Pub Date : 2024-05-30DOI: 10.1007/s00041-024-10089-9
E. K. Narayanan, P. K. Sanjay, K. T. Yasser
We establish injectivity results for three different spherical means on an H-type group, G. The first one is the standard spherical mean operator, which is defined as the average of a function over the spheres in the complement of the center, the second one is the average over the product of spheres in the center and its complement, and the third one is the average over the spheres defined by a homogeneous norm on G. If m is the dimension of the center of G, injectivity of these spherical means is proved for the range (1 le p le frac{2m}{m-1}). Examples are provided to show the sharpness of our results in the first two cases.
我们建立了 H 型群 G 上三种不同球均值的注入性结果。第一种是标准球均值算子,它被定义为函数在中心补集球面上的平均值;第二种是在中心球面与其补集球面乘积上的平均值;第三种是在 G 上由同质规范定义的球面上的平均值。如果 m 是 G 的中心维数,那么在 (1 le p le frac{2m}{m-1}) 的范围内,这些球面均值的注入性将得到证明。举例说明了我们在前两种情况下的结果的尖锐性。
{"title":"Injectivity of Spherical Means on H-Type Groups","authors":"E. K. Narayanan, P. K. Sanjay, K. T. Yasser","doi":"10.1007/s00041-024-10089-9","DOIUrl":"https://doi.org/10.1007/s00041-024-10089-9","url":null,"abstract":"<p>We establish injectivity results for three different spherical means on an <i>H</i>-type group, <i>G</i>. The first one is the standard spherical mean operator, which is defined as the average of a function over the spheres in the complement of the center, the second one is the average over the product of spheres in the center and its complement, and the third one is the average over the spheres defined by a homogeneous norm on <i>G</i>. If <i>m</i> is the dimension of the center of <i>G</i>, injectivity of these spherical means is proved for the range <span>(1 le p le frac{2m}{m-1})</span>. Examples are provided to show the sharpness of our results in the first two cases.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141193183","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}
Let (Omega ) be a homogeneous function of degree zero, have vanishing moment of order one on the unit sphere (mathbb {S}^{d-1})((dge 2)). In this paper, our object of investigation is the following rough non-standard singular integral operator
where A is a function defined on ({mathbb {R}}^d) with derivatives of order one in ({textrm{BMO}}({mathbb {R}}^d)). We show that (T_{Omega ,,A}) enjoys the endpoint (Llog L) type estimate and is (L^p) bounded if (Omega in L(log L)^{2}({mathbb {S}}^{d-1})). These results essentially improve the previous known results given by Hofmann (Stud Math 109:105–131, 1994) for the (L^p) boundedness of (T_{Omega ,,A}) under the condition (Omega in L^{q}({mathbb {S}}^{d-1}))((q>1)), Hu and Yang (Bull Lond Math Soc 35:759–769, 2003) for the endpoint weak (Llog L) type estimates when (Omega in textrm{Lip}_{alpha }({mathbb {S}}^{d-1})) for some (alpha in (0,,1]).
让 (Omega )是一个零度均质函数,在单位球上有一阶消失矩 (mathbb {S}^{d-1})((dge 2)).在本文中,我们的研究对象是下面这个粗糙的非标准奇异积分算子 $$begin{aligned}T_{Omega ,,A}f(x)=mathrm{p.,v.}int _{{{mathbb {R}}^d}frac{Omega (x-y)}{|x-y|^{d+1}}big (A(x)-A(y)-nabla A(y)(x-y)big )f(y)textrm{d}y、end{aligned}$where A is a function defined on ({mathbb {R}}^d) with derivatives of order one in ({textrm{BMO}}({mathbb {R}}^d)).我们证明,如果 (Omega in L(log L)^{2}({mathbb {S}}^{d-1})) 享有端点 (Llog L) 类型估计,并且是 (L^p) 有界的。这些结果基本上改进了霍夫曼(Stud Math 109:105-131,1994)之前给出的在 (Omega in L^{q}({mathbb {S}}^{d-1}) 条件下 (T_{Omega ,,A})的(L^p)有界性的已知结果。)Hu and Yang (Bull Lond Math Soc 35:759-769, 2003) for the endpoint weak (Llog L) type estimates when (Omega in textrm{Lip}_{alpha }({mathbb {S}}^{d-1})) for some (alpha in (0,,1]).
{"title":"On the Boundedness of Non-standard Rough Singular Integral Operators","authors":"Guoen Hu, Xiangxing Tao, Zhidan Wang, Qingying Xue","doi":"10.1007/s00041-024-10086-y","DOIUrl":"https://doi.org/10.1007/s00041-024-10086-y","url":null,"abstract":"<p>Let <span>(Omega )</span> be a homogeneous function of degree zero, have vanishing moment of order one on the unit sphere <span>(mathbb {S}^{d-1})</span>(<span>(dge 2)</span>). In this paper, our object of investigation is the following rough non-standard singular integral operator </p><span>$$begin{aligned} T_{Omega ,,A}f(x)=mathrm{p.,v.}int _{{mathbb {R}}^d}frac{Omega (x-y)}{|x-y|^{d+1}}big (A(x)-A(y)-nabla A(y)(x-y)big )f(y)textrm{d}y, end{aligned}$$</span><p>where <i>A</i> is a function defined on <span>({mathbb {R}}^d)</span> with derivatives of order one in <span>({textrm{BMO}}({mathbb {R}}^d))</span>. We show that <span>(T_{Omega ,,A})</span> enjoys the endpoint <span>(Llog L)</span> type estimate and is <span>(L^p)</span> bounded if <span>(Omega in L(log L)^{2}({mathbb {S}}^{d-1}))</span>. These results essentially improve the previous known results given by Hofmann (Stud Math 109:105–131, 1994) for the <span>(L^p)</span> boundedness of <span>(T_{Omega ,,A})</span> under the condition <span>(Omega in L^{q}({mathbb {S}}^{d-1}))</span> <span>((q>1))</span>, Hu and Yang (Bull Lond Math Soc 35:759–769, 2003) for the endpoint weak <span>(Llog L)</span> type estimates when <span>(Omega in textrm{Lip}_{alpha }({mathbb {S}}^{d-1}))</span> for some <span>(alpha in (0,,1])</span>.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140939910","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}
Pub Date : 2024-05-07DOI: 10.1007/s00041-024-10088-w
Yan-Song Fu, Min-Wei Tang
In this paper we investigate the harmonic analysis of infinite convolutions generated by admissible pairs on Euclidean space ({{mathbb {R}}}^n). Our main results give several sufficient conditions so that the infinite convolution (mu ) to be a spectral measure, that is, its Hilbert space (L^2(mu )) admits a family of orthonormal basis of exponentials. As a concrete application, we give a complete characterization on the spectral property for certain infinite convolution on the plane ({{mathbb {R}}}^2) in terms of admissible pairs.
{"title":"Existence of Exponential Orthonormal Bases for Infinite Convolutions on $${{mathbb {R}}}^n$$","authors":"Yan-Song Fu, Min-Wei Tang","doi":"10.1007/s00041-024-10088-w","DOIUrl":"https://doi.org/10.1007/s00041-024-10088-w","url":null,"abstract":"<p>In this paper we investigate the harmonic analysis of infinite convolutions generated by admissible pairs on Euclidean space <span>({{mathbb {R}}}^n)</span>. Our main results give several sufficient conditions so that the infinite convolution <span>(mu )</span> to be a spectral measure, that is, its Hilbert space <span>(L^2(mu ))</span> admits a family of orthonormal basis of exponentials. As a concrete application, we give a complete characterization on the spectral property for certain infinite convolution on the plane <span>({{mathbb {R}}}^2)</span> in terms of admissible pairs.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940018","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}
Pub Date : 2024-05-02DOI: 10.1007/s00041-024-10085-z
Isroil A. Ikromov, Dildora I. Ikromova
In this article, we studied the convolution operators (M_k) with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator (M_k) is associated to the characteristic hypersurfaces(Sigma subset {mathbb {R}}^3) of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order (-k) for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point (vin Sigma ) at which, exactly one of the principal curvatures of the surface (Sigma ) does not vanish. Such surfaces exhibit singularities of the type A in the sense of Arnold’s classification. Denoting by (k_p) the minimal number such that (M_k) is (L^pmapsto L^{p'})-bounded for (k>k_p,) we showed that the number (k_p) depends on some discrete characteristics of the surface (Sigma ).
{"title":"On the Sharp Estimates for Convolution Operators with Oscillatory Kernel","authors":"Isroil A. Ikromov, Dildora I. Ikromova","doi":"10.1007/s00041-024-10085-z","DOIUrl":"https://doi.org/10.1007/s00041-024-10085-z","url":null,"abstract":"<p>In this article, we studied the convolution operators <span>(M_k)</span> with oscillatory kernel, which are related to the solutions of the Cauchy problem for the strictly hyperbolic equations. The operator <span>(M_k)</span> is associated to the characteristic hypersurfaces<span>(Sigma subset {mathbb {R}}^3)</span> of a hyperbolic equation and smooth amplitude function, which is homogeneous of the order <span>(-k)</span> for large values of the argument. We investigated the convolution operators assuming that the corresponding amplitude function is contained in a sufficiently small conic neighborhood of a given point <span>(vin Sigma )</span> at which, exactly one of the principal curvatures of the surface <span>(Sigma )</span> does not vanish. Such surfaces exhibit singularities of the type <i>A</i> in the sense of Arnold’s classification. Denoting by <span>(k_p)</span> the minimal number such that <span>(M_k)</span> is <span>(L^pmapsto L^{p'})</span>-bounded for <span>(k>k_p,)</span> we showed that the number <span>(k_p)</span> depends on some discrete characteristics of the surface <span>(Sigma )</span>.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887480","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}
Pub Date : 2024-05-02DOI: 10.1007/s00041-024-10087-x
Emiel Lorist, Zoe Nieraeth
We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator T in the weighted Lebesgue scale and the compactness of T in the unweighted Lebesgue scale yields compactness of T on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.
我们证明了满足某些凸性和凹性条件的巴拿赫函数空间上算子的紧凑性定理的外推法。特别是,我们证明了算子 T 在加权 Lebesgue 标度上的有界性和 T 在非加权 Lebesgue 标度上的紧凑性,从而得到了 T 在一类非常普遍的巴拿赫函数空间上的紧凑性。作为我们的主要新工具,我们利用一种新颖的稀疏自改进技术,证明了哈代-利特尔伍德最大算子在这类空间及其关联空间上的有界性的各种特征。我们将我们的主要结果应用于证明奇异积分算子换元的紧凑性,以及在加权可变 Lebesgue 空间等上与平均振荡消失的函数进行点相乘。
{"title":"Extrapolation of Compactness on Banach Function Spaces","authors":"Emiel Lorist, Zoe Nieraeth","doi":"10.1007/s00041-024-10087-x","DOIUrl":"https://doi.org/10.1007/s00041-024-10087-x","url":null,"abstract":"<p>We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator <i>T</i> in the weighted Lebesgue scale and the compactness of <i>T</i> in the unweighted Lebesgue scale yields compactness of <i>T</i> on a very general class of Banach function spaces. As our main new tool, we prove various characterizations of the boundedness of the Hardy-Littlewood maximal operator on such spaces and their associate spaces, using a novel sparse self-improvement technique. We apply our main results to prove compactness of the commutators of singular integral operators and pointwise multiplication by functions of vanishing mean oscillation on, for example, weighted variable Lebesgue spaces.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887467","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}
Pub Date : 2024-04-23DOI: 10.1007/s00041-024-10080-4
D. Leviatan, O. Motorna, I. Shevchuk
{"title":"Fast Decreasing Trigonometric Polynomials and Applications","authors":"D. Leviatan, O. Motorna, I. Shevchuk","doi":"10.1007/s00041-024-10080-4","DOIUrl":"https://doi.org/10.1007/s00041-024-10080-4","url":null,"abstract":"","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140669285","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}
Pub Date : 2024-04-18DOI: 10.1007/s00041-024-10082-2
Jacopo Bellazzini, Vladimir Georgiev
We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite (L^2) norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.
{"title":"Compact Embeddings for Fractional Super and Sub Harmonic Functions with Radial Symmetry","authors":"Jacopo Bellazzini, Vladimir Georgiev","doi":"10.1007/s00041-024-10082-2","DOIUrl":"https://doi.org/10.1007/s00041-024-10082-2","url":null,"abstract":"<p>We prove compactness of the embeddings in Sobolev spaces for fractional super and sub harmonic functions with radial symmetry. The main tool is a pointwise decay for radially symmetric functions belonging to a function space defined by finite homogeneous Sobolev norm together with finite <span>(L^2)</span> norm of the Riesz potentials. As a byproduct we prove also existence of maximizers for the interpolation inequalities in Sobolev spaces for radially symmetric fractional super and sub harmonic functions.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140630054","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}
Pub Date : 2024-04-17DOI: 10.1007/s00041-024-10083-1
Aleksandar Bulj, Vjekoslav Kovač
The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by standard ellipsoids or axes-parallel rectangles. Along the lines of the same proof, we also establish a d-parameter Menshov–Paley–Zygmund-type theorem for the Fourier transform on ({mathbb {R}}^d). Such a result is interesting for (dgeqslant 2) because, in a sharp contrast with the one-dimensional case, the corresponding endpoint ({text {L}}^2) estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.
{"title":"Multi-parameter Maximal Fourier Restriction","authors":"Aleksandar Bulj, Vjekoslav Kovač","doi":"10.1007/s00041-024-10083-1","DOIUrl":"https://doi.org/10.1007/s00041-024-10083-1","url":null,"abstract":"<p>The main result of this note is the strengthening of a quite arbitrary a priori Fourier restriction estimate to a multi-parameter maximal estimate of the same type. This allows us to discuss a certain multi-parameter Lebesgue point property of Fourier transforms, which replaces Euclidean balls by standard ellipsoids or axes-parallel rectangles. Along the lines of the same proof, we also establish a <i>d</i>-parameter Menshov–Paley–Zygmund-type theorem for the Fourier transform on <span>({mathbb {R}}^d)</span>. Such a result is interesting for <span>(dgeqslant 2)</span> because, in a sharp contrast with the one-dimensional case, the corresponding endpoint <span>({text {L}}^2)</span> estimate (i.e., a Carleson-type theorem) is known to fail since the work of C. Fefferman in 1970. Finally, we show that a Strichartz estimate for a given homogeneous constant-coefficient linear dispersive PDE can sometimes be strengthened to a certain pseudo-differential version.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140608891","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}
Pub Date : 2024-04-15DOI: 10.1007/s00041-024-10081-3
Ushangi Goginava, Farrukh Mukhamedov
Revisiting the main point of the almost everywhere convergence, it becomes clear that a weak (1,1)-type inequality must be established for the maximal operator corresponding to the sequence of operators. The better route to take in obtaining almost everywhere convergence is by using the uniform boundedness of the sequence of operator, instead of using the mentioned maximal type of inequality. In this paper it is proved that a sequence of operators, defined by matrix transforms of the Walsh–Fourier series, is convergent almost everywhere to the function (fin L_{1}) if they are uniformly bounded from the dyadic Hardy space (H_{1} left( {mathbb {I}}right) ) to (L_{1}left( mathbb {I}right) ). As a further matter, the characterization of the points are put forth where the sequence of the operators of the matrix transform is convergent.
{"title":"Uniform Boundedness of Sequence of Operators Associated with the Walsh System and Their Pointwise Convergence","authors":"Ushangi Goginava, Farrukh Mukhamedov","doi":"10.1007/s00041-024-10081-3","DOIUrl":"https://doi.org/10.1007/s00041-024-10081-3","url":null,"abstract":"<p>Revisiting the main point of the almost everywhere convergence, it becomes clear that a weak (1,1)-type inequality must be established for the maximal operator corresponding to the sequence of operators. The better route to take in obtaining almost everywhere convergence is by using the uniform boundedness of the sequence of operator, instead of using the mentioned maximal type of inequality. In this paper it is proved that a sequence of operators, defined by matrix transforms of the Walsh–Fourier series, is convergent almost everywhere to the function <span>(fin L_{1})</span> if they are uniformly bounded from the dyadic Hardy space <span>(H_{1} left( {mathbb {I}}right) )</span> to <span>(L_{1}left( mathbb {I}right) )</span>. As a further matter, the characterization of the points are put forth where the sequence of the operators of the matrix transform is convergent.</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571292","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}
for given data in the Sobolev space (H^s(mathbb R)). This IVP is known to be locally well-posed for given data with Sobolev regularity (s>-frac{1}{4}) and globally well-posed for (sge 0) (Carvajal in Electron J Differ Equ 2004:1–10, 2004). For given data in (H^s(mathbb R)), (0>s> -frac{1}{4}) no global well-posedness result is known. In this work, we derive an almost conserved quantity for such data and obtain a sharp global well-posedness result. Our result answers the question left open in (Carvajal in Electron J Differ Equ 2004:1–10, 2004).
{"title":"Sharp Global Well-Posedness for the Cubic Nonlinear Schrödinger Equation with Third Order Dispersion","authors":"X. Carvajal, M. Panthee","doi":"10.1007/s00041-024-10084-0","DOIUrl":"https://doi.org/10.1007/s00041-024-10084-0","url":null,"abstract":"<p>We consider the initial value problem (IVP) associated to the cubic nonlinear Schrödinger equation with third-order dispersion </p><span>$$begin{aligned} partial _{t}u+ialpha partial ^{2}_{x}u- partial ^{3}_{x}u+ibeta |u|^{2}u = 0, quad x,t in mathbb R, end{aligned}$$</span><p>for given data in the Sobolev space <span>(H^s(mathbb R))</span>. This IVP is known to be locally well-posed for given data with Sobolev regularity <span>(s>-frac{1}{4})</span> and globally well-posed for <span>(sge 0)</span> (Carvajal in Electron J Differ Equ 2004:1–10, 2004). For given data in <span>(H^s(mathbb R))</span>, <span>(0>s> -frac{1}{4})</span> no global well-posedness result is known. In this work, we derive an <i>almost conserved quantity</i> for such data and obtain a sharp global well-posedness result. Our result answers the question left open in (Carvajal in Electron J Differ Equ 2004:1–10, 2004).</p>","PeriodicalId":15993,"journal":{"name":"Journal of Fourier Analysis and Applications","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140571449","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}