By Alexandre Almeida, Luís Castro, Frank-Olme Speck
This quantity is devoted to Professor Stefan Samko at the party of his 70th birthday. The contributions exhibit the variety of his medical pursuits in harmonic research and operator thought. specific attention is paid to fractional integrals and derivatives, singular, hypersingular and power operators in variable exponent areas, pseudodifferential operators in a variety of glossy functionality and distribution areas, besides as related purposes, to say yet a number of. such a lot contributions have been to start with offered in meetings at Lisbon and Aveiro, Portugal, in June‒July 2011.
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Extra resources for Advances in harmonic analysis and operator theory : the Stefan Samko anniversary volume
These studies were continued in , 2011, where there was obtained the boundedness of singular integral operators in the variable exponent Lebesgue spaces ????????(⋅) (Γ, ????) on a class of composed Carleson curves Γ with the weights ???? having a ﬁnite set of oscillating singularities. The proof was based on the boundedness of Mellin PDO in the spaces ????????(⋅) (ℝ+ , ????????), ???????? = ???????? ???? . This allowed in  to obtain a criterion of local invertibility of singular integral operators with piecewise slowly oscillating coeﬃcients acting on the space ????????(⋅) (Γ, ????) via the corresponding criteria of local invertibility at the point 0 of Mellin PDO and local invertibility of SIO on ℝ.
Russian) Dokl. Akad. Nauk SSSR 221:6 (1975), 1260–1263. Transl. in Soviet Math. Dokl. 16:2 (1975), 510–514. K. G. Samko, Singular convolution operators with a discontinuous symbol. (Russian) Sibirsk. Mat. Zh 16:1 (1975), 44–61. Transl. in Siberian Math. J. 16:1 (1975), 35-48. 38 V. K. G. Samko, Singular integral operators with Carleman shift in the case of piece-wise continuous coeﬃcients, I. (Russian) Izv. Vysch. Uchebn. , Matematika. 1975:2 (1975), 43–54. K. G. Samko, Singular integral operators with Carleman shift in the case of piece-wise continuous coeﬃcients, II.
18) in the sequel) adjusted for the ﬁnite interval (????, ????) with the weight (???? − ????)????(????) (???? − ????)???? (????) where the inﬂuence of the both end points on the conditions on the exponents had to be separated in terms of the decay conditions at the end-points. Stefan G. Samko – Mathematician, Teacher and Man A similar equation ∫ ????(????, ????) ????(????)???????? = ???? (????) , ????−???? ℝ???? ∣???? − ????∣ ???? ∈ ℝ???? , 25 0 < ???? < 1, of the 1st kind over the whole space ℝ???? was later investigated in , 2010, and , 2011, where it was shown that the method of reducing such equations with ???????? -solutions to equations of the second kind with an operator compact in ???????? (ℝ???? ) remains valid for the variable exponent Lebesgue spaces ????????(⋅) (ℝ???? ) under the usual local and decay log-conditions on ????(????).