Maximum principles, extension problem and inversion for nonlocal one-sided equations

Abstract

Abstract. We study one-sided nonlocal equations of the formon the real line. Notice that to compute this nonlocal operator of order 0 < α < 1 at a point x 0 we need to know the values of u(x) to the right of x 0 , that is, for x ≥ x 0 . We show that the operator above corresponds to a fractional power of a one-sided first order derivative. Maximum principles and a characterization with an extension problem in the spirit of Caffarelli-Silvestre and StingaTorrea are proved. It is also shown that these fractional equations can be solved in the general setting of weighted one-sided spaces. In this regard we present suitable inversion results. Along the way we are able to unify and clarify several notions of fractional derivatives found in the literature.

Regularity Theory and Extension Problem for Fractional Nonlocal Parabolic Equations and the Master Equation

“…However we stress that, for the fractional heat operator, our new extension problem is parabolic. The extension problem for the Marchaud fractional derivative has been originally obtained in [5], where the extension PDE is also of parabolic type. For the definition of e −τ H u(t, x) see Section 2.…”

“…that is, u is the Weyl fractional integral of f , see [5,29]. Observe that (∂ t ) −s is the inverse of the Marchaud fractional derivative appearing in Corollary 1.4.…”

“…Observe that (∂ t ) −s is the inverse of the Marchaud fractional derivative appearing in Corollary 1.4. In fact, a Fractional Fundamental Theorem of Calculus on general one-sided weighted Lebesgue spaces was recently proved in [5].…”

“…It is worth noticing that the appearance of the Marchaud fractional derivative from the past is related to the fact that the heat equation is not time reversible, see also the detailed analysis in [5].…”

Boundedness of Differential Transforms for One-Sided Fractional Poisson-Type Operator Sequence

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Let P α τ f be given byThis is a fractional Poisson-type operator on the line, which can be found in [3]. It is known that the Poisson-type operator appeared when solving the extension problem, see [5,12,13].

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