Laplace-Bessel Differential Operator Research Articles

$B-$ Riezs potential in $B-$ local Morrey-Lorentz spaces

In this paper, the Riesz potential (B−Riesz potential) which are generated by the Laplace-Bessel differential operator will be studied. We obtain the necessary and sufficient conditions for the boundedness of the B−Riesz potential $I_{\gamma }^{\alpha }$ in the B-local Morrey-Lorentz spaces $M_{p,q,\lambda,\gamma }^{loc}(\mathbb{R}_{k,+}^{n})$ with the use of the rearrangement inequalities and boundedness of the Hardy operators $H_{\upsilon }^{\beta }$ and $\mathcal{H}_{\upsilon}^{\beta }$ with power weights.

MAXIMAL OPERATOR IN WEIGHTED MORREY SPACES, ASSOCIATED WITH THE LAPLACE-BESSEL DIFFERENTIAL OPERATOR

We consider the generalized shift operator, associated with the Laplace-Bessel differential operator. The maximal operator Mγ(B-maximal operator), associated with the generalized shift operator is investigated. At first, we prove that the boundedness of maximal operator weighted Morrey spaces Lp,λ,ω1,γ(Rn k,+) to the spaces Lp,λ,ω2,γ(Rnk,+), where 1 < p < ∞,(ω1, ω2) ∈ Fp,γ(Rnk,+), associated with the Laplace-Bessel differential operator

Weak-Type Estimate for the Littlewood-Paley Operators Associated with Laplace-Bessel Differential Operator

Weak-Type Estimate for the Littlewood-Paley Operators Associated with Laplace-Bessel Differential Operator

A characterization for $B$-singular integral operator and its commutators on generalized weighted $B$-Morrey spaces

We study the maximal operator $M_{\gamma}$ and the singular integral operator $A_{\gamma}$, associated with the generalized shift operator. The generalized shift operators are associated with the Laplace-Bessel differential operator. Our analysis is based on two weighted inequalities for the maximal operator, singular integral operators, and their commutators, related to the Laplace-Bessel differential operator in generalized weighted $B$-Morrey spaces.

Some results concerning Riesz–Bessel transforms of high order

In the present study, we introduce the sharp function related to the Laplace–Bessel differential operator and investigate its properties on the variable Lebesgue spaces. Moreover, we obtain that Riesz–Bessel transforms of high order are bounded on variable Lebesgue spaces .

On Flett potentials associated with the Laplace–Bessel differential operator

On Flett potentials associated with the Laplace–Bessel differential operator

Two-weighted inequalities of maximal commutators in the modified Morrey spaces, associated with the Laplace-Bessel differential operator

We consider the generalized shift operator, associated with the LaplaceBessel differential operator ∆B =∑ni=1∂2∂x2i+∑ki=1γixi∂∂xi. The maximal commutators Mb,γ, associated with the generalized shift operator are investigated. At first, we prove that the maximal commutators is bounded from the modified Morrey space Lep,λ,φ1,γ(Rnk,+) to Lep,λ,φ1,γ(R n k,+) for all 1 < p < ∞, b ∈ BMOγ(Rnk,+), (φ1, φ2) ∈ Aep,γ(Rnk,+).

Equivalence of K-functional and modulus of smoothness constructed by Fourier-Bessel transform in the space L2,γ(ℝn+)

Using a generalized shift operator, we define generalized modulus of smothness in the space L2,γ(ℝn+). Based on the Laplace-Bessel differential operator we define Sobolev-type space and K-functionals. In this paper paper we prove the equivalence theorem for a K-functional and a modulus of smoothness for the Fourier-Bessel transformation on ℝn+.

Strong- and weak-type estimate for Littlewood–Paley operators associated with Laplace–Bessel differential operator

Strong- and weak-type estimate for Littlewood–Paley operators associated with Laplace–Bessel differential operator

On the BMO spaces associated with the Laplace-Bessel differential operator

In this paper, the characteristic properties of the space of functions of bounded mean oscillation called the $B$-$BMO$ associated with the Laplace-Bessel differential operator are obtained. The John-Nirenberg type inequality on the $B$-$BMO$ space and a relation between the $B$-Poisson integral and the $B$-$BMO$ functions are proved.

A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

Abstract In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

A generalization of parabolic potentials associated to Laplace–Bessel differential operator and its behavior in the weighted Lebesque spaces

A generalization of parabolic potentials associated to Laplace–Bessel differential operator and its behavior in the weighted Lebesque spaces

On the boundedness of B-maximal commutators, commutators of B-Riesz potentials and B-singular integral operators in modified B-Morrey spaces

In this paper we consider the generalized shift operator associated to the Laplace–Bessel differential operator ∆B and investigate B-maximal commutators, commutators of B-Riesz potentials and commutators of B-singular integral operators associated to the generalized shift operator. The boundedness of the B-maximal commutator Mb,γ and the commutator [b, Aγ] of the B-singular integral operator on the modified B-Morrey spaces Lep,λ,γ(R n k,+) for all 1 < p < ∞ when b ∈ BMOγ(R n k,+) are proved. In addition, we obtain that the commutator [b, Iα,γ] of the B-Riesz potential Iα,γ is bounded from the modified B-Morrey space Lep,λ,γ(R n k,+) to Leq,λ,γ(R n k,+), 1 < p < n+|γ|−λ n+|γ| ≤ 1 p − 1 q ≤ n+|γ|−λ and from the space Le1,λ,γ(R n k,+) to WLeq,λ,γ(R n k,+), n+|γ| ≤ 1 − 1 q ≤ n+|γ|−λ

Bn -maximal operator and Bn -singular integral operators on variable exponent Lebesgue spaces

Abstract In this paper, we prove the boundedness of the Bn maximal operator and Bn singular integral operators associated with the Laplace-Bessel differential operator Δ Bn on variable exponent Lebesgue spaces.

Fourier-Bessel Transforms of Dini-Lipschitz Functions on Lebesgue Spaces $L_{p,\gamma}(\mathbb{R}^{n}_{+})$

In this paper, we prove a generalization of Titchmarsh's theorem for the Laplace-Bessel differential operator in the space $L_{p,\gamma}(\mathbb{R}^{n}_{+})$ for functions satisfying the $(\psi,p)$-Laplace-Bessel Lipschitz condition for $1 0 $.

Commutators of the B-maximal operator and B-maximal commutators

In this paper we consider the commutator of the B-maximal operator and the B-maximal commutator associated with the Laplace-Bessel differential operator. The boundedness of the commutator of the B-maximal operator with BMO symbols on weighted Lebesque space and weak-type inequality for the commutator of the B-maximal operator are proved.

Some generalizations of Bessel and Flett potentials associated to the Laplace–Bessel differential operator

ABSTRACTWe introduce the notion of bi-parametric potential-type operators, which generalize the Bessel and Flett potentials associated to singular Laplace–Bessel differential operator. Explicit inversion formulas for them are obtained by using a special wavelet-like transform, generated by the so-called ‘modified beta-semigroup’.

Two weighted inequalities for B-fractional integrals

In this paper we prove a two weighted inequality for Riesz potentials $I_{\alpha,\gamma} f$ (B-fractional integrals) associated with the Laplace-Bessel differential operator $\Delta_{B}=\sum_{i=1}^{n} \frac{\partial^{2}}{\partial x_{i}^{2}} + \sum_{j=1}^{k} \frac{\gamma _{j}}{x_{j}}\frac{\partial}{\partial x_{j}}$ . This result is an analog of Heinig’s result (Indiana Univ. Math. J. 33(4):573-582, 1984) for the B-fractional integral. Further, the Stein-Weiss inequality for B-fractional integrals is proved as an application of this result.

Square-like functions generated by the Laplace-Bessel differential operator

Abstract We introduce a wavelet-type transform associated with the Laplace-Bessel differential operator Δ B = ∑ k = 1 n ∂ 2 ∂ x k + 2 ν k ∂ x k ∂ ∂ x k and the relevant square-like functions. An analogue of the Calderón reproducing formula and the L 2 , ν boundedness of the square-like functions are obtained. MSC:47G10, 42C40, 44A35.

The boundedness of the generalized anisotropic potentials with rough kernels in the Lorentz spaces

In this paper, we study the generalized anisotropic potential integral K α, γ⊗ f and anisotropic fractional integral I Ω,α, γ f with rough kernels, associated with the Laplace–Bessel differential operator Δ B . We prove that the operator f→K α, γ⊗ f is bounded from the Lorentz spaces to for 1≤p<q≤∞, 1≤r≤s≤∞. As a result of this, we get the necessary and sufficient conditions for the boundedness of I Ω,α, γ from the Lorentz spaces to , 1<p<q<∞, 1≤r≤s≤∞ and from to , 1<q<∞, 1≤r≤∞. Furthermore, for the limiting case p=Q/α, we give an analogue of Adams’ theorem on the exponential integrability of I Ω,α, γ in .