L(log L) spaces and weights for the strong maximal function

1984 ◽  
Vol 44 (1) ◽  
pp. 21-31 ◽  
Author(s):  
Richard J. Bagby ◽  
Douglas S. Kurtz
Keyword(s):  
Maximal Function ◽  
L Log L

scholarly journals On the strong maximal function and Zygmund's class $L(log^{+} L)^{n}$

1981 ◽  
Vol 69 (2) ◽  
pp. 155-158 ◽  
Author(s):  
N. Fava ◽  
E. Gatto ◽  
C. Gutiérrez
Keyword(s):  
Maximal Function ◽  
L Log L

scholarly journals Some estimates for the commutators of multilinear maximal function on Morrey-type space

2021 ◽  
Vol 19 (1) ◽  
pp. 515-530
Author(s):  
Xiao Yu ◽  
Pu Zhang ◽  
Hongliang Li
Keyword(s):  
Maximal Function ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Bounded Mean Oscillation ◽  
Generalized Morrey Spaces ◽  
Type Space ◽  
Bmo Function ◽  
Mean Oscillation ◽  
Multilinear Maximal Function ◽  
Equivalent Conditions

Abstract In this paper, we study the equivalent conditions for the boundedness of the commutators generated by the multilinear maximal function and the bounded mean oscillation (BMO) function on Morrey space. Moreover, the endpoint estimate for such operators on generalized Morrey spaces is also given.

scholarly journals A note on singular integrals with angular integrability

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Feng Liu
Keyword(s):  
Singular Integral ◽  
Singular Integrals ◽  
Function Spaces ◽  
Riesz Transforms ◽  
Hilbert Transforms ◽  
Rotation Method ◽  
Vector Valued Function ◽  
Vector Valued ◽  
L Log L

Abstract In this note we study the rough singular integral $$ T_{\varOmega }f(x)=\mathrm{p.v.} \int _{\mathbb{R}^{n}}f(x-y)\frac{\varOmega (y/ \vert y \vert )}{ \vert y \vert ^{n}}\,dy, $$ T Ω f ( x ) = p . v . ∫ R n f ( x − y ) Ω ( y / | y | ) | y | n d y , where $n\geq 2$ n ≥ 2 and Ω is a function in $L\log L(\mathrm{S} ^{n-1})$ L log L ( S n − 1 ) with vanishing integral. We prove that $T_{\varOmega }$ T Ω is bounded on the mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}( \mathbb{R}^{n})$ L | x | p L θ p ˜ ( R n ) , on the vector-valued mixed radial-angular spaces $L_{|x|}^{p}L_{\theta }^{\tilde{p}}(\mathbb{R}^{n},\ell ^{\tilde{p}})$ L | x | p L θ p ˜ ( R n , ℓ p ˜ ) and on the vector-valued function spaces $L^{p}(\mathbb{R}^{n}, \ell ^{\tilde{p}})$ L p ( R n , ℓ p ˜ ) if $1<\tilde{p}\leq p<\tilde{p}n/(n-1)$ 1 < p ˜ ≤ p < p ˜ n / ( n − 1 ) or $\tilde{p}n/(\tilde{p}+n-1)< p\leq \tilde{p}<\infty $ p ˜ n / ( p ˜ + n − 1 ) < p ≤ p ˜ < ∞ . The same conclusions hold for the well-known Riesz transforms and directional Hilbert transforms. It should be pointed out that our proof is based on the Calderón–Zygmund’s rotation method.

scholarly journals The Dirichlet problem for the Jacobian equation in critical and supercritical Sobolev spaces

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
André Guerra ◽  
Lukas Koch ◽  
Sauli Lindberg
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Spaces ◽  
Regularity Of Solutions ◽  
Jacobian Equation ◽  
L Log L

AbstractWe study existence and regularity of solutions to the Dirichlet problem for the prescribed Jacobian equation, $$\det D u =f$$ det D u = f , where f is integrable and bounded away from zero. In particular, we take $$f\in L^p$$ f ∈ L p , where $$p>1$$ p > 1 , or in $$L\log L$$ L log L . We prove that for a Baire-generic f in either space there are no solutions with the expected regularity.

The Rearrangement Inequality for the Ergodic Maximal Function

2001 ◽  
Vol 8 (4) ◽  
pp. 727-732
Author(s):  
L. Ephremidze
Keyword(s):  
Maximal Function ◽  
Rearrangement Inequality ◽  
Decreasing Rearrangement ◽  
Exact Constants

Abstract The equivalence of the decreasing rearrangement of the ergodic maximal function and the maximal function of the decreasing rearrangement is proved. Exact constants are obtained in the corresponding inequalities.

scholarly journals The Boundedness of the Hardy-Littlewood Maximal Operator and Multilinear Maximal Operator in Weighted Morrey Type Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Takeshi Iida
Keyword(s):  
Maximal Function ◽  
Maximal Operator ◽  
Morrey Spaces ◽  
Weighted Morrey Spaces ◽  
Type Spaces ◽  
Multilinear Maximal Function ◽  
Multilinear Maximal Operator ◽  
Littlewood Maximal Operator

The aim of this paper is to prove the boundedness of the Hardy-Littlewood maximal operator on weighted Morrey spaces and multilinear maximal operator on multiple weighted Morrey spaces. In particular, the result includes the Komori-Shirai theorem and the Iida-Sato-Sawano-Tanaka theorem for the Hardy-Littlewood maximal operator and multilinear maximal function.

scholarly journals Convolution operator and maximal function for the Dunkl transform

2005 ◽  
Vol 97 (1) ◽  
pp. 25-55 ◽  
Author(s):  
Sundaram Thangavelu ◽  
Yuan Xu
Keyword(s):  
Maximal Function ◽  
Convolution Operator ◽  
Dunkl Transform
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