Maximal function on Musielak–Orlicz spaces and generalized Lebesgue spaces

By Lars Diening. 2005

367 citations recorded by the source corpus.

Abstract We consider the Hardy–Littlewood maximal operator M on Musielak–Orlicz Spaces L φ ( R d ) . We give a necessary condition for the continuity of M on L φ ( R d ) which generalizes the concept of Muckenhoupt classes. In the special case of generalized Lebesgue spaces L p ( ⋅ ) ( R d ) we show that this condition is also sufficient. Moreover, we show that the condition is “left-open” in the sense that not only M but also M q is continuous for some q > 1 , where M q f = ( M ( | f | q ) ) 1 q .

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/dce82f8d1eee347dc44408d70523fc7803b58970

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Characterization of Riesz and Bessel potentials on variable Lebesgue spaces (2006, background): “Diening [6] obtained a certain necessary and sufficient condition for the exponent p(·) to be in P(Rn).”
An atomic decomposition of variable Besov and Triebel-Lizorkin spaces (2009, background): “Moreover, these spaces have been applied to partial differential equations and the calculus of variation, see [1, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 17, 18, 19, 21, 22, 29] and references therein.”
New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators (2011, methodology): “Remark that the class of pointwise multipliers for BMO(R) have also recently been used by Lerner [37] for solving a conjecture of Diening (see [17]) on the boundedness of the Hardy–Littlewood maximal operator on the generalized Lebesgue spaces L(R) (a special case of Musielak–Orlicz spaces, for the ”
The relation between variable Bessel potential spaces and Triebel–Lizorkin spaces (2008, methodology): “gif"/> Since we do not use them in the paper, we omit the details here, we recommend references 2–5 , 9 .”
VARIABLE BESOV AND TRIEBEL-LIZORKIN SPACES (2008, background): “omit the details here, one can see [3], [4], [5], [7], [14], [15].”
THE HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENT AND THEIR APPLICATIONS (2012, background): “There exist some sufficient conditions on p(·) such that the maximal operator M is bounded on Lp(·)(Rn), see [1-4, 11, 12].”
THE BOUNDEDNESS OF CLASSICAL OPERATORS ON VARIABLE L p SPACES (2006, background): “His exact condition is somewhat technical and we refer the reader to [14] for details.”
Harmonic type approximation lemmas (2009, background): “Moreover, last years have witnessed a large number of studies dedicated to various aspects of the functional Dp(x); we refer the reader to [4,6,15] as far as regularity is concerned; see also [20,48,66, 45] for more abstract function spaces theory problems.”
Extrapolation and weighted norm inequalities in the variable Lebesgue spaces (2014, neutral): “LE LEBESGUE SPACES 5 (equivalently, w−1 ∈ A p′(·)) and p− > 1. Conversely, if p(·) ∈ LH, with p− > 1, then for any w ∈ Ap(·), (p(·),w) is an M-pair. Remark 2.5. By a very deep result of Diening [22, 23], if 1 < p− ≤ p+ < ∞, M is bounded on Lp(·) if and only if it is bounded on Lp′(·). We”
Hardy spaces associated to operators satisfying Davies–Gaffney estimates and bounded holomorphic functional calculus☆ (2013, background): “[30], [12]), that may vary in the spatial variables and possess some control conditions.”
Musielak–Orlicz–Hardy Spaces Associated with Operators and Their Applications (2012, background): “It is worth pointing out that Musielak-Orlicz functions are the natural generalization of Orlicz functions (see, for example, [31, 32, 63, 73, 76]) and the motivation to study function spaces of Musielak-Orlicz type is attributed to their extensive applications to many branches of physics and mathem”