The Stein-Weiss type inequalities for the B-Riesz potentials

Abstract

We establish two inequalities of Stein-Weiss type for the Riesz potential operator I α,γ ( B− Riesz potential operator) generated by the Laplace-Bessel differential operator Δ B in the weighted Lebesgue spaces L p,|x| β ,γ . We obtain necessary and sufficient conditions on the parameters for the boundedness of I α,γ from the spaces L p,|x| β ,γ to L q,|x| −λ ,γ , and from the spaces L 1,|x| β ,γ to the weak spaces W L q,|x| −λ ,γ . In the limiting case p = Q/α we prove that the modified B− Riesz potential operator I α,γ is bounded from the spaces L p,|x| β ,γ to the weighted B − BMO spaces BMO |x| −λ ,γ .As applications, we get the boundedness of I α,γ from the weighted B -Besov spaces B s pθ ,|x| β ,γ to the spaces B s qθ ,|x| −λ ,γ . Furthermore, we prove two Sobolev embedding theorems on weighted Lebesgue L p,|x| β ,γ and weighted B -Besov spaces B s pθ ,|x| β ,γ by using the fundamental solution of the B -elliptic equation Δ α/2 B . Mathematics subject classification (2010): 42B20, 42B25, 42B35.

Riesz potential and maximal function for Dunkl transform

“…Let us mention that for the so-called B-Riesz potentials the results that are similar to Theorem 1.4 were established in [9]. Hence, by positivity of the operator T t and (6.2),…”

“…Littlewood [11] for d = 1, S. Sobolev [25] for d > 1 and γ = β = 0, E.M. Stein and G. Weiss [26] in the general case. The conditions for weak boundedness can be found in [24,9].…”

Weighted generalized functions and hyperbolic Riesz B‐potential

“…Estimates for potential‐type operators and its inversion can be found, for example, in other studies . Sobolev theorems for potential generated by Bessel operator with Euclidean distance in the Morrey‐Bessel and BMO‐Bessel spaces were considered in previous studies . Riesz B‐potential with Euclidean distance in the weighted Lebegue spaces and its inverse are given in references herein .…”

“…Taking the factor −i by the bracket, we obtain (18). The resulting formula is valid for any quadratic form whose imaginary part is positive definite due to the fact that analytic continuation is unique.…”

“…[9][10][11][12][13][14] Sobolev theorems for potential generated by Bessel operator with Euclidean distance in the Morrey-Bessel and BMO-Bessel spaces were considered in previous studies. [15][16][17][18][19][20] Riesz B-potential with Euclidean distance in the weighted Lebegue spaces and its inverse are given in references herein. [21][22][23][24][25][26][27] In Sarikaya et al, 28 for the operator similar to operator (3), some properties other than given in this article were obtained.…”