Nonlinear parabolic problems in Musielak–Orlicz spaces

Świerczewska-Gwiazda, Agnieszka

doi:10.1016/j.na.2013.11.026

Cited by 38 publications

(31 citation statements)

References 30 publications

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“…For more examples, see Example 2.16 below. The Musielak-Orlicz spaces not only have their own interest, but they are also very useful in partial differential equations [2,5,28,26], in calculus of variations [13], in image restoration [27,43] and in fluid dynamics [73,51].…”

Section: Introductionmentioning

confidence: 99%

New martingale inequalities and applications to Fourier analysis

et al. 2019

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Let (Ω, F , P) be a probability space and ϕ : Ω × [0, ∞) → [0, ∞) be a Musielak-Orlicz function. In this article, the authors prove that the Doob maximal operator is bounded on the Musielak-Orlicz space L ϕ (Ω). Using this and extrapolation method, the authors then establish a Fefferman-Stein vector-valued Doob maximal inequality on L ϕ (Ω). As applications, the authors obtain the dual version of the Doob maximal inequality and the Stein inequality for L ϕ (Ω), which are new even in weighted Orlicz spaces. The authors then establish the atomic characterizations of martingale Musielak-Orlicz Hardy spaces H s ϕ (Ω), P ϕ (Ω), Q ϕ (Ω), H S ϕ (Ω) and H M ϕ (Ω). From these atomic characterizations, the authors further deduce some martingale inequalities between different martingale Musielak-Orlicz Hardy spaces, which essentially improve the corresponding results in Orlicz space case and are also new even in weighted Orlicz spaces. By establishing the Davis decomposition on H S ϕ (Ω) and H M ϕ (Ω), the authors obtain the Burkholder-Davis-Gundy inequality associated with Musielak-Orlicz functions. Finally, using the previous martingale inequalities, the authors prove that the maximal Fejér operator is bounded from H ϕ [0, 1) to L ϕ [0, 1), which further implies some convergence results of the Fejér means; these results are new even for the weighted Hardy spaces.

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Section: Introductionmentioning

confidence: 99%

New martingale inequalities and applications to Fourier analysis

et al. 2019

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“…Motivated by this, the study for the real-variable theory of various function spaces on R n or domains in R n , especially, the Hardy-type spaces, associated with different differential operators, has inspired great interests in recent years; see, for example, [3,4,31,32,34,46,49,52,73,78] for the case of Hardy spaces, [11,61,72] for the case of weighted Hardy spaces, [1,88,89,90] for the case of variable exponent Hardy spaces and [2,50,51,80,82,83,85] for the case of (Musielak-)Orlicz Hardy spaces. Recall that the Musielak-Orlicz space was originated by Nakano [67] and developed by Musielak and Orlicz [64,65], which is a natural generalization of many important spaces such as (weighted) Lebesgue spaces, variable Lebesgue spaces and Orlicz spaces and not only has its own interest, but is also very useful in partial differential equations [6,7,44,40], in calculus of variations [27], in image restoration [43,54] and in fluid dynamics [77,62]. The Musielak-Orlicz Hardy space on R n has proved useful in harmonic analysis (see, for example, [56,18,57,79]) and, especially, naturally appears in the endpoint estimate for both the div-curl lemma and the commutator of Cald...…”

Section: Introductionmentioning

confidence: 99%

Atomic and maximal function characterizations of Musielak–Orlicz–Hardy spaces associated to non-negative self-adjoint operators on spaces of homogeneous type

Yang

2019

Collect. Math.

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Let X be a metric space with doubling measure and L a non-negative self-adjoint operator on L 2 (X) whose heat kernels satisfy the Gaussian upper bound estimates. Assume that the growth function ϕ : X×[0, ∞) → [0, ∞) satisfies that ϕ(x, ·) is an Orlicz function and ϕ(·, t) ∈ A ∞ (X) (the class of uniformly Muckenhoupt weights). Let H ϕ, L (X) be the Musielak-Orlicz-Hardy space defined via the Lusin area function associated with the heat semigroup of L. In this article, the authors characterize the space H ϕ, L (X) by means of atoms, non-tangential and radial maximal functions associated with L. In particular, when µ(X) < ∞, the local non-tangential and radial maximal function characterizations of H ϕ, L (X) are obtained. As applications, the authors obtain various maximal function and the atomic characterizations of the "geometric" Musielak-Orlicz-Hardy spaces H ϕ, r (Ω) and H ϕ, z (Ω) on the strongly Lipschitz domain Ω in R n associated with second-order self-adjoint elliptic operators with the Dirichlet and the Neumann boundary conditions; even when ϕ(x, t) := t for any x ∈ R n and t ∈ [0, ∞), the equivalent characterizations of H ϕ, z (Ω) given in this article improve the known results via removing the assumption that Ω is unbounded.(1.2)V(x, λr) ≤ Cλ n V (x, r) 2010 Mathematics Subject Classification. Primary 42B25; Secondary 42B35, 46E30, 30L99. Key words and phrases. Musielak-Orlicz-Hardy space, atom, maximal function, non-negative self-adjoint operator, Gaussian upper bound estimate, space of homogeneous type, strongly Lipschitz domain.

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“…In addition to being a natural generalization which covers results from both variable exponent and Orlicz spaces, the study of generalized Orlicz spaces can be motivated by applications to image processing [1,5,15], fluid dynamics [31] and differential equations.…”

Section: Introductionmentioning

confidence: 99%