Spectral Multipliers for Multidimensional Bessel Operators

Castro, Alejandro J.; Curbelo, Jezabel

doi:10.1007/s00041-010-9162-1

Cited by 23 publications

(42 citation statements)

References 21 publications

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“…This type of Hankel multipliers were studied in [3] and [9]. Proceeding as in [3,Theorem 1.2], for every f ∈ C ∞ c (0, ∞), we have that Here α denotes a bounded function on (0, ∞) and I ν is the modified Bessel function of the first kind and order ν. By [9, Theorem 1.2], ∆ iω λ f can be extended to L p (0, ∞) as a bounded operator from L p (0, ∞) into itself.…”

Section: 3mentioning

confidence: 99%

Square functions and spectral multipliers for Bessel operators in UMD spaces

Castro¹,

Rodríguez-Mesa²

2016

Banach J. Math. Anal.

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Abstract. In this paper we consider square functions (also called Littlewood-Paley g-functions) associated to Hankel convolutions acting on functions in the Bochner Lebesgue spacewhere B is a UMD Banach space. As special cases we study square functions defined by fractional derivatives of the Poisson semigroup for the Bessel operatorWe characterize the UMD property for a Banach space B by using L p ((0, ∞), B)-boundedness properties of g-functions defined by Bessel-Poisson semigroups. As a by product we prove that the fact that the imaginary power ∆ iω λ , ω ∈ R \ {0}, of the Bessel operator ∆ λ is bounded in L p ((0, ∞), B), 1 < p < ∞, characterizes the UMD property for the Banach space B. As applications of our results for square functions we establish the boundedness in L p ((0, ∞), B) of spectral multipliers m(∆ λ ) of Bessel operators defined by functions m which are holomorphic in sectors Σ ϑ .

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Section: 3mentioning

confidence: 99%

Square functions and spectral multipliers for Bessel operators in UMD spaces

Castro¹,

Rodríguez-Mesa²

2016

Banach J. Math. Anal.

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“…This, as well as many earlier results (see the references in [9]), was done in dimension one. Recently, harmonic analysis in the context of Bessel operators was developed in higher dimensions, see Betancor, Castro and Curbelo [6,7], Betancor, Castro and Nowak [8], and Castro and Szarek [13]. More recently, in a similar spirit Castro and Szarek [14] investigated fundamental harmonic analysis operators in a wider Hankel-Dunkl setting.…”

Section: Introductionmentioning

confidence: 99%

Potential operators associated with Hankel and Hankel-Dunkl transforms

Nowak

Stempak

2017

JAMA

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We study Riesz and Bessel potentials in the settings of Hankel transform, modified Hankel transform and Hankel-Dunkl transform. We prove sharp or qualitatively sharp pointwise estimates of the corresponding potential kernels. Then we characterize those 1 ≤ p, q ≤ ∞, for which the potential operators satisfy L p − L q estimates. In case of the Riesz potentials, we also characterize those 1 ≤ p, q ≤ ∞, for which two-weight L p − L q estimates, with power weights involved, hold. As a special case of our results, we obtain a full characterization of two power-weight L p − L q bounds for the classical Riesz potentials in the radial case. This complements an old result of Rubin and its recent reinvestigations by De Nápoli, Drelichman and Durán, and Duoandikoetxea. A. NOWAK AND K. STEMPAKMoreover, in case of the Riesz potentials, we also determine those 1 ≤ p, q ≤ ∞, for which two powerweight L p −L q bounds hold, see Theorems 2.5, 2.10 and 2.15. All these results are Hankel or Hankel-Dunkl counterparts of a series of recent sharp results concerning potential operators in several classic settings related to discrete orthogonal expansions: Hermite function expansions [28], Jacobi and Fourier-Bessel expansions [26], and Laguerre function and Dunkl-Laguerre expansions [29]. The frameworks studied in this paper correspond to continuous orthogonal expansions, and the general approach elaborated in the above mentioned articles applies here as well. However, the present L p − L q results have somewhat different flavor.An interesting by-product of our results is an alternative proof of a radial analogue of the celebrated two power-weighted L p − L q estimates for the classical Riesz potentials due to Stein and Weiss [34]. Such an analogue was obtained by Rubin [32] in the eighties of the last century. Rubin's result, being apparently overlooked, was recently reinvestigated and refined by De Nápoli, Drelichman and Durán [15], and Duoandikoetxea [17]. In fact, Corollary 2.6 below slightly extends the above mentioned results, see the related comments following the statement.Crucial aspects of our results are their sharpness and completeness. The latter means, in particular, that in each of the contexts we treat the full admissible range of the associated parameter of type, which in the Hankel-Dunkl setting manifests in including an 'exotic' case of negative multiplicity functions. Some parts of our results were obtained earlier, by various authors, which is always commented in the relevant places according to our best knowledge. In this connection, we mention again the article of Muckenhoupt and Stein [25], and the works of Gadjiev and Aliev [18] where Riesz and Bessel potentials in the context of the modified Hankel transform were investigated, Thangavelu and Xu [36] where Riesz and Bessel potentials for the Dunkl transform were introduced and studied, Hassani, Mustapha and Sifi [21] where for the Riesz potentials the subject was continued, Betancor, Martínez and Rodríguez-Mesa [10] where Riesz potentials for the Hankel transfo...

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“…If, for every t > 0, W µ t is given as in (1.4), {W µ t } t>0 also defines a semigroup of operators on L p ((0, ∞)), for each 1 < p < ∞ when µ > −1/2 and for each 1 < p < ∞ such that −µ − 1/2 < 1 p < µ + 3/2, when −1 < µ ≤ −1/2. Harmonic analysis associated with Bessel operator (Riesz transforms, maximal operators, Littlewood-Paley functions, fractional Bessel operators, Hardy spaces,..) has been developed in the last years ( [4], [5], [6], [7], [8], [9], [19] and [20]) although the first results about this topic had been obtain by Muckenhoupt and Stein ([37]) in the sixties of the last century as the paper about parabolic singular integrals mentioned at the beginning.…”

Section: Introductionmentioning

confidence: 99%

Parabolic Equations Involving Bessel Operators and Singular Integrals

León-Contreras

2018

Integr. Equ. Oper. Theory

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In this paper we consider the evolution equation ∂tu = ∆µu + f and the corresponding Cauchy problem, where ∆µ represents the Bessel operator ∂ 2x + ( 1 4 − µ 2 )x −2 , for every µ > −1. We establish weighted and mixed weighted Sobolev type inequalities for solutions of Bessel parabolic equations. We use singular integrals techniques in a parabolic setting. Recently, Jones' results have been extended to weighted and mixed weighted L p spaces by Ping, Stinga and Torrea ([42, Theorem 2.4]). In [42], they also considered the equation (1.1) on the whole space R n+1 . In [42, Theorem 2.3], it was proved that if f is a C 2 -function defined on R n+1 with compact support, when n ≥ 2, the function u given byis a solution of (1.1) on R n+1 , where ∂ 2 xi,xj u(t, x) = lim ǫ→0 + Ωǫ

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Spectral Multipliers for Multidimensional Bessel Operators

Cited by 23 publications

References 21 publications

Square functions and spectral multipliers for Bessel operators in UMD spaces

Square functions and spectral multipliers for Bessel operators in UMD spaces

Potential operators associated with Hankel and Hankel-Dunkl transforms

Parabolic Equations Involving Bessel Operators and Singular Integrals

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