A weighted norm inequality for the Hankel transformation

Author(s):  
P. Heywood ◽  
P. G. Rooney

SynopsisWe give conditions on pairs of non-negative weight functions U and V which are sufficient that for 1<p≤<∞where Hλ is the Hankel transformation.The technique of proof is a variant of Muckenhoupt's recent proof for the boundedness of the Fourier transformation between weighted Lp spaces, and we can also use this variant to prove a somewhat different boundedness theorem for the Fourier transformation.

1986 ◽  
Vol 103 (3-4) ◽  
pp. 325-333 ◽  
Author(s):  
S. A. Emara ◽  
H. P. Heinig

SynopsisWe give conditions on pairs of non-negative weight functions u and v which are sufficient that, for 1<p, q <∞,where T is the Hankel-or the K-transformation.The proofs are based on a weighted Marcinkiewicz interpolation theorem for linear operators. In the case that T is the Hankel transformation and 1<p≦q <∞, the result is similar to a weighted estimate of Heywood and Rooney [9], but with different weight conditions.


2013 ◽  
Vol 89 (3) ◽  
pp. 397-414
Author(s):  
HIROKI SAITO ◽  
HITOSHI TANAKA

AbstractLet $\Omega $ be the set of unit vectors and $w$ be a radial weight on the plane. We consider the weighted directional maximal operator defined by $$\begin{eqnarray*}{M}_{\Omega , w} f(x): = \sup _{x\in R\in \mathcal{B} _{\Omega }}\frac{1}{w(R)} \int \nolimits \nolimits_{R} \vert f(y)\vert w(y)\hspace{0.167em} dy,\end{eqnarray*}$$ where ${ \mathcal{B} }_{\Omega } $ denotes the set of all rectangles on the plane whose longest side is parallel to some unit vector in $\Omega $ and $w(R)$ denotes $\int \nolimits \nolimits_{R} w$. In this paper we prove an almost-orthogonality principle for this maximal operator under certain conditions on the weight. The condition allows us to get the weighted norm inequality $$\begin{eqnarray*}\Vert {M}_{\Omega , w} f\mathop{\Vert }\nolimits_{{L}^{2} (w)} \leq C\log N\Vert f\mathop{\Vert }\nolimits_{{L}^{2} (w)} ,\end{eqnarray*}$$ when $w(x)= \vert x\hspace{-1.2pt}\mathop{\vert }\nolimits ^{a} $, $a\gt 0$, and when $\Omega $ is the set of unit vectors on the plane with cardinality $N$ sufficiently large.


Author(s):  
Man Kam Kwong ◽  
A. Zettl

SynopsisHere we obtain the inequalityunder very general conditions on the non-negative weight functions u, v, w, for general p, l≦p<∞ and for both bounded and unbounded intervals I.


1953 ◽  
Vol 5 ◽  
pp. 273-288 ◽  
Author(s):  
Israel Halperin

This paper is the first in a series dealing with Banach spaces L whose elements are functions on a measure space S. If W is a family of non-negative weight functions wα we sometimes write LWp when the norm is given as


1999 ◽  
Vol 51 (2) ◽  
pp. 141-161 ◽  
Author(s):  
Dashan Fan ◽  
Yibiao Pan ◽  
Dachun Yang

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