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A weighted norm inequality for the Hankel transformation. (English) Zbl 0564.44003

The authors present conditions on non-negative functions U and V which are sufficient for \[ (1)\quad [\int^{\infty}_{0}| U(x)(H_{\lambda}f)(x)|^ q]^{1/q}\leq K[\int | V(x)f(x)|^ p]^{1/p} \] where \(H_{\lambda}\) is the Hankel transform, and \(1<p\leq q<\infty\). Let \(\| f\|_{\mu,v,p}=[\int^{\infty}_{0}| x^{\mu}v(x)f(x)|^ pdx/x]^{1/p}\) where v(x) is non-negative. Let \(L_{\mu,v,p}(R^+)\) be the set of all measurable complex-valued functions in (0,\(\infty)\) for which \(\| f\|_{\mu,v,p}<\infty\). Similarly let \(L_{\mu,v,p}(R^ u)\) be the set of all functions where dx/\(| x|\) substitutes dx/x in the definition of \(\| f\|_{\mu,v,p}\). Let A(p,q,\(\alpha)\) denote the set u,v of non-negative measurable functions on \(R^+\) such that there exist constants B and C such that for every positive s, \[ [\int_{u(x)>Bs}\{x^{\alpha}u(x)\}^ qdx/x]^{1/q}[\int_{v(x)<s}\{x^{\alpha}/v(x)\}^{p'}dx/x]^{1/p'}\l e C \] (with p’ and q’ satisfying \(p\leq q<p'\) and \(q'<p\leq q)\). The authors prove the following: Theorem 1. Suppose that \(1<p\leq q<\infty\), \(\lambda\geq -\), \(1\leq \eta \leq \lambda +3/2\), \(\lambda (p,q)\leq \mu <\lambda +3/2\) and (u,v)\(\in A(p,q,\eta -\mu)\). Then \(H_{\lambda}\) can be extended to a bounded operator in \([L_{\mu,v,p}(R^+)]\); that is there is a positive number K such that for every \(f\in L_{\mu,v,p}(R^+)\), \[ [\int^{\infty}_{0}| x^{1-\mu} u(x)(H_{\lambda}f(x)|^ q dx/x]^{1/q}\leq K[\int^{\infty}_{0}| x^{\mu}v(x)f(x)|^ p]^{1/p}dx/x]^{1/p}. \] Theorem 2. Let U and V be non-negative measurable functions on \(R^ n\), and suppose that \(1<p\leq q\leq 1\) and that there are positive constants B and C such that \[ (\int_{| x|^{(1/p-1/q')^ n}U(x)>Bs}\{U(x)\}^ qdx)^{1/q}(\int_{V(x)<s}\{V(x)\}^{-p'}dx)^{1/p'}\leq C \] for every positive s. Then the Fourier transform can be extended to \(L_{1/p,V,p}(R^ n)\) as a bounded operator and (1) holds with K depending on B, C, p and q.
Reviewer: L.Arteaga

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
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References:

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