The boundedness of the generalized anisotropic potentials with rough kernels in the Lorentz spaces

References

• Adams , D. 1988 . A sharp inequality of J. Moser for high order derivatives . Ann. Math , 128 : 385 – 398 .
   Web of Science ®Google Scholar
• Aliyev , I. A. and Gadjiev , A. D. 1988 . On classes of operators of potential types, generated by a generalized shift . Reports of enlarged Session of the Seminars of I.N. Vekua Institute of Applied Mathematics, Tbilisi , 3 ( 2 ) : 21 – 24 . (Russian)
   Google Scholar
• Besov , O. V. , Il'in , V. P. and Lizorkin , P. I. 1966 . The L p -estimates of a certain class of non-isotropically singular integrals . Dokl. Akad. Nauk. SSSR , 169 : 1250 – 1253 . (Russian)
   Google Scholar
• Bramanti , M. and Cerutti , M. C. 1996 . Commutators of singular integrals on homogeneous spaces . Boll. Un. Mat. Ital. B , 10 ( 7 ) : 843 – 883 .
   Google Scholar
• Fabes , E. B. and Riviere , N. 1966 . Singular integrals with mixed homogeneity . Studia Math , 27 : 19 – 38 .
   Web of Science ®Google Scholar
• Gadjiev , A. D. and Guliyev , V. S. 2008 . The Stein–Weiss type inequality for fractional integrals, associated with the Laplace–Bessel differential operator . Fract. Calc. Appl. Anal , 11 ( 1 ) : 77 – 90 .
   Google Scholar
• Guliyev , V. S. 1998 . Sobolev theorems for the Riesz B-potentials . Dokl. Acad. Nauk. Russia , 358 ( 4 ) : 450 – 451 . (Russian)
   Web of Science ®Google Scholar
• Guliyev , V. S. 1999 . Sobolev theorems for anisotropic Riesz–Bessel potentials on Morrey-Bessel spaces . Dokl. Acad. Nauk. Russia , 367 ( 2 ) : 155 – 156 . (Russian)
   Google Scholar
• Guliyev , V. S. 2000 . Some properties of the anisotropic Riesz–Bessel potential . Anal. Math , 26 ( 2 ) : 99 – 118 .
   Google Scholar
• Guliyev , V. S. 2003 . On maximal function and fractional integral, associated the Bessel differential operator . Math. Inequal. Appl , 6 ( 2 ) : 317 – 330 .
   Web of Science ®Google Scholar
• Guliyev , V. S. , Serbetci , A. and Ekincioglu , I. 2007 . Necessary and sufficient conditions for the boundedness of rough B-fractional integral operators in the Lorentz spaces . J. Math. Anal. Appl , 336 ( 1 ) : 425 – 437 .
   Google Scholar
• Guliyev , V. S. , Serbetci , A. and Ekincioglu , I. 2007 . On boundedness of the generalized B-potential integral operators in the Lorentz spaces . Integral Transforms Spec. Funct , 18 ( 12 ) : 885 – 895 .
   Web of Science ®Google Scholar
• Guliyev , V. S. , Garakhanova , N. N. and Zeren , Y. 2008 . Pointwise and integral estimates for the Riesz B-potential in terms of B-maximal and B-fractionally maximal functions . Sib. Mat. Zh , 49 ( 6 ) : 1263 – 1279 . (Russian); translation in Sib. Math. J. 49(6) (2008), pp. 1008–1022
   Google Scholar
• Guliyev , V. S. and Garakhanova , N. N. 2009 . The Sobolev-Il'in theorem for the B-Riesz potential . Siberian Math. J , 50 ( 1 ) : 49 – 59 .
   Google Scholar
• Kipriyanov , I. A. 1967 . Fourier–Bessel transformations and imbedding theorems for weight classes . Trudy Math. Inst. Steklov , 89 : 130 – 213 .
   Google Scholar
• Levitan , B. M. 1951 . Bessel function expansions in series and Fourier integrals . Uspekhi Mat. Nauk , 6 (2(42)) : 102 – 143 . (Russian)
   Google Scholar
• Lyakhov , L. N. 1996 . Multipliers of the mixed Fourier–Bessel transform . Proc. Steklov Inst. Math , 214 ( 3 ) : 227 – 242 .
   Google Scholar
• Lyakhov , L. N. 2007 . B-hypersingular integrals and their applications , Lipetsk, , Russia : LPSU (Lipetsk State Pedagogical University) .
   Google Scholar
• Maz'ya , V. G. 1985 . Sobolev Spaces , Berlin : Springer-Verlag .
   Google Scholar
• Opic , B. and Kufner , A. 1990 . “ Hardy-type Inequalities ” . Harlow : Longman Scientific and Technical . Pitman Research Notes in Mathematics Series 219
   Google Scholar
• Serbetci , A. and Ekincioglu , I. 2004 . Boundedness of Riesz potential generated by generalized shift operator on Ba spaces . Czech. Math. J , 54 ( 3 ) : 579 – 589 .
   Web of Science ®Google Scholar
• Ziemer , W. P. 1989 . Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , New York : Springer-Verlag . Graduate Texts in Mathematics 120
   Google Scholar