Considering potentials defined by Bessel kernel with Bessel convolution a Kerman-Sawyer type characterization of trace inequality is given. As an application an estimate on the least eigenvalue of Schrödinger-Bessel operators is derived.
The aim of this paper is to establish Turán -type inequality for the Hahn-Exton $q$-Bessel functions. The result is obtained by the use of limit transition.
We define and examine nonlinear potential by Bessel convolution with Bessel kernel. We investigate removable sets with respect to Laplace-Bessel inequality. By studying the maximal and fractional maximal measure, a Wolff type inequality is proved. Finally the relation of B-$p$ capacity and B-Lipschitz mapping, and the B-$p$ capacity and weighted Hausdorff measure and the B-$p$ capacity of Cantor sets are examined.
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.
In this paper, an analogous of Heisenberg inequality is established for Laguerre-Bessel transform. Also, a local uncertainty principle for this transform is investigate
We unify the three distinct inequality sequences (Abramowitz ans Stegun (1972) [1,9.5.2]) of positive real zeros of Bessel functions into a single one.
In the paper, the authors verify the complete monotonicity of the difference $e^{1/t}-ψ'(t)$ on $(0,\infty)$, compute the completely monotonic degree and establish integral representations of the remainder of the Laurent series expansion of $e^{1/z}$, and derive an inequality which gives a lower bound for the first order modified Bessel function of the first kind.
In this work we prove analogues of Bessel inequality and Riesz-Fisher theorem in Hilbert spaces with respect to sequences. We apply our generalized Bessel inequality to the Hilbert spaces associated with the Normal, Beta, Gamma and certain discrete probability distributions to show how to generate certain type of inequalities for special functions systematically.
N. Kishore, Proc. Amer. Math. Soc. 14 (1963), 523, considered the Rayleigh functions sigma_n, sums of the negative even powers of the (non-zero) zeros of the Bessel function J_nu(z) and provided a convolution type sum formula for finding sigma_n in terms of sigma_1, ...,sigma_{n-1}. His main tool was the recurrence relation for Bessel functions. Here we extend this result to a larger class of functions by using Riccati ifferential equations. We get new results for the zeros of certain combinations of Bessel functions and their first and second derivatives as well as recovering some results of …
Bessel-type convolution algebras of bounded Borel measures on the matrix cones of positive semidefinite $q\times q$-matrices over $\mathbb R, \mathbb C, \mathbb H$ were introduced recently by Rösler. These convolutions depend on some continuous parameter, generate commutative hypergroup structures and have Bessel functions of matrix argument as characters. Here, we first study the rich algebraic structure of these hypergroups. In particular, the subhypergroups and automorphisms are classified, and we show that each quotient by a subhypergroup carries a hypergroup structure of the same type. Th…
In this paper we introduce probability-preserving convolution algebras on cones of positive semidefinite matrices over one of the division algebras $\b F = \b R, \b C$ or $\b H$ which interpolate the convolution algebras of radial bounded Borel measures on a matrix space $M_{p,q}(\b F)$ with $p\geq q$. Radiality in this context means invariance under the action of the unitary group $U_p(\b F)$ from the left. We obtain a continuous series of commutative hypergroups whose characters are given by Bessel functions of matrix argument. Our results generalize well-known structures in the rank one cas…
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.
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.
We study Bessel and Dunkl processes $(X_{t,k})_{t\ge0}$ on $\mathbb R^N$ with possibly multivariate coupling constants $k\ge0$. These processes describe interacting particle systems of Calogero-Moser-Sutherland type with $N$ particles. For the root systems $A_{N-1}$ and $B_N$ these Bessel processes are related with $β$-Hermite and $β$-Laguerre ensembles. Moreover, for the frozen case $k=\infty$, these processes degenerate to deterministic or pure jump processes. We use the generators for Bessel and Dunkl processes of types A and B and derive analogues of Wigner's semicircle and Marchenko-Pastu…
Consider the trilinear form for twisted convolution on $\mathbb{R}^{2d}$: \begin{equation*} \mathcal{T}_t(\mathbf{f}):=\iint f_1(x)f_2(y)f_3(x+y)e^{itσ(x,y)}dxdy,\end{equation*} where $σ$ is a symplectic form and $t$ is a real-valued parameter. It is known that in the case $t\neq0$ the optimal constant for twisted convolution is the same as that for convolution, though no extremizers exist. Expanding about the manifold of triples of maximizers and $t=0$ we prove a sharpened inequality for twisted convolution with an arbitrary antisymmetric form in place of $σ$.
In this paper we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner-Lebesgue space $L^p((0,\infty),B)$, where $B$ is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operator $Δ_λ=-x^{-λ}\frac{d}{dx}x^{2λ}\frac{d}{dx}x^{-λ}$, $λ>0$. We characterize the UMD property for a Banach space $B$ by using $L^p((0,\infty),B)$-boundedness properties of g-functions defined by Bessel-Poisson semigroups. As a by product we prove that t…
The boundary behaviour of convolutions with Poisson kernel and with square root from Poisson kernel is essentially differs. The first ones have only nontangential limit. For the last ones the convergence is over domains admittings a logarithmic order of the contact with the boundary . This result was generalized by authors on the spaces of homogeneous type. Here we prove the boundedness and some weighted estimates for the corresponding maximal operator. Earlier it was known only weak type inequality.
The convolution inequality $h*h(ξ) \leq B |ξ|^θh(ξ)$ defined on $\Rn$ arises from a probabilistic representation of solutions of the $n$-dimensional Navier-Stokes equations, $n \geq 2$. Using a chaining argument, we establish the nonexistence of strictly positive fully supported solutions of this inequality if $θ\geq n/2$, in all dimensions $n \geq 1$. We use this result to describe a chain of continuous embeddings from spaces associated with probabilistic solutions to the spaces $BMO^{-1}$ and $BMO_T^{-1}$ associated with the Koch-Tataru solutions of the Navier-Stokes equations.
A different application of the familiar integral representation for the modifed Bessel function drives to a new Kontorovich-Lebedev-like integral transformation of a general complex index. Mapping and operational properties, a convolution operator and inversion formula are established. Solvability conditions and explicit solutions of the corresponding class of convolution integral equations are exhibited.
In this paper, we consider the normalized Bessel function of index $α> -\frac{1}{2}$, we find an integral representation of the term $x^nj_{α+n}(x)j_α(y)$. This allows us to establish a product formula for the generalized Hankel function $B^{κ,n}_λ$ on $\mathbb{R}$. $B^{κ,n}_λ$ is the kernel of the integral transform $\mathcal{F}_{κ,n}$ arising from the Dunkl theory. Indeed we show that $B^{κ,n}_λ(x)B^{κ,n}_λ(y)$ can be expressed as an integral in terms of $B^{κ,n}_λ(z)$ with explicit kernel invoking Gegenbauer polynomials for all $n\in\mathbb{N}^\ast$. The obtained result generalizes the prod…
We give necessary and sufficient conditions for Borel measures to satisfy the inequality introduced by Komisarski, Rajba (2018). This inequality is a generalization of the convex order inequality for binomial distributions, which was proved by Mrowiec, Rajba, Wąsowicz (2017), as a probabilistic version of the inequality for convex functions, that was conjectured as an old open problem by I.~Raşa. We present also further generalizations using convex order inequalities between convolution polynomials of finite Borel measures. We generalize recent results obtained by B.~Gavrea (2018) in the discr…
Young's convolution inequality provides an upper bound for the convolution of functions in terms of $L^p$ norms. It is known that for certain groups, including Heisenberg groups, the optimal constant in this inequality is equal to that for Euclidean space of the same topological dimension, yet no extremizing functions exist. For Heisenberg groups we characterize ordered triples of functions that nearly extremize the inequality. The analysis relies on a characterization of approximate solutions of a certain class of functional equations. A result of this type is developed for a class of such eq…
Those functions which nearly extremize Young's convolution inequality are characterized for discrete groups which have no nontrivial finite subgroups. Near-extremizers of the Hausdorff-Young inequality are characterized for Z^d.
We establish the general form of a geometric comparison principle for $n$-fold convolutions of certain singular measures in $\mathbb{R}^d$ which holds for arbitrary $n$ and $d$. This translates into a pointwise inequality between the convolutions of projection measure on the paraboloid and a perturbation thereof, and we use it to establish a new sharp Fourier extension inequality on a general convex perturbation of a parabola. Further applications of the comparison principle to sharp Fourier restriction theory are discussed in a companion paper.
Let $d\ge 2$ and $T$ be the convolution operator $Tf(x)=\int_{\reals^{d-1}} f(x'-t,x_d-|t|^2)\,dt$, which is is bounded from $L^{(d+1)/d}(\reals^d)$ to $L^{d+1}(\reals^d)$. We show that any critical point $f\in L^{(d+1)/d}$ of the functional $\norm{Tf}_{d+1}/\norm{f}_{(d+1)/d}$ is infinitely differentiable, and that $|x|^δf\in L^{(d+1)/d}$ for some $δ>0$. In particular, this holds for all extremizers of the associated inequality. This is done by exploiting a generalized Euler-Lagrange equation, and certain weighted norm inequalities for $T$.
The Riesz-Sobolev inequality relates the convolution of nonnegative functions on Euclidean space to the convolution of their symmetric nonincreasing rearrangements. We show that for dimension one, for indicator functions of sets, if the inequality is sufficiently close to an equality then the sets in question must nearly coincide with intervals.
We establish a weighted inequality for fractional maximal and convolution type operators, between weak Lebesgue spaces and Wiener amalgam type spaces on $ \mathbb R $ endowed with a measure which needs not to be doubling.
The Hölder-Brascamp-Lieb inequalities are a collection of multilinear inequalities generalizing a convolution inequality of Young and the Loomis-Whitney inequalities. The full range of exponents was classified in Bennett et al. (2008). In a setting similar to that of Ivanisvili and Volberg (2015), we introduce a notion of size for these inequalities which generalizes $L^p$ norms. Under this new setup, we then determine necessary and sufficient conditions for a generalized Hölder-Brascamp-Lieb type inequality to hold and establish sufficient conditions for extremizers to exist when the underlyi…
