scholarly journals Weighted norm inequality for the singular integral with variable kernel and fractional differentiation

2015 ◽  
Vol 423 (2) ◽  
pp. 1610-1629 ◽  
Author(s):  
Yanping Chen ◽  
Kai Zhu
Keyword(s):  
Singular Integral ◽  
Norm Inequality ◽  
Weighted Norm Inequality ◽  
Weighted Norm ◽  
Fractional Differentiation ◽  
Variable Kernel

scholarly journals Weak Estimates of Singular Integrals with Variable Kernel and Fractional Differentiation on Morrey-Herz Spaces

2017 ◽  
Vol 2017 ◽  
pp. 1-10
Author(s):  
Yanqi Yang ◽  
Shuangping Tao
Keyword(s):  
Integral Operator ◽  
Singular Integral ◽  
Singular Integrals ◽  
Differentiation Operator ◽  
Fractional Differentiation ◽  
Herz Spaces ◽  
Variable Kernel ◽  
Harmonic Decomposition ◽  
Spherical Harmonic Decomposition ◽  
Fractional Differentiation Operator

Let T be the singular integral operator with variable kernel defined by Tf(x)=p.v.∫Rn(Ω(x,x-y)/x-yn)f(y)dy and let Dγ  (0≤γ≤1) be the fractional differentiation operator. Let T⁎and T♯ be the adjoint of T and the pseudoadjoint of T, respectively. In this paper, the authors prove that TDγ-DγT and (T⁎-T♯)Dγ are bounded, respectively, from Morrey-Herz spaces MK˙p,1α,λ(Rn) to the weak Morrey-Herz spaces WMK˙p,1α,λ(Rn) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product T1T2 and the pseudoproduct T1∘T2 are also given.

A weighted norm inequality for the Hankel transformation

1984 ◽  
Vol 99 (1-2) ◽  
pp. 45-50 ◽  
Author(s):  
P. Heywood ◽  
P. G. Rooney
Keyword(s):  
Fourier Transformation ◽  
Norm Inequality ◽  
Weight Functions ◽  
Weighted Norm Inequality ◽  
Weighted Norm ◽  
Hankel Transformation ◽  
Negative Weight ◽  
Boundedness Theorem ◽  
Image Position

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.

scholarly journals DIRECTIONAL MAXIMAL OPERATORS AND RADIAL WEIGHTS ON THE PLANE

2013 ◽  
Vol 89 (3) ◽  
pp. 397-414
Author(s):  
HIROKI SAITO ◽  
HITOSHI TANAKA
Keyword(s):  
Maximal Operator ◽  
Unit Vector ◽  
Norm Inequality ◽  
Maximal Operators ◽  
Weighted Norm Inequality ◽  
Weighted Norm ◽  
Radial Weight ◽  
Orthogonality Principle ◽  
Image Position ◽  
Unit Vectors

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.

scholarly journals A weighted norm inequality for rough singular integrals

1999 ◽  
Vol 51 (2) ◽  
pp. 141-161 ◽  
Author(s):  
Dashan Fan ◽  
Yibiao Pan ◽  
Dachun Yang
Keyword(s):  
Singular Integrals ◽  
Norm Inequality ◽  
Weighted Norm Inequality ◽  
Weighted Norm
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