Fractional Hardy–Sobolev type inequalities for half spaces and John domains

Bartłomiej Dyda; Juha Lehrbäck; Antti V. Vähäkangas

doi:10.1090/proc/14051

Fractional Hardy–Sobolev type inequalities for half spaces and John domains

Proceedings of the American Mathematical Society ◽

10.1090/proc/14051 ◽

2018 ◽

Vol 146 (8) ◽

pp. 3393-3402 ◽

Cited By ~ 4

Author(s):

Bartłomiej Dyda ◽

Juha Lehrbäck ◽

Antti V. Vähäkangas

Keyword(s):

Sobolev Type ◽

John Domains

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B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces

Open Mathematics ◽

10.1515/math-2020-0033 ◽

2020 ◽

Vol 18 (1) ◽

pp. 715-730

Author(s):

Javanshir J. Hasanov ◽

Rabil Ayazoglu ◽

Simten Bayrakci

Keyword(s):

Differential Operator ◽

Singular Integral ◽

Riesz Potential ◽

Type Theorem ◽

Integral Operators ◽

Morrey Spaces ◽

Singular Integral Operators ◽

Sobolev Type ◽

Riesz Potentials ◽

Bessel Differential Operator

Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .

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Results on approximate controllability of Sobolev-type fractional neutral differential inclusions of Clarke subdifferential type

Chaos Solitons & Fractals ◽

10.1016/j.chaos.2021.111264 ◽

2021 ◽

Vol 151 ◽

pp. 111264

Author(s):

K. Kavitha ◽

V. Vijayakumar ◽

Anurag Shukla ◽

Kottakkaran Sooppy Nisar ◽

R. Udhayakumar

Keyword(s):

Differential Inclusions ◽

Approximate Controllability ◽

Clarke Subdifferential ◽

Sobolev Type

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Regularization of backward time-fractional parabolic equations by Sobolev-type equations

Journal of Inverse and Ill-Posed Problems ◽

10.1515/jiip-2020-0062 ◽

2020 ◽

Vol 28 (5) ◽

pp. 659-676

Author(s):

Dinh Nho Hào ◽

Nguyen Van Duc ◽

Nguyen Van Thang ◽

Nguyen Trung Thành

Keyword(s):

Error Estimates ◽

Parabolic Equations ◽

Regularization Parameter ◽

A Priori ◽

Sobolev Type ◽

A Posteriori ◽

Parameter Choice ◽

Numerical Tests ◽

Ill Posed ◽

Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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