In this paper we prove that the Hankel multipliers of Laplace transform type on $(0,1)^n$ are of weak type (1,1). Also we analyze Lp-boundedness properties for the imaginary powers of Bessel operator on $(0,1)^n$.
We study the problem of characterizing Hankel-Schur multipliers and Toeplitz-Schur multipliers of Schatten-von Neumann class $\bS_p$ for $0<p<1$. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of $\bS_p$. We also give a characterization of the Hankel-Schur multipliers of $\bS_p$ whose symbols have lacunary power series. Then the results on Hankel-Schur multipliers are used to obtain a characterization of the Toeplitz-Schur multipliers of $\bS_p$. Finally, we return to Hankel-Schur multipliers and obtain new results in the cas…
We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of Hankel multipliers.
Connections between Hankel transforms of different order for $L^p$-functions are examined. Well known are the results of Guy [Guy] and Schindler [Sch]. Further relations result from projection formulae for Bessel functions of different order. Consequences for Hankel multipliers are exhibited and implications for radial Fourier multipliers on Euclidean spaces of different dimensions indicated.
We give characterizations of radial Fourier multipliers as acting on radial L^p-functions, 1<p<2d/(d+1), in terms of Lebesgue space norms for Fourier localized pieces of the convolution kernel. This is a special case of corresponding results for general Hankel multipliers. Besides L^p-L^q bounds we also characterize weak type inequalities and intermediate inequalities involving Lorentz spaces. Applications include results on interpolation of multiplier spaces.
The complete Lipschitz-Hankel integrals (LHIs) include the Laplace transforms of the Bessel functions, multiplied by powers. Such Laplace transforms can be evaluated using associated Legendre functions. It is noted that there are errors in published versions of these evaluations, and a merged and emended list of seven transforms is given. Errata for standard reference works, such as the table of Gradshteyn and Ryzhik, are also given. Most of the errors are attributable to inconsistent normalization of the Legendre functions. These transforms can be viewed as limits of incomplete LHIs, which fi…
We investigate Laplace type and Laplace-Stieltjes type multipliers in the $d$-dimensional setting of the Dunkl harmonic oscillator with the associated group of reflections isomorphic to $\mathbb{Z}_2^d$ and in the related context of Laguerre function expansions of convolution type. We use Calderón-Zygmund theory to prove that these multiplier operators are bounded on weighted $L^p$, $1<p<\infty$, and from $L^1$ to weak $L^1$.
Fourier multiplier analysis is developed for nonlocal peridynamic-type Laplace operators, which are defined for scalar fields in $\mathbb{R}^n$. The Fourier multipliers are given through an integral representation. We show that the integral representation of the Fourier multipliers is recognized explicitly through a unified and general formula in terms of the hypergeometric function $_2F_3$ in any spatial dimension $n$. Asymptotic analysis of $_2F_3$ is utilized to identify the asymptotic behavior of the Fourier multipliers $m(ν)$ as $\|ν\|\rightarrow \infty$. We show that the multipliers are …
We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.
The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(α,β)}=J^{(α,β)}-I$, where $J^{(α,β)}$ is the three-term recurrence relation for the normalized Jacobi polynomials and $I$ is the identity operator, we define the corresponding Littlewood-Paley-Stein $g_k^{(α,β)}$-functions associated with it and we prove an equivalence of norms with weights for them. As a consequence, we deduce a result for Laplace type multipliers.
In the present paper the authors show that iterations of the Hankel transform with $\mathscr{K}_ν$-transform is a constant multiple of the Widder transform. Using these iteration identities, several Parseval-Goldstein type theorems for these transforms are given. By making use of these results a number of new Goldstein type exchange identities are obtained for these and the Laplace transform. The identities proven in this paper are shown to give rise to useful corollaries for evaluating infinite integrals of special functions. Some examples are also given as illustration of the results present…
In this paper we establish $L^p$-boundedness properties for Laplace type transform spectral multipliers associated with the Schrödinger operator $\mathcal{L}=-Δ+V$. We obtain for this type of multipliers pointwise representation as principal value integral operators. We also characterize the UMD Banach spaces in terms of the $L^p$-boundedness of the imaginary powers $\mathcal{L}^{iγ}$, $γ\in \mathbb{R}$, of $\mathcal{L}$.
Following a symmetrization procedure proposed recently by Nowak and Stempak, we consider the setting of symmetrized Jacobi expansions. In this framework we investigate mapping properties of several fundamental harmonic analysis operators, including Riesz transforms, Poisson semigroup maximal operator, Littlewood-Paley-Stein square functions and multipliers of Laplace and Laplace-Stieltjes transform type. Our paper delivers also some new results in the original setting of classical Jacobi expansions.
We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices $α$ in this context. This generalizes a simpler method existing in the literature, but being valid for a restricted range of $α$. As an application, we prove that several fundamental operators in harmonic analysis of the Laguerre expansions, including maximal operators related to the heat and Poisson semigroups, Riesz transforms, Littlewood-Paley-Stein type square functions and multipliers of Laplace and Laplace-Stieltjes tra…
We study several fundamental harmonic analysis operators in the multi-dimensional context of the Dunkl harmonic oscillator and the underlying group of reflections isomorphic to $\mathbb{Z}_2^d$. Noteworthy, we admit negative values of the multiplicity functions. Our investigations include maximal operators, $g$-functions, Lusin area integrals, Riesz transforms and multipliers of Laplace and Laplace-Stieltjes type. By means of the general Calderón-Zygmund theory we prove that these operators are bounded on weighted $L^p$ spaces, $1 < p < \infty$, and from weighted $L^1$ to weighted weak $L^1$. …
In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),γ_α)$ to $L^1((0,\infty),γ_α)$ and also from $L^\infty((0,\infty),γ_α)$ to $\BMO((0,\infty),γ_α)$, with respect to the probability measure $dγ_α(x)=\frac{2}{Γ(α+1)} x^{2α+1} e^{-x^2} dx$ on $(0,\infty)$ when ${α>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.
The localized Fourier-Laplace transform of the Gauß-Manin system of $f\colon \mathbb{G}_m \to \mathbb{A}^1,\ x \mapsto x + x^{-3}$ is a $\mathcal{D}_{\mathbb{G}_m}$-module, having a regular singularity at $0$ and an irregular one at $\infty$. By mirror symmetry, it is closely related to the quantum connection of the weighted projective line $\mathbb{P}(1,3)$. Following results of A. D'Agnolo, M. Hien, G. Morando and C. Sabbah from 2017, we compute its Stokes multipliers at $\infty$ by purely topological methods. We compare it to the Gram matrix of the Euler-Poincaré pairing on $D^b(\text{Coh}(…
In the paper we study applications of integral transforms composition method (ITCM) for obtaining transmutations via integral transforms. It is possible to derive wide range of transmutation operators by this method. Classical integral transforms are involved in the integral transforms composition method (ITCM) as basic blocks, among them are Fourier, sine and cosine-Fourier, Hankel, Mellin, Laplace and some generalized transforms. The ITCM and transmutations obtaining by it are applied to deriving connection formulas for solutions of singular differential equations and more simple non-singula…
