Weak Type (1, 1) Estimates for Heat Kernel Maximal Functions on Lie Groups

1991 ◽  
Vol 323 (2) ◽  
pp. 637 ◽  
Author(s):  
Michael Cowling ◽  
Garth Gaudry ◽  
Saverio Giulini ◽  
Giancarlo Mauceri
Keyword(s):  
Heat Kernel ◽  
Lie Groups ◽  
Weak Type ◽  
Maximal Functions

scholarly journals Weak type $(1,1)$ estimates for heat kernel maximal functions on Lie groups

1991 ◽  
Vol 323 (2) ◽  
pp. 637-649
Author(s):  
Michael Cowling ◽  
Garth Gaudry ◽  
Saverio Giulini ◽  
Giancarlo Mauceri
Keyword(s):  
Heat Kernel ◽  
Lie Groups ◽  
Weak Type ◽  
Maximal Functions

scholarly journals Hardy-Littlewood maximal functions on some solvable Lie groups

1988 ◽  
Vol 45 (1) ◽  
pp. 78-82 ◽  
Author(s):  
G. Gaudry ◽  
S. Giulini ◽  
A. Hulanicki ◽  
A. M. Mantero
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Weak Type ◽  
Maximal Functions ◽  
Solvable Lie Groups ◽  
Littlewood Maximal Function ◽  
The Family ◽  
Fixed Power ◽  
Simply Connected

AbstractLet N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

The Weak Type (1, 1) Estimates of Maximal Functions on the Laguerre Hypergroup

2010 ◽  
Vol 53 (3) ◽  
pp. 491-502 ◽  
Author(s):  
Jizheng Huang ◽  
Liu Heping
Keyword(s):  
Maximal Function ◽  
Weak Type ◽  
Maximal Functions ◽  
Laguerre Hypergroup ◽  
Littlewood Maximal Function

AbstractIn this paper, we discuss various maximal functions on the Laguerre hypergroup K including the heat maximal function, the Poisson maximal function, and the Hardy–Littlewood maximal function which is consistent with the structure of hypergroup of K. We shall establish the weak type (1, 1) estimates for these maximal functions. The Lp estimates for p > 1 follow fromthe interpolation. Some applications are included.

BOUNDEDNESS OF MAXIMAL FUNCTIONS ON NONDOUBLING PARABOLIC MANIFOLDS WITH ENDS

2020 ◽  
pp. 1-25
Author(s):  
HONG CHUONG DOAN
Keyword(s):  
Heat Kernel ◽  
Maximal Function ◽  
Upper Bound ◽  
Nonnegative Integer ◽  
Weak Type ◽  
Adjoint Operator ◽  
Beltrami Operator ◽  
Maximal Functions ◽  
Heat Semigroup ◽  
Doubling Condition

Let $M$ be a nondoubling parabolic manifold with ends. First, this paper investigates the boundedness of the maximal function associated with the heat semigroup ${\mathcal{M}}_{\unicode[STIX]{x1D6E5}}f(x):=\sup _{t>0}|e^{-t\unicode[STIX]{x1D6E5}}f(x)|$ where $\unicode[STIX]{x1D6E5}$ is the Laplace–Beltrami operator acting on $M$ . Then, by combining the subordination formula with the previous result, we obtain the weak type $(1,1)$ and $L^{p}$ boundedness of the maximal function ${\mathcal{M}}_{\sqrt{L}}^{k}f(x):=\sup _{t>0}|(t\sqrt{L})^{k}e^{-t\sqrt{L}}f(x)|$ on $L^{p}(M)$ for $1<p\leq \infty$ where $k$ is a nonnegative integer and $L$ is a nonnegative self-adjoint operator satisfying a suitable heat kernel upper bound. An interesting thing about the results is the lack of both doubling condition of $M$ and the smoothness of the operators’ kernels.

scholarly journals Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Heisenberg Group ◽  
Weak Type ◽  
Riesz Transforms ◽  
Heisenberg Groups ◽  
Related Measure

AbstractIn the Heisenberg group of dimension $$2n+1$$ 2 n + 1 , we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$ L p bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

scholarly journals Weak type estimates for the Hardy-Littlewood maximal functions

1974 ◽  
Vol 49 (3) ◽  
pp. 217-223 ◽  
Author(s):  
Luis Caffarelli ◽  
Calixto Calderón
Keyword(s):  
Weak Type ◽  
Maximal Functions ◽  
Weak Type Estimates
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