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Fock Space, the Heisenberg Group, Heat Flow, and Toeplitz Operators
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
Proof The proof uses the Hardy-Littlewood maximal function f* to estimate f˜(t). A technical argument shows that limt→0f˜(t)(w)=f(w) except on a set of measure 0. The rest follows since it is standard that ||f˜(t)||∞≤||Tf(t)||t≤||f||∞.
Fundamental Theorem of Calculus
Published in Kenneth Kuttler, Modern Analysis, 2017
Definition 14.3f∈Lloc1(ℝn) means fXB(0,R) ∈ L1(ℝn) for all R > 0. For f∈Lloc1(ℝn), the Hardy Littlewood Maximal Function, Mf, is defined by
Applications of Singular Integral Operators and Commutators
Published in Michael Ruzhansky, Hemen Dutta, Advanced Topics in Mathematical Analysis, 2019
Recently, in Ragusa and Scapellato (2017) the authors defined a new class of functions. These spaces generalize Morrey spaces and give a refinement of the classical Lebesgue spaces. The authors also proved some embeddings between these new classes and, as application, they obtain some regularity results for solutions to partial differential equations of parabolic type in non-divergence form. Preparatory to achieving these results is the study of the behavior of Hardy-Littlewood maximal function, Riesz potential, sharp and fractional maximal functions, singular integral operators with Calderón-Zygmund kernel and commutators.
Maximal and Calderón–Zygmund operators on the local variable Morrey–Lorentz spaces and some applications
Published in Applicable Analysis, 2023
A. Kucukaslan, V. S. Guliyev, C. Aykol, A. Serbetci
Let f be a locally integrable function on . Hardy–Littlewood maximal function Mf is defined by Maximal operators play an important role in the differentiability properties of functions, singular integrals and partial differential equations. They often provide a deeper and more simplified approach to understanding problems in these areas. For the operator M, the rearrangement inequality holds, where c and C are independent of f. Here denotes the right continuous non-increasing rearrangement of f: and is the distribution function of the function f.
Boundedness of paraproducts on spaces of homogeneous type II
Published in Applicable Analysis, 2022
Der-Chen Chang, Xing Fu, Dachun Yang
Let , p, g and ν be as in this proposition. As in [4, (5.214)], for any , define First, we claim that, for any , where M is the Hardy–Littlewood maximal function as in (2.7). Indeed, for any with , letting for some , from [1, Definition 2.6(i)], (1.1) and (1.9), we deduce that To estimate , by , we find that Then we estimate . By and (1.1), we know that which, combined with the estimate of , then completes the proof of the above claim (3.4).
Bessel type Kolmogorov inequalities on weighted Lebesgue spaces
Published in Applicable Analysis, 2021
Some of the most significant and studied operators in Harmonic Analysis are the Hardy Littlewood maximal function, the Hilbert transform and the Riesz transforms. Riesz transforms which are convolution type singular integral operators are known as the important operator in analysis and its applications. In this study, it is studied the singular integral operators of convolution type which are related to the Laplace-Bessel differential operator Riesz transforms have been the research areas of many mathematicians such as Muckenhoupt and Stein [1–3], Kipriyanov and Klyuchantsev [4–6], Trimeche [7], Stempak [8], Aliev and Gadjiev [9], Guliyev [10], Gadjiev and Guliyev [11], Ekincioglu [12] and others.