One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces

Stinga, P. R.; Vaughan, Mary

doi:10.1016/j.na.2019.04.004

Cited by 8 publications

(6 citation statements)

References 17 publications

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“…We obtain a variety of PDE and analysis results, including maximum principles, extension problems, characterization of associated Sobolev spaces by limits of fractional power operators in the spirit of Bourgain-Brezis-Mironescu [8], and the fundamental theorem of calculus for our fractional derivatives and integrals. Our results generalize and complement the recent exhaustive analysis of fractional powers of the first derivative on the line that has been developed in [4,22]. In Section 3, we show that our general first order operators contain as particular cases operators that have appeared in relation with Ornstein-Uhlenbeck, Hermite, Laguerre and Jacobi expansions.…”

Section: Introductionsupporting

confidence: 80%

“…, by a standard approximation argument (see [22]), it follows that E −1 (D left ) α (E −1 u) coincides with the pointwise formula in the statement, for all x ∈ I. Moreover,…”

Section: 3mentioning

confidence: 67%

“…It has been observed in [4] that (D left ) α and (D right ) α do not preserve the Schwartz class S(R). Several attempts have been made to find an appropriate space of distributions to define these operators, see, for instance, [4,22]. We shall present a quick overview of the strategy in [22] for the left-sided fractional derivative.…”

Section: 3mentioning

confidence: 99%

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Fractional powers of first order differential operators and new families of polynomials associated to inverse measures

Mazzitelli¹,

Stinga²,

Torrea³

2022

Preprint

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First, we establish the theory of fractional powers of first order differential operators with zero order terms, obtaining PDE properties and analyzing the corresponding fractional Sobolev spaces. In particular, our study shows that Lebesgue and Sobolev spaces with inverse measures (like the inverse Gaussian measure) play a fundamental role in the theory of fractional powers of the first order operators. Second, and motivated in part by such a theory, we lay out the foundations for the development of the harmonic analysis for inverse measures. We discover new families of polynomials related to the inverse Gaussian, Laguerre, and Jacobi measures, and characterize them using generating and Rodrigues formulas, and three-term recurrence relations. Moreover, we prove boundedness of several fundamental singular integral operators in these inverse measure settings.

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Section: Introductionsupporting

confidence: 80%

“…, by a standard approximation argument (see [22]), it follows that E −1 (D left ) α (E −1 u) coincides with the pointwise formula in the statement, for all x ∈ I. Moreover,…”

Section: 3mentioning

confidence: 67%

See 1 more Smart Citation

Fractional powers of first order differential operators and new families of polynomials associated to inverse measures

Mazzitelli¹,

Stinga²,

Torrea³

2022

Preprint

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“…The distributional space must reflect the one-sided nature of fractional derivatives. It was shown in [SV20] that the appropriate test functions must be supported on intervals of the form (−∞, 𝐴], so as to look at (𝐷 left ) 𝛼 𝑢 from the left. Since (𝐷 right ) 𝛼 𝜑 will also have support in (−∞, 𝐴], in the action 𝑢((𝐷 right ) 𝛼 𝜑) the only values of 𝑢 involved are those to the left.…”

Section: Theory Of Left-sided Fractional Derivativesmentioning

confidence: 99%

Fractional Derivatives: Fourier, Elephants, Memory Effects, Viscoelastic Materials, and Anomalous Diffusions

Stinga¹

2023

Notices Amer. Math. Soc.

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“…We recall that the well-known classical fractional derivative concepts include Riemann-Liouville, Caputo, Fourier, and Grünwald-Letnikov fractional order derivatives (cf. [15,16,6]). The second main issue, which is also a technical obstruction, is the compatibility between these classical fractional derivatives D α v and the (energy) space V α in (1.3).…”

Section: Introductionmentioning

confidence: 99%

On a New Class of Fractional Calculus of Variations and Related Fractional Differential Equations

Feng¹,

Sutton²

2021

Preprint

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This paper is concerned with analyzing a class of fractional calculus of variations problems and their associated Euler-Lagrange (fractional differential) equations. Unlike the existing fractional calculus of variations which is based on the classical notion of fractional derivatives, the fractional calculus of variations considered in this paper is based on a newly developed notion of weak fractional derivatives and their associated fractional order Sobolev spaces. Since fractional derivatives are direction-dependent, using one-sided fractional derivatives and their combinations leads to new types of calculus of variations and fractional differential equations as well as nonstandard Neumann boundary operators. The primary objective of this paper is to establish the well-posedness and regularities for a class of fractional calculus of variations problems and their Euler-Lagrange (fractional differential) equations. This is achieved first for one-sided Dirichlet energy functionals which lead to one-sided fractional Laplace equations, then for more general energy functionals which give rise to more general fractional differential equations.

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One-sided fractional derivatives, fractional Laplacians, and weighted Sobolev spaces

Cited by 8 publications

References 17 publications

Fractional powers of first order differential operators and new families of polynomials associated to inverse measures

Fractional powers of first order differential operators and new families of polynomials associated to inverse measures

Fractional Derivatives: Fourier, Elephants, Memory Effects, Viscoelastic Materials, and Anomalous Diffusions

On a New Class of Fractional Calculus of Variations and Related Fractional Differential Equations

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