Lp− Boundedness for Integral Transforms Associated with Singular Partial Differential Operators

We define fractional transforms R μ and H μ , μ > 0 on the space R × R n . First, we study these transforms on regular function spaces and we establish that these operators are topological isomorphisms and we give the inverse operators as integro differential operators. Next, we study the L p -boundedness of these operators. Namely, we give necessary and sufficient condition on the parameter μ for which the transforms R μ and H μ are bounded on the weighted spaces L p ( [ 0 , + ∞ [ × R n , r 2 a d r ⊗ d x ) and we give their norms.