Geodesic
Approximating functions in the power-type weighted variable exponent Sobolev space by the hardy averaging operator
Filomat, Tome 36 (2022) no. 10, p. 3321

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We investigate the problem of approximating function f in the power-type weighted variable exponent Sobolev space W r,p(.) α(.) (0, 1), (r = 1, 2, ...), by the Hardy averaging operator A f (x) = 1 x x 0 f (t)dt. If the function f lies in the power-type weighted variable exponent Sobolev space W r,p(.) α(.) (0, 1), it is shown that A f − f p(.),α(.)−rp(.) ≤ C f (r) p(.),α(.) , where C is a positive constant. Moreover, we consider the problem of boundedness of Hardy averaging operator A in power-type weighted variable exponent grand Lebesgue spaces L p(.),θ α(.) (0, 1). The sufficient criterion established on the power-type weight function α(.) and exponent p(.) for the Hardy averaging operator to be bounded in these spaces.
DOI : 10.2298/FIL2210321A
Classification : 42A05, 42B25, 26D10, 35A23
Keywords: Approximation, Hardy averaging operator, Power-type weighted Sobolev spaces with variable exponent, Power-type weighted grand Lebesgue spaces with variable exponent 
Rabil Ayazoglu; Ismail Ekincioglu; S Şule Şener. Approximating functions in the power-type weighted variable exponent Sobolev space by the hardy averaging operator. Filomat, Tome 36 (2022) no. 10, p. 3321 . doi: 10.2298/FIL2210321A
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     author = {Rabil Ayazoglu and Ismail Ekincioglu and S \c{S}ule \c{S}ener},
     title = {Approximating functions in the power-type weighted variable exponent {Sobolev} space by the hardy averaging operator},
     journal = {Filomat},
     pages = {3321 },
     year = {2022},
     volume = {36},
     number = {10},
     doi = {10.2298/FIL2210321A},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.2298/FIL2210321A/}
}
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