On the harmonic extension approach to fractional powers in Banach spaces

@article{Meichsner2019OnTH,
  title={On the harmonic extension approach to fractional powers in Banach spaces},
  author={Jan Meichsner and Christian Seifert},
  journal={Fractional Calculus and Applied Analysis},
  year={2019},
  volume={23},
  pages={1054 - 1089}
}
Abstract We show that fractional powers of general sectorial operators on Banach spaces can be obtained by the harmonic extension approach. Moreover, for the corresponding second order ordinary differential equation with incomplete data describing the harmonic extension we prove existence and uniqueness of a bounded solution (i.e., of the harmonic extension). 
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8 Citations
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References

SHOWING 1-10 OF 29 REFERENCES

Fractional powers of non-negative operators in Banach spaces via the Dirichlet-to-Neumann operator

We consider fractional powers of non-densely defined non-negative operators in Banach spaces defined by means of the Balakrishnan operator. Under mild assumptions on the operator we show that the…

Extension problem and fractional operators: semigroups and wave equations

We extend results of Caffarelli–Silvestre and Stinga–Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of…

An Extension Problem Related to the Fractional Laplacian

The operator square root of the Laplacian (− ▵)1/2 can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the…

Analytic Semigroups and Optimal Regularity in Parabolic Problems

Introduction.- 0 Preliminary material: spaces of continuous and Holder continuous functions.- 1 Interpolation theory.- Analytic semigroups and intermediate spaces.- 3 Generation of analytic…

Extension Problem and Harnack's Inequality for Some Fractional Operators

The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional…

Fractional powers of operators, III Negative powers

This is a continuation of the author's work on fractional powers of operators A in a Banach space X whose resolvent (2+A)-1 exists for 2 > 0 and satisfies II A(2+A)-1 II <_ M < oo, 0 < 2 < oo. This…

Functional calculi of second-order elliptic partial differential operators with bounded measurable coefficients

Consider a second-order elliptic partial differential operatorL in divergence form with real, symmetric, bounded measurable coefficients, under Dirichlet or Neumann conditions on the boundary of a…

Fractional powers of sectorial operators via the Dirichlet-to-Neumann operator

ABSTRACT In the very influential paper [4] Caffarelli and Silvestre studied regularity of (−Δ)s, 0<s<1, by identifying fractional powers with a certain Dirichlet-to-Neumann operator. Stinga and…

Fractional powers of closed operators and the semigroups generated by them.

Fractional powers of closed linear operators were first constructed by Bochner [2] and subsequently Feller [3], for the Laplacian operator. These constructions depend in an essential way on the fact…

Note on Fractional Powers of Linear Operators

In the preceding paper by K. Yosida, it is shown that the fractional power A, 0<a<l, of a linear operator A in a Banach space X can be constructed whenever--A is the infinitesimal generator of a…