Boundedness of operators on Hardy spaces via atomic decompositions

By Marcin Bownik. 2005

149 citations recorded by the source corpus.

An example of a linear functional defined on a dense subspace of the Hardy space H 1 (R n ) is constructed. It is shown that despite the fact that this functional is uniformly bounded on all atoms, it does not extend to a bounded functional on the whole H 1 . Therefore, this shows that in general it is not enough to verify that an operator or a functional is bounded on atoms to conclude that it extends boundedly to the whole space. The construction is based on the fact due to Y. Meyer which states that quasi-norms corresponding to finite and infinite atomic decompositions in H p , 0 < p < 1, are not equivalent.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/749ca9dbd2735cbbd0765ff4c54f93ed252eb135

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In Applied Mathematics

1. Dachun Yang
2. Yuan Zhou
3. Wen Yuan
4. Luong Dang Ky
5. Liguang Liu
6. Ferenc Weisz
7. Loukas Grafakos
8. Jun Liu

In Mathematical Physics

1. Dachun Yang
2. Yuan Zhou
3. Luong Dang Ky
4. Wen Yuan
5. Liguang Liu
6. Loukas Grafakos
7. Marcin Bownik
8. Baode Li

Citations in the corpus

• New Hardy Spaces of Musielak–Orlicz Type and Boundedness of Sublinear Operators (2011, background): “By Theorem 2 in [8], one has to be careful with the argument “the operator T is uniformly bounded in H w(R ) (H(R) here φ(x, t) = w(x).”
• Bilinear decompositions and commutators of singular integral operators (2011, background): “Indeed, as is pointed out in [6], there exists a linear operator U defined on the space of all finite linear combinations of (1,∞)-atoms satisfying sup{‖U(a)‖L1 : a is a (1,∞)−atom} < ∞, but which does not admit an extension to a bounded operator from H(1)(R) into L(1)(R).”
• Boundedness of sublinear operators in Hardy spaces on RD-spaces via atoms (2008, background): “513] (see also [1,7]) gave an example of f ∈ H 1(Rn) whose norm cannot be achieved by its finite atomic decompositions via (1,∞)-atoms.”
• Summability of Multi-Dimensional Trigonometric Fourier Series (2012, background): “Unfortunately, for a linear operator V , the uniform boundedness of the Lp0-norms of V a is not enough for the boundedness V : H p0(T ) → Lp0(T) (see [13, 64, 65, 14, 70]).”
• On the ¹–¹ boundedness of operators (2008, background): “Bownik [3] showed that there exists a linear functional F defined on finite linear combinations of (1,∞)-atoms in R with the property that sup{|F (a)| : a is a (1,∞)-atom} <∞, but which does not admit a continuous extension to H(1)(R).”
• Boundary value problems for the Stokes system in arbitrary Lipschitz domains (2011, unknown): “, the discussion in [7].”
• Maximal function characterizations of Hardy spaces on RD-spaces and their applications (2008, background): “However, Meyer in [26, p. 513] (see also [27, 28 ]) gave an example of f ∈ H 1 (R n ) whose norm cannot be achieved by its finite atomic decompositions via (1, ∞)-atoms.”
• Localized Hardy Spaces $H^1$ Related to Admissible Functions on RD-Spaces and Applications to Schr\ (2009, unknown): “See [3, 55, 38, 21] for further details and examples.”
• Calderón–Zygmund operators on product Hardy spaces (2010, methodology): “This is nontrivial, in the light of the Meyer–Bownik example [1].”
• Littlewood–Paley characterizations for Hardy spaces on spaces of homogeneous type (2006, background): “Recently Bownik proved in [2] that there exists a linear functional which is uniformly bounded on all atoms of the Hardy space H(1)(R), however, it does not extend to a bounded functional on the whole H(1)(R).”
• BOUNDEDNESS OF SINGULAR INTEGRALS IN HARDY SPACES ON SPACES OF HOMOGENEOUS TYPE (2009, background): “If X = Rm, ρ(x) = |x|m for all x ∈ Rm, μ is the m-dimensional Lebesgue measure, and K ∈ Dρ(1, η) with ηk = C2−k for k ∈ N, certain ∈ (0, 1/m] and constant C > 0, Alverez [2] then proved that every (p,∞)ρ-atom was mapped by T into L(R) for 1/(1 + ) < p ≤ 1 or into weak-L(R) for p = 1/(1 + ) with unif”
• H1 and BMO for certain locally doubling metric measures spaces (2009, background): “Recently, M. Bownik [5], following up earlier work of Y. Meyer, produced in the classical Euclidean setting an example of an operator T B defined on ( 1 , ∞ ) -atoms with sup {(cid:13) T B a L 1 (λ) : a is a ( 1 , ∞ ) -atom } ∞ , that does not extend to a bounded operator from H 1 (λ) to L 1 (λ) :…”
• A Boundedness Criterion via Atoms for Linear Operators in Hardy Spaces (2009, result): “However, Meyer [14, p. 513] (see also [ 2 , 9]) gave an example of f ∈ H 1 (R n )”
• Weighted anisotropic product Hardy spaces and boundedness of sublinear operators (2009, methodology): “Another key tool used in this paper are the dyadic cubes of Christ [15], which substitute the role played by dilated balls and cubes in [3, 4, 5], and are used in deriving the atomic decomposition of product Hardy spaces via the Lusin-area function.”