Abstract
Lp− Boundedness for Integral Transforms Associated with Singular Partial Differential Operators
Highlights
Let Dj, 1 ≤ j ≤ n, and Ξμ, μ > 0, be the singular partial differential operators defined by ∂ Dj = ∂xj 2μ ∂ n ∂ Ξμ =( )2 ++ ( )[2]; (r, x) ∈]0, +∞[×Rn, μ > 0. ∂r r ∂r j=1 ∂xj
We show that the fractional transform Hμ can be extended to μ ∈ R and that for every μ ∈ R, Hμ is a topological isomorphism from the Schwartz’s space Se R × Rn onto itself whose inverse operator is Hμ−1 = H−μ
We study the Lp− boundedness of the operators Rμ and Hμ on the weighted spaces Lp [0, +∞[×Rn, r2adr⊗dx, p ∈ [1, +∞]. We recall in this context, that studing the Lp− boundedness of integral transforms connected with differential systems is an interesting subject because knowing the range of parameters μ, p for which an operator is bounded on Lebesgue space gives quantitative information about the rate of growth of the transformed functions ([15, 16, 17])
Summary
Let Dj, 1 ≤ j ≤ n, and Ξμ, μ > 0, be the singular partial differential operators defined by. Using the relation (4), we define the fractional transform Rμ on Ce(R × Rn) (the space of continuous functions on R × Rn, even with respect to the first variable) by r. The relations (2), (6) and (7) show that for all integrable functions f, g on [0, +∞[×Rn with respect to the measure dνμ(r, x), we have. We study the Lp− boundedness of the operators Rμ and Hμ on the weighted spaces Lp [0, +∞[×Rn, r2adr⊗dx , p ∈ [1, +∞] We recall in this context, that studing the Lp− boundedness of integral transforms connected with differential systems is an interesting subject because knowing the range of parameters μ, p for which an operator is bounded on Lebesgue space gives quantitative information about the rate of growth of the transformed functions ([15, 16, 17]). We give the best (the smallest) contants Cp,a,μ and Dp,a,μ that satisfy the relations (12) and (13)
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