Sever Silvestru Dragomir
Some sharp Bessel type inequalities for non-orthonormal families of vectors in inner product spaces are given. Applications for complex numbers are also provided.
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Reverses of Schwarz, triangle and Bessel inequalities in inner product spaces that improve some earlier results are pointed out. They are applied to obtain new Gruss type inequalities in inner product spaces. Some natural applications for integral inequalities are also pointed out.
New reverses of the Schwarz, triangle and Bessel inequalities in inner product spaces are pointed out. These results complement the recent ones obtained by the author in an earlier paper. Further, they are employed to establish new Gruss type inequalities. Finally, some natural integral inequalities are stated as well.
Some new counterparts of Bessel's inequality for orthornormal families in real or complex inner product spaces are pointed out. Applications for some Gruss type inequalities are also empahsized.
A new counterpart of Bessel's inequality for orthornormal families in real or complex inner product spaces is obtained. Applications for some Gruss type results are also provided.
The main aim of this monograph is to survey some recent results obtained by the author related to reverses of the Schwarz, triangle and Bessel inequalities. Some Gruss' type inequalities for orthonormal families of vectors in real or complex inner product spaces are presented as well. Generalizations of the Boas-Bellman, Bombieri, Selberg, Heilbronn and Pecaric inequalities for finite sequences of vectors that are not necessarily orthogonal are also provided. Two extensions of the celebrated Ostrowski's inequalities for sequences or real numbers and the generalization of Wagner's inequality in…
We review recent results on functional inequalities for systems of orthonormal functions. The key finding is that for various operators the orthonormality leads to a gain over a simple application of the triangle inequality. The operators under consideration are either related to Sobolev type inequalities or to Fourier restriction type inequalities.
In this paper we show that every g-frame for an \linebreak infinite dimensional Hilbert space $\mathcal{H}$ can be written as a sum of three g-orthonormal bases for $\mathcal{H}$. Also, we prove that every g-frame can be represented as a linear combination of two g-orthonormal bases if and only if it is a g-Riesz basis. Further, we show each g-Bessel multiplier is a Bessel multiplier and investigate the inversion of g-frame multipliers. Finally, we introduce the concept of controlled g-frames and weighted g-frames and show that the sequence induced by each controlled g-frame (resp. weighted g-…
In this paper, we prove restriction theorems for the Fourier-Laguerre transform and establish Strichartz estimates for the Schrödinger propagator $e^{-itL_α}$ for the Laguerre operator $L_α=-Δ-\sum_{j=1}^{n}(\dfrac{2α_j+1}{x_j}\dfrac{\partial}{\partial x_j})+\dfrac{|x|^2}{4}$, $α=(α_1,α_2,\cdots,α_n)\in{(-\frac{1}{2},\infty)^n}$ on $\mathbb{R}_+^n$ involving systems of orthonormal functions.
In this paper, we develop a thorough analysis of the boundedness properties of the maximal operator for the Bochner-Riesz means related to the Fourier-Bessel expansions. For this operator, we study weighted and unweighted inequalities in the spaces L^p((0,1),x^{2ν+1}dx). Moreover, weak and restricted weak type inequalities are obtained for the critical values of p. As a consequence, we deduce the almost everywhere pointwise convergence of these means.
A counterpart of the famous Bessel's inequality for orthornormal families in real or complex inner product spaces is given. Applications for some Gruss type inequalities are also provided.
Regarding the generalizations of the Bessel inequality in Hilbert spaces which are due to Bombiari and Boas--Bellman, we obtain a version of the Bessel inequality and some generalizations of this inequality in the framework of Hilbert $C^*$-modules.
Companion results to the Bombieri generalisation of Bessel's inequality in inner product spaces are given.
Let $Δ_{k,a}$ and $Δ_k $ be the $(k,a)$-generalized Laguerre operator and the Dunkl Laplacian operator on $\mathbb{R}^n$, respectively. The aim of this article is twofold. First, we prove a restriction theorem for the Fourier-$Δ_{k,a}$ transform. Next, as an application of the restriction problem, we establish Strichartz estimates for orthonormal families of initial data for the Schrödinger propagator $e^{-i t Δ_{k, a}} $ associated with the operator $ Δ_{k, a}$. Further, using the classical Strichartz estimates for the free Schrödinger propagator $e^{-i t Δ_{k, a}} $ for orthonormal systems o…
The notion of multipliers in Hilbert space was introduced by Schatten in 1960 using orthonormal sequences and was generalized by Balazs in 2007 using Bessel sequences. This was extended to Banach spaces by Rahimi and Balazs in 2010 using p-Bessel sequences. In this paper, we further extend this by considering Lipschitz functions. On the way we define frames for metric spaces which extends the notion of frames and Bessel sequences for Banach spaces. We show that when the symbol sequence converges to zero, the multiplier is a Lipschitz compact operator. We study how the variation of parameters i…
Some inequalities in 2-inner product spaces generalizing Bessel's result that are similar to the Boas-Bellman inequality from inner product spaces, are given. Applications for determinantal integral inequalities are also provided.
We generalize the theorems of Stein--Tomas and Strichartz about surface restrictions of Fourier transforms to systems of orthonormal functions with an optimal dependence on the number of functions. We deduce the corresponding Strichartz bounds for solutions to Schrödinger equations up to the endpoint, thereby solving an open problem of Frank, Lewin, Lieb and Seiringer. We also prove uniform Sobolev estimates in Schatten spaces, extending the results of Kenig, Ruiz, and Sogge. We finally provide applications of these results to a Limiting Absorption Principle in Schatten spaces, to the well-pos…
Orthonormal Strichartz inequalities and their applications on abstract measure spaces
本文将基本不等式从单函数推广至规范正交系统，在抽象测度空间上建立了Schrödinger传播子与色散半群的规范正交Strichartz估计，并应用于证明Hartree方程适定性及非紧超曲面上Fourier变换的规范正交限制定理。
In this paper we obtain the non - asymptotic estimations for Riesz's and Bessel's potential integral operators in the so - called Bilateral Grand Lebesgue Spaces. We also give examples to show the sharpness of these inequalities.
The motivation for this paper comes from the following question on comparison of norms of conformal martingales $X$, $Y$ in $\R^d$, $d\geq 2$. Suppose that $Y$ is differentially subordinate to $X$. For $0<p<\infty$, what is the optimal value of the constant $C_{p,d}$ in the inequality $$ Y_p\leq C_{p,d}X_p ?$$ We answer this question by considering a more general related problem for nonnegative submartingales. This enables us to study extension of the above inequality to the case when $d>1$ is not an integer, which has further interesting applications to stopped Bessel processes and to the beh…
Grothendieck's theorem for Bessel sequences
本文建立贝塞尔序列的精确格罗滕迪克定理，将其应用于肯定回答Olevskii的延拓问题，且附带Lean 4对主要结果的形式化验证。
Semiclassical Moser--Trudinger type inequalities for orthonormal families in $\mathbb{R}^2$
本文针对二维空间中H¹标准正交族，证明了带最优系数4π的全空间端点Moser-Trudinger不等式，通过热半群改进Rumin密度方法处理相关问题，还揭示了不等式中秩依赖项N-1的不可或缺性。
A div-curl inequality for orthonormal functions
本文证明了关于无旋场与无散场乘积之和的一个界，在适当的正交性条件下，该界关于项数呈次线性缩放，其精神与 Lieb--Thirring 不等式相似。
We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidea…
We review some recent progress on Lieb-Thirring inequalities, focusing on direct methods to kinetic estimates for orthonormal functions and applications for many-body quantum systems.
We prove various extensions of the Loomis-Whitney inequality and its dual, where the subspaces on which the projections (or sections) are considered are either spanned by vectors $w_i$ of a not necessarily orthonormal basis of $\mathbb{R}^n$, or their orthogonal complements. In order to prove such inequalities we estimate the constant in the Brascamp-Lieb inequality in terms of the vectors $w_i$. Restricted and functional versions of the inequality will also be considered.
This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the boundary of the domain. One-dimensional $L_1$ and $L_p$ inequalities and their multidimensional analogues are proved. We consider spatial inequalities in open convex domains with the finite inner radius. Constants in these inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases.
We prove some vector-valued inequalities for fractional integrals defined in the context of two different orthonormal systems of Laguerre functions. Our results are based on estimates of the corresponding kernels with precise control of the parameters involved. As an application, mixed norm estimates for the fractional integrals related to the harmonic oscillator are deduced.
Continuous generalized fusion frame theory was recently introduced by Rahimi and al. Several equalities and inequalities have been obtained for frame, fusion generalized fusion frame, among others. In the present paper, we continue and extend these results to obtain some important identities and inequalities in the case of continuous generalized fusion frame, Parceval continuous generalized fusion frame, $ λ$-tight continuous generalized fusion frame. Moreover, we obtain some new inequalities for the alternate dual continuous generalized fusion frame. Finally, we obtain frame operator of a pai…
Power-weighted Hardy type integral inequalities with additional terms on finite intervals
本文研究有限区间上的积分Hardy型不等式，证明新的L2和Lp不等式，常数依赖于Bessel函数对应Lamb型方程的首根，并给出极值函数和多维推广。
In this paper, we investigate the wavelet coefficients for function spaces $\mathcal{A}_k^p=\{f: \|(i ω)^k\hat{f}(ω)\|_p\leq 1\}, k\in N, p\in(1,\infty)$ using an important quantity $C_{k,p}(ψ)$. In particular, Bernstein type inequalities associated with wavelets are established. We obtained a sharp inequality of Bernstein type for splines, which induces a lower bound for the quantity $C_{k,p}(ψ)$ with $ψ$ being the semiorthogonal spline wavelets. We also study the asymptotic behavior of wavelet coefficients for both the family of Daubechies orthonormal wavelets and the family of semiorthogona…
Bochkarev's inequalities in the Anisotropic grand Lorentz Spaces
本文研究了各向异性大洛伦茨空间中Bochkarev型不等式，推广了经典结果至任意正交系统。
Properties of local orthonormal systems, Part II: Geometric characterization of Bernstein inequalities
本文研究了二元滤流下局部标准正交系与字典在 $L^p$ 空间中最佳 $n$-项逼近空间的等价性问题，给出了决定该问题的 Bernstein 型不等式的几何刻画，并特化至 Haar 系统与多项式空间等实例。
We present some Caffarelli-Kohn-Nirenberg-type inequalities on Herz-type Besov-Triebel-Lizorkin spaces, Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. More Precisely, we investigate the inequalities \begin{equation*} \big\|f\big\|_{\dot{k}_{v,σ}^{α_{1},r}}\leq c\big\|f\big\|_{\dot{K}_{u}^{α_{2},δ}}^{1-θ}\big\|f\big\|_{\dot{K}_{p}^{α_{3},δ_{1}}A_{β}^{s}}^{θ}, \end{equation*} and \begin{equation*} \big\|f\big\|_{\mathcal{E}_{p,2,u}^{σ}}\leq c\big\|f\big\|_{\mathcal{M}_{μ}^{δ}}^{1-θ}\big\|f\big\|_{\mathcal{N}_{q,β,v}^{s}}^{θ}, \end{equation*} with some appropriate assumptions on the para…
A generic functional inequality and Riccati pairs: an alternative approach to Hardy-type inequalities
本文在黎曼流形上提出加性与乘性泛函不等式，引入Riccati对扩展Bessel对，通过求解Riccati型ODE简洁证明多种Hardy型不等式，并在Cartan-Hadamard流形上证明精确不确定性原理。
Extending several works, we prove a general Adams-Moser-Trudinger type inequality for the embedding of Bessel-potential spaces $\tilde H^{\frac{n}{p},p}(Ω)$ into Orlicz spaces for an arbitrary domain $Ω\subset \mathbb{R}^n$ with finite measure. In particular we prove $$\sup_{u\in \tilde H^{\frac{n}{p},p}(Ω), \;\|(-Δ)^{\frac{n}{2p}}u\|_{L^{p}(Ω)}\leq 1}\int_Ωe^{α_{n,p} |u|^\frac{p}{p-1}}dx \leq c_{n,p}|Ω|, $$ for a positive constant $α_{n,p}$ whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. $(-Δ)^\frac{n}{2p}u\in L^{(p,q)})$. The proofs are simple…
We give necessary and sufficient conditions for interpolation inequalities of the type considered by Marcinkiewicz and Zygmund to be true in the case of Banach space-valued polynomials and Jacobi weights and nodes. We also study the vector-valued expansion problem of $L_p$-functions in terms of Jacobi polynomials and consider the question of unconditional convergence. The notion of type $p$ with respect to orthonormal systems leads to some characterizations of Hilbert spaces. It is also shown that various vector-valued Jacobi means are equivalent.
Hilbert space frames generalize orthonormal bases to allow redundancy in representations of vectors while keeping good reconstruction properties. A frame comes with an associated frame operator encoding essential properties of the frame. We study a polytope that arises in an algorithm for constructing all finite frames with given lengths of frame vectors and spectrum of the frame operator, which is a Gelfand-Tsetlin polytope. For equal norm tight frames, we give a non-redundant description of the polytope in terms of equations and inequalities. From this we obtain the dimension and number of f…
Sharp bounds for non-trace class noise and applications to SPDEs
针对带有色、非迹类时空噪声的随机偏微分方程，建立了高斯级数在Bessel位势空间中收敛的充要条件，并应用于随机热方程得到尖锐估计。
It is now well known that sparse or compressible vectors can be stably recovered from their low-dimensional projection, provided the projection matrix satisfies a Restricted Isometry Property (RIP). We establish new implications of the RIP with respect to nonlinear approximation in a Hilbert space with a redundant frame. The main ingredients of our approach are: a) Jackson and Bernstein inequalities, associated to the characterization of certain approximation spaces with interpolation spaces; b) a new proof that for overcomplete frames which satisfy a Bernstein inequality, these interpolation …
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