Boundedness of operators arising from Schwarz BVP in modified local Morrey-type spaces

References

• Schwarz HA. Zur Integration der partiellen Differentialgleichung. J Reine Angew Math. 1872;74:218–253. German.
   Google Scholar
• Begehr H, Schmersau D. The Schwarz problem for polyanalytic functions. Z Anal Anwendungen. 2005;24(2):341–351.
   Web of Science ®Google Scholar
• Begehr H. Iteration of the Pompeiu integral operator and complex higher order equations. Gen Math. 1999;7(1–4):3–23.
   Google Scholar
• Aksoy Ü, Çelebi AO. Schwarz problem for higher-order complex elliptic partial differential equations. Integral Transforms Spec Funct. 2008;19(5–6):413–428.
   Web of Science ®Google Scholar
• Begehr H. Boundary value problems for the Bitsadze equation. Mem. Diff Equ Math Phys. 2004;33:5–23.
   Google Scholar
• Begehr H. Boundary value problems in complex analysis I. Bol Asoc Mat Venez. 2005;12(1):65–85.
   Google Scholar
• Vekua IN. Generalized analytic functions. London/Reading (MA): Pergamon Press/Addison-Wesley; 1962. xxix+668 pp.
   Google Scholar
• Begehr H, Gilbert RP. Transformations, transmutations, and kernel functions. Vol. 1, (Monographs and surveys in pure and applied mathematics; Vol. 58). Pitman; 1992. x+399pp. Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc.; ISBN: 0-582-02695-4.
   Google Scholar
• Begehr H, Gilbert RP. Transformations, transmutations, and kernel functions. Vol. 2, (Monographs and surveys in pure and applied mathematics; Vol. 59). New York (NY): Pitman; 1993. viii+268pp. Longman Scientific and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc.; ISBN: 0-582-09109-8.
   Google Scholar
• Dzhuraev A. Methods of singular integral equations. Translated from the Russian; revised by the author (Monographs and surveys in pure and applied mathematics; Vol. 68). New York (NY): Pitman; 1992. xii+311 pp. Longman Scientic and Technical, Harlow; copublished in the United States with John Wiley and Sons, Inc.; ISBN: 0-582-08373-7 35J55.
   Google Scholar
• Begehr H. Complex analytic methods for partial differential equations. An introductory text. River Edge (NJ): World Scientific; 1994. x+273 pp. ISBN: 981-02-1550-9.
   Google Scholar
• Begehr H. Pompeiu operators in complex, hypercomplex, and Clifford analysis. Rev Roumaine Math Pures Appl. 2001;46(1):1–11.
   Google Scholar
• Aksoy Ü, Çelebi AO. Norm estimates of a class of Calderón-Zygmund type strongly singular integral operators. Integral Transforms Spec Funct. 2007;18(1–2):87–93.
   Web of Science ®Google Scholar
• Begehr H, Hile GN. A hierarchy of integral operators. Rocky Mountain J Math. 1997;27(3):669–706.
   Web of Science ®Google Scholar
• Calderón AP, Zygmund A. On the existence of certain singular integrals. Acta Math. 1952;88:85–139.
   Web of Science ®Google Scholar
• Calderón AP, Zygmund A. On singular integrals. Am J Math. 1956;78:289–309.
   Web of Science ®Google Scholar
• Stein EM. Singular integrals and differentiability properties of functions. Princeton mathematical series no. 30. Princeton (NJ): Princeton University Press; 1970. xiv+290 pp.
   Google Scholar
• Zhu K. Operator theory in function spaces. 2nd ed., Vol. 138, Mathematical surveys and monographs. Providence (RI): American Mathematical Society; 2007. xvi+348 pp. ISBN: 978-0-8218-3965-2.
   Google Scholar
• Forelli F, Rudin W. Projections on spaces of holomorphic functions in balls. Indiana Univ Math J. 1974/1975;24:593–602.
   Web of Science ®Google Scholar
• Morrey CB Jr. On the solutions of quasi-linear elliptic partial differential equations. Trans Am Math Soc. 1938;43(1):126–166.
   Google Scholar
• Guliyev VS. Integral operators on function spaces on the homogeneous groups and on domains in ℝⁿ [doctoral dissertation]. Russian. Moscow (Russia): Steklov Institute of Mathematics; 1994.
   Google Scholar
• Guliyev VS. Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications. Elm Baku Russian. 1999. p. 332.
   Google Scholar
• Guliev VS, Mustafaev RCh. Fractional integrals in spaces of functions defined on spaces of homogeneous type (Russian). Anal Math. 1998;24(3):181–200.
   Google Scholar
• Burenkov VI, Guliyev HV. Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces. Studia Math. 2004;163(2):157–176.
   Web of Science ®Google Scholar
• Burenkov VI, Guliyev HV, Guliyev VS. Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces. J Comput Appl Math. 2007;208(1):280–301.
   Web of Science ®Google Scholar
• Burenkov VI, Guliyev VS, Serbetci A, et al. Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces. Eurasian Math J. 2010;1(1):32–53.
   Google Scholar
• Burenkov VI, Gogatishvili A, Guliyev VS, et al. Boundedness of the Riesz potential in local Morrey-type spaces. Potential Anal. 2011;35(1):67–87.
   Web of Science ®Google Scholar
• Gogatishvili A, Stepanov VD. Reduction theorems for weighted integral inequalities on the cone of monotone functions. Uspekhi Mat Nauk 68. 2013;412(4):3–68. Russian. translation in Russian Math Surv 68. 2013;4:597–664.
   Google Scholar
• Gogatishvili A, Mustafayev RCh. Weighted iterated Hardy-type inequalities. Math Inequal Appl. Forthcoming. Available from: https://arxiv.org/pdf/1503.04079.pdf
   Google Scholar
• Soria F, Weiss G. A remark on singular integrals and power weights. Indiana Univ Math J. 1994;43(1):187–204. English summary.
   Web of Science ®Google Scholar
• Lu S, Ding Y, Yan D. Singular integrals and related topics. Hackensack (NJ): World Scientific; 2007. viii+272pp. ISBN: 978-981-270-623-2, ISBN: 981-270-623-2.
   Google Scholar
• Ding Y, Yang D, Zhou Z. Boundedness of sublinear operators and commutators on Lp,w(Rn). Yokohama Math J. 1998;46(1):15–27.
   Google Scholar
• Stein EM, Weiss G. Introduction to Fourier analysis on Euclidean spaces. Princeton mathematical series no. 32. Princeton (NJ): Princeton University Press; 1971. x+297pp.
   Google Scholar
• Torchinsky A. Real-variable methods in harmonic analysis. Vol. 123, Pure and applied mathematics. Orlando (FL): Academic Press; 1986. xii+462pp. ISBN: 0-12-695460-7; 0-12-695461-5.
   Google Scholar