# On fractional differentiation and integration on spaces of homogeneous type.

A. Eduardo Gatto; Carlos Segovia; Stephen Vági

Revista Matemática Iberoamericana (1996)

- Volume: 12, Issue: 1, page 111-145
- ISSN: 0213-2230

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topGatto, A. Eduardo, Segovia, Carlos, and Vági, Stephen. "On fractional differentiation and integration on spaces of homogeneous type.." Revista Matemática Iberoamericana 12.1 (1996): 111-145. <http://eudml.org/doc/39512>.

@article{Gatto1996,

abstract = {In this paper we define derivatives of fractional order on spaces of homogeneous type by generalizing a classical formula for the fractional powers of the Laplacean [S1], [S2], [SZ] and introducing suitable quasidistances related to an approximation of the identity. We define integration of fractional order as in [GV] but using quasidistances related to the approximation of the identity mentioned before.We show that these operators act on Lipschitz spaces as in the classical cases. We prove that the composition Tα of a fractional integral Iα and a fractional derivative Dα of the same order and its transpose (a fractional derivative composed with a fractional integral of the same order) are Calderón-Zygmund operators. We also prove that for small order α, Tα is an invertible operator in L2. In order to prove that Tα is invertible we obtain Nahmod type representations for Iα and Dα and then we follow the method of her thesis [N1], [N2].},

author = {Gatto, A. Eduardo, Segovia, Carlos, Vági, Stephen},

journal = {Revista Matemática Iberoamericana},

keywords = {Operadores diferenciales; Operadores integrales; Espacio de Lipschitz; Espacio homogéneo; Kernel; derivatives of fractional order; spaces of homogeneous type; quasidistances; integration of fractional orders; operators; Lipschitz spaces; Calderón-Zygmund operators},

language = {eng},

number = {1},

pages = {111-145},

title = {On fractional differentiation and integration on spaces of homogeneous type.},

url = {http://eudml.org/doc/39512},

volume = {12},

year = {1996},

}

TY - JOUR

AU - Gatto, A. Eduardo

AU - Segovia, Carlos

AU - Vági, Stephen

TI - On fractional differentiation and integration on spaces of homogeneous type.

JO - Revista Matemática Iberoamericana

PY - 1996

VL - 12

IS - 1

SP - 111

EP - 145

AB - In this paper we define derivatives of fractional order on spaces of homogeneous type by generalizing a classical formula for the fractional powers of the Laplacean [S1], [S2], [SZ] and introducing suitable quasidistances related to an approximation of the identity. We define integration of fractional order as in [GV] but using quasidistances related to the approximation of the identity mentioned before.We show that these operators act on Lipschitz spaces as in the classical cases. We prove that the composition Tα of a fractional integral Iα and a fractional derivative Dα of the same order and its transpose (a fractional derivative composed with a fractional integral of the same order) are Calderón-Zygmund operators. We also prove that for small order α, Tα is an invertible operator in L2. In order to prove that Tα is invertible we obtain Nahmod type representations for Iα and Dα and then we follow the method of her thesis [N1], [N2].

LA - eng

KW - Operadores diferenciales; Operadores integrales; Espacio de Lipschitz; Espacio homogéneo; Kernel; derivatives of fractional order; spaces of homogeneous type; quasidistances; integration of fractional orders; operators; Lipschitz spaces; Calderón-Zygmund operators

UR - http://eudml.org/doc/39512

ER -

## Citations in EuDML Documents

top- A. Gatto, Stephen Vági, On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type
- Silvia I. Hartzstein, Beatriz E. Viviani, Integral and derivative operators of functional order on generalized Besov and Triebel-Lizorkin spaces in the setting of spaces of homogeneous type
- Silvia I. Hartzstein, Beatriz E. Viviani, On the composition of the integral and derivative operators of functional order
- Toni Heikkinen, Juha Lehrbäck, Juho Nuutinen, Heli Tuominen, Fractional Maximal Functions in Metric Measure Spaces