl":"10.1016/j.bulsci.2025.103613","url":null,"abstract":"<div><div>Let Ω be homogeneous of degree zero, integrable and have mean zero on <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>, <span><math><mi>P</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> be a real-valued polynomial on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Define the oscillatory singular integral <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>P</mi></mrow></msubsup></math></span> by<span><span><span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>P</mi></mrow></msubsup><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mrow><mi>p</mi><mo>.</mo><mi>v</mi><mo>.</mo></mrow><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></munder><msup><mrow><mi>e</mi></mrow><mrow><mi>i</mi><mi>P</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></msup><mfrac><mrow><mi>Ω</mi><mo>(</mo><mi>x</mi><mo>−</mo><mi>y</mi><mo>)</mo></mrow><mrow><mo>|</mo><mi>x</mi><mo>−</mo><mi>y</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>n</mi></mrow></msup></mrow></mfrac><mi>f</mi><mo>(</mo><mi>y</mi><mo>)</mo><mi>d</mi><mi>y</mi><mo>.</mo></math></span></span></span> This paper is devoted to studying the compactness of <span><math><mo>[</mo><mi>b</mi><mo>,</mo><msubsup><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>P</mi></mrow></msubsup><mo>]</mo></math></span>, the commutators formed by <span><math><msubsup><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>P</mi></mrow></msubsup></math></span> with <span><math><mi>b</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mtext>loc</mtext></mrow><mrow><mn>1</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo></math></span>. Several compactness criteria of <span><math><mo>[</mo><mi>b</mi><mo>,</mo><msubsup><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>P</mi></mrow></msubsup><mo>]</mo></math></span> on Lebesgue spaces and Morrey spaces are given. As applications, the compactness of <span><math><mo>[</mo><mi>b</mi><mo>,</mo><msubsup><mrow><mi>T</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mi>P</mi></mrow></msubsup><mo>]</mo></math></span> with rough kernels on Lebesgue spaces and Morrey spaces is obtained.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103613"},"PeriodicalIF":1.3,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143682034","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Sharp estimates for Hausdorff operators on complementary local Morrey-type spaces","authors":"Mingquan Wei ,&nbsp;Xiaoyu Liu ,&nbsp;Dunyan Yan","doi":"10.1016/j.bulsci.2025.103614","DOIUrl":"10.1016/j.bulsci.2025.103614","url":null,"abstract":"<div><div>In this paper, some necessary and sufficient conditions for the boundedness of some linear and multilinear Hausdorff operators are established and the corresponding sharp constants are also given. Our main results can apply to some concrete operators, such as the Hardy operator and its adjoint operator, the weighted Hardy–Littlewood average, the multilinear Hardy operator and so on.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103614"},"PeriodicalIF":1.3,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143682116","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On near orthogonality of the Banach frames for the wave packet spaces","authors":"Dimitri Bytchenkoff","doi":"10.1016/j.bulsci.2025.103611","DOIUrl":"10.1016/j.bulsci.2025.103611","url":null,"abstract":"&lt;div&gt;&lt;div&gt;In solving scientific, engineering or pure mathematical problems one is frequently faced with a need to approximate the function of a given class with a specified precision by the linear combination of a preferably small number of simpler functions. This can often achieved by choosing the simpler functions localised one way or another both in the time and frequency domain. Constructing a set of linearly independent functions, let alone a basis, with a given time-frequency localisation is a formidable and often unsolvable problem, though. A much better chance one stands in building a set of time-frequency localised functions that constitutes a so-called frame – a generalisation of the notion of the basis, whose elements need not be linear independent, rather than a basis.&lt;/div&gt;&lt;div&gt;Over the last seventy years or so, a range of frames have been designed to allow the decomposition and synthesis of functions of various classes. The most prominent examples of such systems are Gabor functions, wavelets, ridgelets, curvelets, shearlets and wave atoms. We recently introduced a family of quasi-Banach spaces – which we called &lt;span&gt;&lt;math&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; – that encompasses all those classes of functions whose elements have sparse expansions in one of the above-mentioned frames, supplied them with Banach frames – the kind of frames that ensure that any element of the class of functions for which a frame was designed can be decomposed and reconstructed using that frame – and provided their atomic decomposition. Herein we prove that the Banach frames for and sets of atoms of the wave packet spaces – which we call &lt;span&gt;&lt;math&gt;&lt;mi&gt;w&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; &lt;span&gt;&lt;math&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi&gt;s&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; – are indeed localised in the time and frequency domain or, more specifically, that they are near orthogonal; and therefore so are all of the above-mentioned examples of frames.&lt;/div&gt;&lt;div&gt;We shall also show that, unlike those examples, the wave packet system can be made to assume a wide range of types and degrees of time-frequency localisation by the suitable choice of values of the parameters of the system. This, we believe, makes the wave packet systems not only suitable for decomposing, synthesising or approximating functions of a wide range of quasi-Banach function spaces in an efficient and effective way, but also for their use for representing linear bounded operators on the quasi-Banach spaces by sparse and well structured matrices using the Galerkin method. This, in its turn, should allow one to design efficient computer programs for solving corresponding operator equations on t","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103611"},"PeriodicalIF":1.3,"publicationDate":"2025-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143643317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Inequalities for eigenvalues of the poly-Laplacian with arbitrary order on spherical domains","authors":"Yue He,&nbsp;Huan Wang","doi":"10.1016/j.bulsci.2025.103608","DOIUrl":"10.1016/j.bulsci.2025.103608","url":null,"abstract":"<div><div>In this paper, we are devoted to the study of universal inequalities for eigenvalues of the poly-Laplacian with arbitrary order on bounded domains in <span><math><msup><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mn>1</mn><mo>)</mo></math></span>, respectively, and then establish some new universal inequalities that are different from those already present in the literature. In particular, our results can reveal the relationship between the <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-th eigenvalue and the first <em>k</em> eigenvalues relatively quickly, and some methods used in this paper might be applied to other eigenvalue problems.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103608"},"PeriodicalIF":1.3,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143600544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weak weak approximation for certain quadric surface bundles","authors":"Nick Rome","doi":"10.1016/j.bulsci.2025.103601","DOIUrl":"10.1016/j.bulsci.2025.103601","url":null,"abstract":"<div><div>We investigate weak approximation away from a finite set of places for a class of biquadratic fourfolds inside <span><math><msup><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>×</mo><msup><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>, some of which appear in the recent work of Hassett–Pirutka–Tschinkel <span><span>[16]</span></span>.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103601"},"PeriodicalIF":1.3,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the essential norms of Toeplitz operators with symbols in C + H∞ acting on abstract Hardy spaces built upon translation-invariant Banach function spaces","authors":"Oleksiy Karlovych ,&nbsp;Eugene Shargorodsky","doi":"10.1016/j.bulsci.2025.103599","DOIUrl":"10.1016/j.bulsci.2025.103599","url":null,"abstract":"<div><div>Let <em>X</em> be a translation-invariant Banach function space on the unit circle and let <span><math><mi>H</mi><mo>[</mo><mi>X</mi><mo>]</mo></math></span> be the abstract Hardy space built upon <em>X</em>. We suppose the Riesz projection <em>P</em> is bounded on <em>X</em> and estimate the essential norms <span><math><msub><mrow><mo>‖</mo><mi>T</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mi>B</mi><mo>(</mo><mi>H</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo><mo>,</mo><mi>e</mi></mrow></msub></math></span> of Toeplitz operators <span><math><mi>T</mi><mo>(</mo><mi>a</mi><mo>)</mo><mi>f</mi><mo>:</mo><mo>=</mo><mi>P</mi><mo>(</mo><mi>a</mi><mi>f</mi><mo>)</mo></math></span> with <span><math><mi>a</mi><mo>∈</mo><mi>C</mi><mo>+</mo><msup><mrow><mi>H</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>. We prove that in this case<span><span><span><math><msub><mrow><mo>‖</mo><mi>a</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>≤</mo><msub><mrow><mo>‖</mo><mi>T</mi><mo>(</mo><mi>a</mi><mo>)</mo><mo>‖</mo></mrow><mrow><mi>B</mi><mo>(</mo><mi>H</mi><mo>[</mo><mi>X</mi><mo>]</mo><mo>)</mo><mo>,</mo><mi>e</mi></mrow></msub><mo>≤</mo><mi>min</mi><mo>⁡</mo><mrow><mo>{</mo><mn>2</mn><mo>,</mo><msub><mrow><mo>‖</mo><mi>P</mi><mo>‖</mo></mrow><mrow><mi>B</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></msub><mo>}</mo></mrow><msub><mrow><mo>‖</mo><mi>a</mi><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>,</mo></math></span></span></span> extending the results by the second author <span><span>[27]</span></span> for classical Hardy spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>=</mo><mi>H</mi><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup><mo>]</mo></math></span>, <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span>. In contrast to our previous works <span><span>[27]</span></span> and <span><span>[16]</span></span>, we do not assume that <em>X</em> is reflexive or separable, which complicates the matters, but allows us to include the Hardy-Lorentz spaces <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>=</mo><mi>H</mi><mo>[</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></msup><mo>]</mo></math></span> with <span><math><mn>1</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mo>∞</mo></math></span> and <span><math><mi>q</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo>∞</mo></math></span> into consideration.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103599"},"PeriodicalIF":1.3,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551727","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Schwarz lemma and Schwarz-Pick lemma for solutions of the α-harmonic equation","authors":"Ming Li ,&nbsp;Xiu-Shuang Ma ,&nbsp;Li-Mei Wang","doi":"10.1016/j.bulsci.2025.103598","DOIUrl":"10.1016/j.bulsci.2025.103598","url":null,"abstract":"<div><div>In this paper, the Schwarz type and Schwarz-Pick type inequalities for solutions of <em>α</em>-harmonic equation (<span><math><mi>α</mi><mo>&gt;</mo><mo>−</mo><mn>1</mn></math></span>) are investigated. By making use of the integral of trigonometric functions, we obtain these two types of inequalities in terms of hypergeometric functions which improve the corresponding results due to Khalfallah et al. (Complex Var. Elliptic Equ., 2023) and Li et al. (Bull. Malays. Math. Sci. Soc., 2022).</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103598"},"PeriodicalIF":1.3,"publicationDate":"2025-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143551791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the representation of large even integers as the sum of eight primes from positive density sets","authors":"Meng Gao","doi":"10.1016/j.bulsci.2025.103597","DOIUrl":"10.1016/j.bulsci.2025.103597","url":null,"abstract":"<div><div>Let <span><math><mi>P</mi></math></span> denote the set of all primes. We have proved that if <em>A</em> is a subset of <span><math><mi>P</mi></math></span>, and the lower density of <em>A</em> in <span><math><mi>P</mi></math></span> is larger than 1/2, then every sufficiently large even integer <em>n</em> can be expressed in the form <span><math><mi>n</mi><mo>=</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>p</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mn>8</mn></mrow></msub><mo>∈</mo><mi>A</mi></math></span>. The constant 1/2 in this statement is the best possible.</div></div>","PeriodicalId":55313,"journal":{"name":"Bulletin des Sciences Mathematiques","volume":"201 ","pages":"Article 103597"},"PeriodicalIF":1.3,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}