Fractional integrals and hypersingular integrals in variable order Holder spaces on homogeneous spaces

• Autor(a) principal: Samko, Natasha
• Data de Publicação: 2010
• Outros Autores: Samko, Stefan, Vakulov, Boris
• Tipo de documento: Artigo
• Idioma: eng
• Título da fonte: Repositório Científico de Acesso Aberto de Portugal (Repositórios Cientìficos)
• Texto Completo: http://hdl.handle.net/10400.1/11921
• topic: Non doubling measures; Spherical potentials; Complex order; Operators

We consider non-standard Holder spaces H(lambda(.))(X) of functions f on a metric measure space (X, d, mu), whose Holder exponent lambda(x) is variable, depending on x is an element of X. We establish theorems on mapping properties of potential operators of variable order alpha(x), from such a variable exponent Holder space with the exponent lambda(x) to another one with a "better" exponent lambda(x) + alpha(x), and similar mapping properties of hypersingular integrals of variable order alpha(x) from such a space into the space with the "worse" exponent lambda(x) - alpha(x) in the case alpha(x) < lambda(x). These theorems are derived from the Zygmund type estimates of the local continuity modulus of potential and hypersingular operators via such modulus of their densities. These estimates allow us to treat not only the case of the spaces H(lambda(.))(X), but also the generalized Holder spaces H(w(.,.))(X) of functions whose continuity modulus is dominated by a given function w(x, h), x is an element of X, h > 0. We admit variable complex valued orders alpha(x), where R alpha(x) may vanish at a set of measure zero. To cover this case, we consider the action of potential operators to weighted generalized Holder spaces with the weight alpha(x).